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Vol. 9 (2004), Paper no. 22, pages 634-673.

Journal URL

http://www.math.washington.edu/˜ejpecp/

Countable Systems of Degenerate Stochastic Differential Equations

with Applications to Super-Markov Chains

Richard F. Bass1

and

Edwin A. Perkins2

Abstract

We prove well-posedness of the martingale problem for an infinite-dimensional degenerate

elliptic operator under appropriate H¨older continuity conditions on the coefficients. These

martingale problems include large population limits of branching particle systems on a

countable state space in which the particle dynamics and branching rates may depend on

the entire population in a H¨older fashion. This extends an approach originally used by the

authors in finite dimensions.

1. Research supported in part by an NSF grant DMS0244737.

2. Research supported in part by an NSERC Research grant.

Keywords: Martingale problem, infinite dimensional operators, elliptic operators, degenerate, Bessel, stochastic

differential equations, superprocesses

Classification: Primary 60H10

Running Head: Degenerate SDE’s .

Submitted to EJP on January 19, 2004. Final version accepted on August 23, 2004.

634

1. Introduction.

We prove existence and uniqueness of the martingale problem for the infinite-dimensional degenerate operator

Lf (x) =

X

[xi γi (x)fii (x) + bi (x)fi (x)]

i∈S

under suitable H¨older continuity assumptions on the coefficients γi and bi . Here S is a

countably infinite discrete set, we write x = (xi )i∈S with xi ≥ 0 for each i, L operates

on the class of finite-dimensional cylindrical functions, and f i and fii denote the first and

second partials of f in the direction xi .

In the last ten years there has been considerable interest in infinite-dimensional

operators whose coefficients are only H¨older continuous rather than Lipschitz continuous.

See [CD96], [D96], [L96], [Z00], and [DZ02], for example, which consider operators that

are perturbations of either the infinite-dimensional Laplacian or of the infinite-dimensional

Ornstein-Uhlenbeck operator. The operator L given above is not only infinite-dimensional,

but also degenerate, due to the xi factor in the second order term. This degeneracy also

means that the diffusion coefficient will not have a Lipschitz square root even for smooth

γi , invalidating the standard fixed point approaches.

The principal motivation for this work is the question of uniqueness for measurevalued diffusions which behave locally like a superprocess. In general assume S is a Polish

space and let MF (S) denote the space of finite measures on S with the weak topology.

R

Write m(f ) = f dm for m ∈ MF (S) and an appropriate R-valued f on S. Assume

{Ax : x ∈ MF (S)} is a collection of generators, all defined on an appropriate domain D 0 of

bounded continuous functions on S, and γ : S × MF (S) → R+ . Let Ω be C(R+ , MF (S)),

equipped with the topology of uniform convergence on bounded intervals, its Borel σ-field

F, canonical right-continuous filtration Ft , and coordinate maps Xt (ω) = ωt .

For each law µ on MF (S), a probability P on (Ω, F) is a solution of the martingale

problem associated with A, γ and initial law µ, written M P (A, γ, µ), if for each f ∈ D 0 ,

Xt (f ) = X0 (f ) +

Mtf

+

Z

t

Xs (AXs f ) ds,

0

where M f is a continuous Ft -martingale such that

f

hM it =

Z

t

Xs (2γ(·, Xs )f 2 ) ds.

0

Under appropriate continuity conditions on (Ax , γ) one can usually construct solutions to M P (A, γ, µ) as the weak limit points of large population (N ), small mass (N −1 )

systems of branching particle systems. In these approximating systems a particle at x

635

in population Xt branches into a mean 1 number of offspring with rate N γ(x, Xt ), and

between branch times particles evolve like a Markov process with generator A Xt (see e.g.

[MR92]). The main difficulty lies in questions of uniqueness of solutions to M P (A, γ, µ).

A case of particular interest is S = Rd and

X

X

AX f (y) =

aij (y, X)fij (y) +

bi (y, X)fi (y),

i,j≤d

i≤d

in which particles evolve according to a state dependent Itˆ

o equation between branch

times. For γ = γ0 constant, uniqueness is proved in [DK98] under appropriate Lipschitz

conditions on a, b, using methods in [P95]. The latter also effectively handles the case

γ(y, X) = γ(y) (and some other special cases of X-dependence) by proving uniqueness for

an associated strong equation and historical martingale problem.

Even in the case where S is finite, the problem of handling general γ was only

recently solved in [ABBP02] and [BP03]. If S = {1, . . . , d}, then M F (S) = Rd+ and

Pd

Ax f (i) = j=1 qij (x)f (j), where for each x ∈ Rd+ , (qij (x)) is a Q-matrix of a Markov

P

chain on S, that is, qij ≥ 0 for i 6= j and qii = − j6=i qij . Then X solves M P (A, γ, µ) if

and only if Xt ∈ Rd+ solves the degenerate stochastic differential equation

Xti

=

X0i

+

Z tX

d

0 j=1

Xsj qji (Xs ) ds

+

Z

0

t

(2γi (Xs )Xsi )1/2 dBsi ,

i = 1 . . . d.

(1.1)

Here γi : Rd+ → [0, ∞), i = 1, . . . , d, B 1 , . . . , B d are independent one-dimensional Brownian

motions, and X0 has law µ for a given probability measure µ on Rd+ . More generally,

consider the generator

Lf (x) =

d

X

[xi γi (x)fii (x) + bi (x)fi (x)]

i=1

for f ∈ Cb2 (Rd+ ), the space of bounded continuous functions on Rd+ whose first and second

P

partials are also bounded and continuous; bi (x) = j xj qji (x) would correspond to (1.1).

If µ is a law on Rd+ , a probability P on C(R+ , Rd+ ) solves M P (L, µ) if and only if for all

f ∈ Cb2 (Rd+ ),

Z t

f

Mt = f (Xt ) − f (X0 ) −

Lf (Xs ) ds

0

is an Ft -martingale under P and X0 has law µ.

Theorem A (Corollary 1.3 of [BP03]). Assume γi : Rd+ → (0, ∞), bi : Rd+ → R are

α-H¨older continuous on compact sets and satisfy

bi (x) ≥ 0 on {xi = 0},

636

(1.2)

|bi (x)| ≤ c(1 + |x|).

(1.3)

Then there is a unique solution to M P (L, µ) for each law µ on Rd+ .

A similar existence and uniqueness theorem was proved in [ABBP02] (see Theorem

A of [BP03]) assuming only continuity of γi and bi but with (1.2) strengthened to

bi (x) > 0 on {xi = 0}.

(1.4)

A simple one-dimensional example shows these results are sharp in the sense that uniqueness fails if only continuity and (1.2) are assumed (see Section 8 of [ABBP03]). Clearly

(1.2) is needed to ensure solutions remain in the positive orthant.

In this work we extend the method of [BP03] to the case where S is a countably

infinite discrete set and hence take a step towards resolving the general uniqueness problem

described above. Both [ABBP02] and [BP03] adapt the perturbation approach of [SV79]

P

to this setting by considering L as a perturbation of L0 f = i xi γi0 fii + b0i fi for some

constants γi0 > 0 and b0i ≥ 0. If Rλ is the resolvent associated with L0 , the key step is to

show that on a suitable Banach space (B, k · k), one has

k(Rλ f )i k + kxi (Rλ f )ii k ≤ Ckf k for all i.

(1.5)

Qd b0 /γ 0 −1

In [ABBP02] the space B is L2 (Rd+ , 1 xi i i dxi ) (here b0i > 0), while in [BP] the space

B is a weighted H¨older space ((4.9) below gives the precise norm). In both cases the constant C in (1.5) is independent of d. The L2 setting in [ABBP02], however, does not appear

to extend readily to infinite dimensions. There is the question of an appropriate measure

on R∞

+ , there are problems extending the Krylov-Safonov type theorems on regularity of

the resolvents which are required to handle all starting points as opposed to almost all

starting points, and, as in the finite-dimensional setting, (1.4) will not hold for the most

natural Q-matrices such as nearest random walk on the discrete circle. We therefore will

extend the weighted H¨older approach in [BP03]. This approach has also been effective in

other (non-singular) infinite-dimensional settings ([ABP04]).

Our main result (Theorem 2.7) states that the natural infinite dimensional analogue

of M P (L, ν) has a unique solution when S is a discrete countably infinite set and X

takes on values in an appropriate space of measures. To understand the nature of the

assumptions made on the coefficients bi and γi , consider the Corollaries 2.10–2.12 when

S = Zd . Basically, we require bi and γi to be H¨older continuous in the j th variable, where

the H¨older constant approaches 0 at a certain polynomial rate as |i − j| approaches ∞.

P

The state space of X will be measures x(·) satisfying i |i|q x(i) < ∞ where q > 0 may

approach zero for α close to 1 but becomes large as α gets small. There are cases where

infinite measures are allowed but they require stronger H¨older conditions on the coefficients

as the mass gets large (see Remark 2.13).

637

The main result and a number of corollaries are stated in Section 2. In Section 3 we

prove a more general existence theorem (Theorem 2.4) by truncating to a finite-dimensional

system and taking weak limits. Although these type of arguments are well-known (see

[SS80]), we could not find the particular result we needed in the literature and have included

the proof for completeness: in addition, there is an unexpected mild condition needed; see

Theorem 2.4 and Remark 2.5. The weighted H¨older spaces are introduced in Section 4

where the infinite dimensional analogues of (1.5) are derived. Since the constants in [BP03]

are independent of dimension this should be easy, but some complications arise in infinite

dimensions since boundedness of the weighted H¨older norms does not imply continuity, in

contrast to the case of finite dimensions. We must establish uniform convergence of the

appropriate derivatives of the resolvent by the corresponding quantities for a sequence of

approximating finite-dimensional functions to carry over the finite-dimensional estimates

from [BP03] and obtain continuity (Proposition 4.8). The key bounds on the weighted

H¨older norm then follow from the finite-dimensional result in [BP] (Corollary 4.10). This

approximation is also used to derive the perturbation equation for the resolvent of strong

Markov solutions of M P (L, µ) in terms of Rλ = (λ − L0 )−1 (Proposition 5.4).

In Section 5 local uniqueness is established (Theorem 5.5), i.e., if γ i and bi are

sufficiently close to constant functions, uniqueness is shown. In this setting our state space

may include counting measure, but as the coefficients become asymptotically constant this

is not surprising. In Section 6 we use the local uniqueness and a localization argument

to prove Theorem 2.7. Localization in infinite dimensions still seems to be an awkward

process and our arguments here are surely not optimal–we believe some of the additional

continuity conditions in Assumption 2.6 may be weakened. Still it is important to note that

the weighted H¨older spaces at least allow for localization. In their ground-breaking paper

[DM95], Dawson and March establish a quite general uniqueness result in the FlemingViot setting but were unable to carry out the localization step. Nonetheless [DM95] still

represents the best available uniqueness result in general infinite dimensional settings albeit

in the Fleming-Viot setting and for close to constant coefficients. Finally in Section 7 we

prove the various corollaries to Theorem 2.7.

2. Notation and statement of results.

We will use the letter c with or without subscripts to denote positive finite constants

whose exact value does not matter and which may change from line to line. We use κ with

subscripts to denote positive finite constants whose value does matter.

Let S be a countable set equipped with a map | · | : S → [0, ∞) such that S n = {i ∈

S : |i| < n} is finite for all n ∈ Z+ . (S0 = ∅). Our prototype is of course S = Zd with | · |

equal to the usual Euclidean length of i. Let ν : S → (0, ∞) and for x ∈ R S let

X

|x|ν =

|xi |νi ∈ [0, ∞].

i∈S

638

ν will be called a weight functon. We will use both νi and ν(i) for the ith coordinate of ν

and similarly for other maps defined on S.

We let

Mν (S) = {x ∈ RS+ : |x|ν < ∞}.

and consider elements of Mν (S) as measures on S with

hx, ϕi =

X

xi ϕ(i)

i∈S

when

ϕ : S → R,

X

i∈S

xi |ϕ(i)| < ∞.

It is easy to see that Mν (S) is a Polish space when equipped with the distance |x − x′ |ν .

If νi ≡ 1, it is easy to check that Mν (S) = MF (S) is the usual space of finite measures on

S equipped with the topology of weak convergence for the discrete topology on S.

R+ = [0, ∞) and Cb2 (RS+n ) is the set of bounded continuous functions f : RS+n → R

whose first and second partial derivatives fi , fij are bounded and continuous. If xi = 0,

then the partials fi , fij are interpreted as right-hand derivatives.

Define the projection operator πn : RS+ → RS+n by

πn x(i) = x(i),

i ∈ Sn ,

(2.1)

and define the lift operator Πn : RS+n → RS+ by

Πn x(i) = 1(i∈Sn ) x(i).

(2.2)

Let

2

Cb,F

(Mν (S)) = {f : Mν (S) → R : ∃n, fn ∈ Cb2 (RS+n ) such that f (x) = fn (πn x)}.

These are the functions which only depend (in a C 2 way) on the coordinates xi with i ∈ Sn .

2

Clearly Cb,F

(Mν (S)) ⊂ Cb (Mν (S)), the space of bounded continuous maps from Mν (S)

to R.

For i ∈ S assume γi : Mν (S) → [0, ∞), bi : Mν (S) → R, and for f : Mν (S) → R

define

X£

¤

xi γi (x)fii (x) + bi (x)fi (x) ,

x ∈ Mν (S),

(2.3)

Lf (x) =

i∈S

provided these partial derivatives exist and the above series is absolutely convergent. Note

2

that this is the case if f ∈ Cb,F

(Mν (S)). Let Ων equal C(R+ , Mν (S)), equipped with the

topology of uniform convergence on bounded intervals. Let Xt (ω) = ω(t) for ω ∈ Ων , let

Fu0 be the universal completion of σ(Xs : s ≤ u), and set Ft = ∩u>t Fu0 , F = ∪t≥0 Ft0 .

639

Definition 2.1. Let µ be a probability on Mν (S). A probability P on (Ων , F) solves

MP(L, µ), the martingale problem for L in Mν (S) started at µ, if P(X0 ∈ ·) = µ(·) and

2

for any f ∈ Cb,F

(Mν (S))

Mtf

= f (Xt ) − f (X0 ) −

Z

t

0

Lf (Xs )ds

is an (Ft )-local martingale under P. MP(L) is well-posed in Mν (S) if and only if there is

a unique solution to MP(L, µ) in Mν (S) for every initial law µ on Mν (S).

Note that s → Lf (Xs ) and t → f (Xt ) and hence t → Mtf are all necessarily

continuous functions.

Remark 2.2 (a) As in Remark 1.1(d) of [BP03] one could also consider test functions

f (x) = fn (πn x) for some fn ∈ Cb2 (RSn ), (i.e., those which extend in a C 2 manner to all of

RSn instead of RS+n ).

(b) Changing the class of test functions changes the martingale problem. The

smaller the class of test functions for which one establish uniqueness, the stronger the

2

theorem. Cb,F

(Mν (S)) is a reasonably small class.

(c) Let {Bi : i ∈ S} be a sequence of independent one-dimensional adapted Brownian motions on a filtered probability space (Ω, F, F t , P) and consider the stochastic differential equation

Yi (t) = Yi (0) +

Z

t

(2Yi (s))γi (Ys ))

1/2

0

dBsi

+

Z

t

bi (Ys )ds,

0

i ∈ S, t ≥ 0.

(2.4)

Here Y (0) is an F 0 -measurable random vector in Mν (S). If Y is a continuous Mν (S)valued solution to (2.4), a simple application of Itˆ

o’s formula shows that the law P Y of Y

is a solution to MP(L, µ) where µ is the law of Y (0). Conversely, given a solution P of

MP(L, µ), a standard construction allows one to build Brownian motions {B i : i ∈ S}

and a solution Y to (2.4) on some Ω such that P is the law of Y .

We introduce conditions on bi , the first of which we assume throughout this work:

Assumption 2.3. (a) There exists a constant κ2.3a (b) such that

X

i∈S

|bi (x)|νi ≤ κ2.3a (b)(|x|ν + 1),

x ∈ Mν (S),

(2.5)

(b) There exists a constant κ2.3b (b) such that

bi (x) ≥ −κ2.3b (b)xi ,

i ∈ S,

640

x ∈ Mν (S).

(2.6)

Assumption 2.3(a) will avoid explosions in finite time while a condition such as

Assumption 2.3(b) is needed to ensure that our solutions have non-negative components

(although a weaker condition bi (x) ≥ 0 if xi = 0 sufficed in finite dimensions – see [BP03].)

Note that no analogue of (2.3) is needed for the diffusion coefficients.

Our focus is on uniqueness in law of solutions to MP(L, δx0 ), but as our setting is

slightly different from that considered in the literature (e.g., Shiga and Shimizu [SS80]),

we state a general existence result. The proof is given in Section 3.

Theorem 2.4. Assume there exists β : S → (0, ∞) such that lim|i|→∞ β(i) = 0 and

for all i ∈ S, bi , γi have (necessarily unique) continuous extensions bi : Mβν (S) → R,

γi : Mβν (S) → [0, ∞). In addition to (2.5) and (2.6) assume there exists a constant κ 2.4 (γ)

such that

sup kγi k∞ = κ2.4 (γ) < ∞.

(2.7)

i

Then for any x0 ∈ Mν (S) there is a solution to MP(L, δx0 ).

Remark 2.5. Note the above continuity condition is trivially satisfied if γ i , bi are given as

continuous functions on RS+ with the product topology (as in Shiga and Shimizu [SS80]).

This condition is only needed to obtain a compact containment condition in the usual

tightness proof and will be easy to verify in the examples of interest. If γ i , bi on Mν (S)

are uniformly continuous with respect to | · |βν , the above extensions exist.

Here is our key H¨older continuity hypothesis on γi and bi :

Assumption 2.6. For some β : S → (0, ∞) satisfying lim|i|→∞ β(i) = 0 and

β(i) ≤ κ2.6a ν(i)−α/2 , for each i ∈ S, bi and γi have (necessarily unique) continuous

extensions bi : Mβν (S) → R, γi : Mβν (S) → (0, ∞). For any x0 ∈ Mν (S), η > 0, there

exists δ0 > 0 and a κ2.6b > 0 such that if x ∈ Mν (S) and |x − x0 |βν < δ0 , then (a) holds

and either (b) or (c) holds, where

(a)

X |γi (x) − γi (x0 )| |bi (x) − bi (x0 )|

+

0,

X |γi (x + hej ) − γi (x)|

i

γi (x0 )

+

|bi (x + hej ) − bi (x)|

−α/2 α

h

≤ κ2.6b γj (x0 )−α/2 xj

γi (x0 )

or

(c) lim|i|→∞ ν(i) = ∞ and for any j ∈ S, h ∈ (0, 1],

X |γi (x + hej ) − γi (x)| |bi (x + hej ) − bi (x)|

−α/2 α

+

≤ κ2.6b γj (x0 )−α/2 (1 + xj

)h .

γ

(x

)

γ

(x

)

i

0

i

0

i

Our main result is the following.

641

Theorem 2.7. Suppose Assumptions 2.3(a) and 2.6 hold, (2.7) holds, and either

bi (x) ≥ 0 for all i ∈ S and x ∈ Mν (S),

(2.9)

or

lim ν(i) = ∞

bi (x) ≥ −κ2.7 xi γi (x), for all i ∈ S and x ∈ Mν (S).

and

|i|→∞

(2.10)

Then MP(L) is well-posed in Mν (S).

We state some corollaries to Theorem 2.7, the proofs of which are given in Section 7.

Corollary 2.8. Assume bi : Mν (S) → R, γi : Mν (S) → R+ are continuous maps for all

i ∈ S, where ν : S → (0, ∞) satisfies lim|i|→∞ ν(i) = ∞. Assume Assumption 2.3(a)-(b),

(2.7),

inf γi (x) = ε0 > 0,

(2.11)

i,x

there exists non-negative constants {C(i, j) : i, j ∈ S} such that

|γi (x + hej ) − γi (x)| + |bi (x + hej ) − bi (x)| ≤ C(i, j)hα ,

and if C(j) =

h > 0,

P

x ∈ Mν (S), (2.12)

C(i, j), then

i

h

X

α

C(j)1/(1−α) ν(j)−α(1− 2 )/(1−α) < ∞.

sup C(j) + 1(α d(1 − α)(α(1 − α2 ))−1 , and, for some

c0 > 0, p > d, set C(i, j) = c0 (|i − j| + 1)−p for i, j ∈ Zd . Let bi : Mν (Zd ) → R and

γi : Mν (Zd ) → (0, ∞) be continuous maps satisfying (2.5), (2.6), (2.7), (2.11), and (2.12).

Then MP(L) is well-posed in Mν (Zd ).

Corollary 2.11. Let p : Zd − {0} → [0, 1] be a probability on Zd − {0} such that mq =

P q

i |i| p(i) < ∞, where q is as in Corollary 2.10. Let q ij = λp(j −i) (i 6= j) be the Q-matrix

of a random walk which takes steps distributed as p with rate λ > 0. Let ν, (bbi ), and (γi )

P

satisfy the hypotheses of Corollary 2.10, and let bi (x) = bbi (x) + j xj qji . Then MP(L)

is well-posed in Mν (Zd ).

−1

Corollary 2.12. Let R > 0, N = {i ∈ Zd : |i| ≤ R} and γ : RN

] be H¨older

+ → [ε, ε

α −1

continuous of index α ∈ (0, 1]. Assume q > d(1 − α)(α(1 − 2 )) and set ν(i) = (|i| + 1)q ,

i ∈ Zd . Let qij = λp(j − i), (i 6= j) where p is a probability on Zd − {0} with finite q th

P

moment. If bi (x) = j xj qji and γi (x) = γ((xi+j : j ∈ N )), then MP(L) is well-posed in

Mν (Zd ).

Corollary 2.8 is a version of Theorem 2.7 where the hypotheses are given in terms

of the H¨older constants of the γi and bi ; Corollary 2.9 applies Corollary 2.8 to the case of

super-Markov chains. Corollaries 2.10–2.12 are the application of Corollaries 2.8 and 2.9

to the case where S = Zd and an explicit bound is given for the H¨older constants of the

γi and bi .

Remark 2.13. If we assume Assumptions 2.6 (a),(b) and (2.9) we may take ν i → 0 so

P

that Mν (S) will contain some infinite measures, that is, points x such that i∈S xi = ∞.

In this case the H¨older condition Assumption 2.6 (b) becomes rather strong if x j gets large.

3. Existence.

If ε = {εn } is a sequence in (0, ∞) decreasing to 0 and S ′ = {Sn′ } is a sequence of

finite subsets of S which increases to S with S0′ = ∅, let

Kε,S ′ = {x ∈ Mν (S) :

X

′

i∈S

/ n

xi νi ≤ εn for all n ∈ Z+ }.

Write Kε for the above in the case when Sn′ = Sn for all n ∈ Z+ .

Lemma 3.1. (a) For any ε, S ′ as above, Kε,S ′ is a compact subset of Mν (S).

(b) If K is a compact subset of Mν (S), there is a sequence εn decreasing to 0 such that

K ⊂ Kε .

Proof. The standard proof is left for the reader.

643

Remark 3.2. If σN : Mν (S) → Mν (S) is defined by σN (x)(i) = 1(i∈SN ) x(i), then for any

sequence εn decreasing to 0, it is easy to check that σN (Kε ) ⊂ Kε .

Proof of Theorem 2.4. First, let Xtn be the solution to

Z t³

hZ t

i

´1/2

n,i

n +

i

bi ((Xs ) )ds +

dBsi .

Xt = x0 + 1(i∈Sn )

γi ((Xsn )+ )|Xsn,i |

0

(3.2)

0

Here (Xsn )+ = ((Xsn,i )+ : i ∈ S) and {B i } is a sequence of independent one-dimensional

Brownian motions on some filtered probability space. Note that

bbi ((xj )j∈S ) = bi ((x+ )j∈S , (xj )j∈S c )

n

n

0

j

n

is a continuous function on RSn with linear growth (by (2.5) and the continuity assumptions

on bi ). The same is true of

´

³

j

+

σ

bi ((xj )j∈Sn ) = γi ((xj )j∈Sn , (x0 )j∈Snc )|xi |

and so the existence of X n follows from Skorokhod’s existence theorem for finite-dimensional SDEs. Using L0t (X n,i ) = 0 and bi (x) ≥ 0 if xi = 0 (by (2.6)), one can use Tanaka’s

formula to see that Xtn,i ≥ 0 for all t ≥ 0 and for all i almost surely, and one may therefore

remove the superscript + ’s in (3.2). Let Tkn = inf{t : |Xtn |ν ≥ k}. For each n, Tkn ↑ ∞ as

Rt

P

k → ∞, since x0 ∈ Mν (S). Define Mtn = i∈Sn νi 0 (γi (Xsn )Xsn,i )1/2 dBsi , and note that

if ν n = maxi∈Sn νi , then

Z tX

n

hMt i =

νi2 γi (Xsn )Xsn,i ds

0 i∈S

n

≤ κ2.4 ν n

Z

t

0

|Xsn |ν ds

≤ κ2.4 ν n kt

for t ≤ Tkn . Let

n

Xt

Z

|Xtn |ν

(3.3)

t

+ κ2.3b

|Xsn |ν ds

0

Z tX

= |x0 |ν +

νi bi (Xsn ) + κ2.3b |Xsn |ν ds + Mtn .

=

0 i∈S

n

n

Mt∧T

n is a martingale by (3.3) and so (2.5) implies

k

n

E X t∧T n

k

Z

t

n

n

E (κ2.3a (|Xs∧T

n |ν + 1) + κ2.3b |Xs∧T n |ν )ds

k

k

0

Z t

n

≤ |x0 |ν + κ2.3a t + c1

E (|X s∧T n |ν )ds.

≤ |x0 |ν +

0

644

k

(3.4)

The left hand side is clearly finite by the definition of Tkn , Gronwall’s lemma implies

n

E (X t∧T n ) ≤ [|x0 |ν + κ2.3a t]ec1 t ,

k

t ≥ 0,

and so Fatou’s lemma gives

n

E (X t ) ≤ [|x0 |ν + κ2.3a t]ec1 t ,

t ≥ 0.

(3.5)

n

Therefore, using (2.6) in (3.4), we see that X t is a submartingale. The weak L1 inequality

and (3.5) imply

n

P(sup |Xtn |ν ≥ k) ≤ P(sup X t ≥ k) ≤ k −1 [|x0 |ν + κ2.3a T ]ec1 T ,

t≤T

(3.6)

t≤T

which implies

lim sup P(Tkn ≤ T ) = 0,

k→∞ n

Define Xtn (ϕ) =

(recall S0c = S)

P

i∈S

T > 0.

(3.7)

Xtn,i ϕ(i) for ϕ : S → R+ and β m = supi∈Sm

c β(i) ↓ 0. Then

c νβ) > ε0 ) ≤ P(sup

P(sup Xtn (1Sm

t≤T

t≤T

≤

X

ν(i)Xtn,i > ε0 /β m )

c

i∈Sm

βm

[|x0 |ν + κ2.3a T ]ec1 T .

ε0

(3.8)

Let ε, T > 0. Choose mk ↑ ∞ and K0 > 0 such that

∞

X

¡

¢1/2

β mk

[|x0 |ν + κ2.3a T ]ec1 T + K0−1 [|x0 |ν + κ2.3a T ]ec1 T < ε,

(3.9)

k=1

and define

K = {x ∈ Mνβ (S) :

X

c

i∈Sm

xi νi βi ≤ (β mk )1/2 for all k ∈ N, |x|ν ≤ K0 }.

k

¡

¢1/2

Then K is compact in Mνβ (S) by Lemma 3.1. By (3.8) with ε0 = β mk

and m = mk

and (3.6) with k = K0 we get that for each n

P(Xtn ∈ K for all t ≤ T )

(3.10)

∞

i

hX

(β mk )1/2 [|x0 |ν + κ2.3a T ]ec1 T + K0−1 [|x0 |ν + κ2.3a T ]ec1 T

≥1−

k=1

≥ 1 − ε,

645

by (3.9). This will give us the compact containment required for the tightness of

{P(X·n ∈ ·) : n ∈ N}.

We claim next that if i ∈ S is fixed, then

{X n,i : n ∈ N} is a tight sequence of processes in C([0, ∞), R).

(3.11)

By (3.7) it suffices to show

{X n,i (· ∧ Tkn ) : n ∈ N} is tight in C([0, ∞), R) for each k ∈ N.

(3.12)

Let Mtn,i denote the stochastic integral on the right hand side of (3.2). Then for s ≤ t,

E

³³ Z

´4 ´

t

s

1(r≤Tkn ) dMrn,i

≤ c2 E

³³ Z

s

2

t

1(r≤Tkn ) Xrn,i γi (Xrn )dr

≤ c3 (κ2.4 ) (k/νi )2 (t − s)2 .

´2 ´

(3.13)

In addition by (2.5),

Z

s

t

1

(r≤Tkn )

|bi (Xrn )|dr

Z

κ2.3a t

≤

1(r≤Tkn ) (|Xrn |ν + 1)dr

νi

s

κ2.3a

≤

(k + 1)(t − s).

νi

This, (3.13), (3.2), and standard arguments now imply (3.12).

(3.10) and (3.11) imply {X·n : n ∈ N} is a tight sequence in C([0, ∞), Mνβ (S)). If

νβ(i) ≡ 1, this is standard, as then Mνβ (S) = MF (S) (see, e.g., Theorem II.4.1 in [Pe02]).

In general, define Φ : Mνβ (S) → MF (S) by

Φ(x)(i) = x(i)νi βi .

Then Φ is an isometry and the above result for MF (S) gives the required implication.

By Skorokhod’s theorem we may first extract a weakly convergent subsequence

nm

{X } and then assume X nm → X a.s. in C([0, ∞), Mνβ (S)). It is easy to use the

continuity of bi , γi on Mνβ (S) to let n = nm → ∞ in (3.2) and conclude

Xti

=

xi0

+

Z

t

0

Z tp

γi (Xs )Xsi dBsi

bi (Xs )ds +

0

for all t ≥ 0 and all i ∈ S, a.s. Fatou’s lemma implies for any t > 0

|Xt |ν =

X

i∈S

lim inf Xtnm ,i νi ≤ lim inf |Xtnm |ν ,

m→∞

m→∞

646

(3.14)

and so an elementary argument implies

P(sup |Xt |ν > k) ≤ P(lim inf sup |Xtnm |ν > k)

m→∞ t≤T

t≤T

≤ lim inf P(sup |Xtnm |ν > k)

m→∞

≤k

−1

t≤T

[|x0 |ν + κ2.3a T ]ec1 T ,

the last by (3.6). This proves

sup |Xt |ν < ∞,

T > 0,

a.s.,

(3.15)

t≤T

and so X· has Mν (S) valued paths a.s.

To show that X· has continuous Mν (S)-valued paths a.s. we use the following

lemma, whose elementary proof is left to the reader.

Lemma 3.3. Suppose x : [0, ∞) → Mν (S) is such that xt (i), i ∈ S, and |xt |ν are all

continuous. Then x is continuous.

Clearly Xt (i) is continuous for all i ∈ S a.s. since it is continuous in Mνβ (S). From

Rtp

P

(3.14) we see that if Mn (t) = i∈Sn νi 0 γi (Xs )Xsi dBsi , then

X

X

Xt (i)νi =

x0 (i)νi +

t

X

νi bi (Xs )ds + Mn (t).

(3.16)

0 i∈S

n

i∈Sn

i∈Sn

Z

(3.15) and (2.5) show that

X

i∈Sn

νi |bi (Xs )| →

X

i∈S

νi |bi (Xs )|,

which is bounded uniformly on compact time intervals a.s. as n → ∞, and so

¯Z

¯

sup¯

t≤T

t

X

0 i∈S

n

≤

Z

T

0

νi bi (Xs )ds −

X

i∈S−Sn

Z tX

0 i∈S

¯

¯

νi bi (Xs )ds¯

νi |bi (Xs )|ds

→0

(3.17)

as n → ∞ by dominated convergence. By (3.16) and (2.5),

sup |Mn (t)| ≤ sup |Xt |ν + |x0 |ν +

t≤T

t≤T

647

Z

0

T

κ2.3a (|Xs |ν + 1)ds.

By (3.15) and the Dubins-Schwartz theorem, this means {hM n iT } remains bounded in

probability as n → ∞. Therefore

Z TX

νi2 γi (Xs )Xs (i)ds = lim hM n iT < ∞,

a.s., T > 0.

0

n→∞

i∈S

A standard square function inequality now implies

sup |Mn (t) − Mm (t)| → 0

t≤T

in probability as m, n → 0 for all T > 0, and so we may take a subsequence such that

Mnk converges uniformly on compact time intervals a.s. Let n = nk → ∞ in (3.16). The

above and (3.17) show that the right hand side of (3.16) converges uniformly on compact

time intervals a.s. to a necessarily continuous process. As the left hand side converges to

|Xt |ν for all t ≥ 0, a.s., it follows that t → |Xt |ν is continuous. Lemma 3.3 therefore shows

t → Xt is a continuous Mν (S)-valued process. By (3.14) and Remark 2.2(c), the law of X

is a solution of the martingale problem for L starting at x0 .

4. Estimates.

We first obtain some key analytic estimates for the special case when γi = γi0 and

bi = b0i are constants. Assume

0 < γi0 ≤ sup γi0′ = κ4.1 (γ 0 ) < ∞,

i′

0 ≤ b0i ,

i ∈ S,

Let

L0 f (x) =

and

X

|b0 |ν =

X

i∈S

i ∈ S,

(4.1)

b0i ν(i) < ∞.

(4.2)

γi0 xi fii (x) + b0i fi (x)

i∈S

for functions f for which the partial derivatives exist and the sum is absolutely convergent.

By Theorem 2.4 there is a solution P0x0 to MP(L, δx0 ) in Mν (S) for each x0 ∈ Mν (S). In

fact it is easy to see that under P0x0 , the processes {Xi : i ∈ S} are independent diffusions

and each X i is a suitably scaled squared Bessel process whose law is that of the pathwise

unique solution to

Z

t

X i (t) = x0 (i) +

0

(X i (s)γi0 )1/2 dBsi + bi0 t,

(4.3)

where the Bsi are independent one dimensional Brownian motions. (Theorem 2.4 is only

needed here to ensure X has paths in Ων .) An explicit formula for the transition kernel

of pit (xi , dyi ) of Xi is given in (2.2) and (2.4) of [BP03]. Let (Pt )t≥0 and (Rλ )λ≥0 be the

semigroup and resolvent, respectively, of the Mν (S)-valued diffusion Xt = (Xti )i∈S .

648

Lemma 4.1. For any compact set K ∈ Mν (S), T > 0, and ε > 0, there is a sequence

η = {ηn } decreasing to zero such that

sup P0x0 (Xt ∈ Kη for t ≤ T ) ≥ 1 − ε.

x0 ∈K

Proof. By Lemma 3.1(b) we may assume K = Kδ for some sequence δn decreasing to

P

0

zero. Set B(N ) = i∈S

/ N ν(i)bi . If

X

Zn,N (t) =

ν(i)Xi (t)

i∈Sn −SN

for n > N ≥ 0 and

ZN (t) =

X

ν(i)Xi (t),

i∈S

/ N

then for x0 ∈ Kδ

E 0x0 [ZN (t)] =

X

i∈S

/ N

x0 (i)ν(i) + b0i ν(i)t ≤ δ(N ) + B(N )t,

where δ(N ), B(N ) ↓ 0 by (4.2). Since Zn,N (t) is a submartingale, the weak L1 inequality

implies that for any x0 ∈ Kδ

P0x0 (sup ZN (t) > A) = lim P0x0 (sup Zn,N (t) > A)

n→∞

t≤T

t≤T

−1

≤ lim A

n→∞

−1

≤A

E 0x0 [Zn,N (t)]

(δ(N ) + B(N )T ).

(4.4)

sup P0x0 (sup ZNk (t) > 2−k ) ≤ 2−k−1 ε

(4.5)

Choose Nk ↑ ∞ such that

x0 ∈Kδ

t≤T

and then η > 1 sufficiently large so that

sup P0x0 (sup |Xt |ν > η) ≤ ε/2.

x0 ∈Kδ

t≤T

The latter is possible by (4.4) with N = 0 since Z0 (t) = |Xt |ν . Now define

ηn =

½

η,

n≤N

−k

2 , Nk ≤ n < Nk+1 .

649

(4.6)

Then for any x0 ∈ Kδ

P0x0 (Xt

∈

Kηc

for some t ≤ T ) ≤

P0x0 (sup |Xt |ν

t≤T

> η) +

∞

X

P0x0 (sup ZNk (t) > 2−k ) < ε

k=1

t≤T

by (4.5) and (4.6).

Define ei ∈ Mν (S) by ei (j) = 1(i=j) . Let α ∈ (0, 1] and for f : Mν (S) → R and

i ∈ S define

α

|f |α,i = sup{|f (x + hei ) − f (x)|x(i) 2 h−α : x ∈ Mν (S), h > 0}.

(4.7)

Set

α

|f |α = sup(γi0 ) 2 |f |α,i ,

kf kα = kf k∞ + |f |α .

i

(4.8)

If

Cα = {f : Mν (S) → R : f continuous , kf kα < ∞},

then it is easy to check that (Cα , k · kα ) is a Banach space.

Remark 4.2. One difference between our infinite dimensional setting and the finite dimensional setting in [BP03] is that supi |f |α,i < ∞ does not imply that f is uniformly

continuous on I = {x ∈ Mν (S) : x(i) > 0 for all i ∈ S} and hence has a continuous

extension to Mν (S). This is true on Rd+ (see Lemma 2.2 of [BP03]). Suppose we define

f (x) to be 1 if infinitely many of the x(i) 6= 2−i /ν(i) and 0 otherwise. Then |f |α,i = 0

but f is discontinuous on I. This complicates things a bit when checking whether various

operators preserve Cα .

Remark 4.3. A key fact in our argument is that the estimates on (Rλ f )i and xi (Rλ f )ii

from [BP03] are independent of the dimension of the space. Recall that the way we obtained the estimates in [BP03] was to first consider the one-dimensional case. If P ti denotes

the semigroup corresponding to the operator xγi0 f ′′ (x) + b0i f ′ (x), then we obtained bounds

on |(Pti f )′ (x)| and on |(Pti f )′ (x + ∆) − (Pti f )′ (x)| in terms of constants depending only on

γi0 and b0i ; see Lemmas 4.3, 4.4, 4.6, and 4.7 of [BP03]. If Pt denotes the semigroup correPd

sponding to L0 f (x) = i=1 [xi γi0 fii (x) + b0i fi (x)], we then derived bounds on |(Pt f )i (x)|

and on |(Pt f )i (x + ∆ej ) − (Pt f )i (x)| with the same constants (Propositions 5.1 and 5.2 of

[BP03]); hence the constants did not depend on the dimension d of the underlying space.

We then deduced estimates on (Rλ f )i . The same reasoning was applied for xi (Rλ f )ii .

Lemma 4.4. Let f ∈ Cα , λ > 0, and i ∈ S. There is a κ4.4 = κ4.4 (α) independent of

f, i, λ such that the following hold:

(a) The partial derivative (Rλ f )i (x) exists for every x ∈ Mν (S) and satisfies

α

α

k(Rλ f )i k∞ ≤ κ4.4 (γi0 ) 2 −1 λ− 2 |f |α,i .

650

(4.9)

(b) The second order derivative (Rλ f )ii (x) exists on {x ∈ Mν (S) : xi > 0} and satisfies

|xi (Rλ f )ii (x)| ≤

κ4.4 (γi0 )−1 |f |α,i

h³ γ 0 ´ α2

i

λ

α

2

i

∧ xi .

(4.10)

In particular, limxi →0 xi (Rλ f )ii (x) = 0 uniformly on Mν (S) and if xi (Rλ f )ii (x) is set to

be this limit on {x ∈ Mν (S) : xi = 0}, then

α

α

kxi (Rλ f )ii (x)k∞ ≤ κ4.4 (γi0 ) 2 −1 λ− 2 |f |α,i .

(4.11)

Proof. We only prove (b) as (a) is similar but easier. Let t > 0. Then argue as in the

finite-dimensional argument (Proposition 5.1 of [BP03]), noting the constants there are

independent of dimension, to see that (Pt f )ii exists on Mν (S) (in fact on RS+ ) and satisfies

´

³x

α

i

(4.12)

|xi (Pt f )ii (x)| ≤ c1 |f |α,i (γi0 t) 2 −1 0 ∧ 1 .

γi t

If xi > 0, this allows one to differentiate through the time integral (by the dominated

convergence and the mean value theorem) and conclude for xi > 0 that (Rλ f )ii exists and

satisfies

Z ∞

e−λt xi (Pt f )ii (x)dt.

xi (Rλ f )ii (x) =

0

A simple calculation using (4.12) leads to (4.10) for xi > 0. The fact that xi (Rλ f )ii

approaches 0 uniformly as xi ↓ 0 is then immediate, as is (4.11).

Lemma 4.5. If f ∈ Cb (Mν (S)) and fn = f ◦ πn , then for any compact subset K of Mν (S)

lim sup |f (x) − fn (x)| = 0.

n→∞ x∈K

Proof. By Lemma 3.1(b) we may assume K = Kη for some sequence η = {ηn } decreasing

to 0. Then

X

sup |x − πn x|ν = sup

xi νi ≤ ηn → 0.

x∈Kη

x∈Kη

c

i∈Sn

Since πn (Kη ) ⊂ Kη (recall Remark 3.2) and f is uniformly continuous on Kη , the result

follows.

Corollary 4.6. If f ∈ Cb (Mν (S)) and fn = f ◦ πn , then for any λ > 0, Rλ fn → Rλ f

uniformly on compact subsets of Mν (S).

Proof. Let K be a compact subset of Mν (S) and ε > 0. Lemma 4.1 shows there is a

compact Kη such that

sup (Rλ 1Kηc )(x) < ε.

x∈K

651

Lemma 4.5 implies that kRλ f 1Kη − Rλ fn 1Kη k∞ → 0 as n → ∞ and so

lim sup sup |Rλ fn (x) − Rλ f (x)| ≤ lim sup sup |Rλ fn 1Kηc (x) − Rλ f 1Kηc (x)|

n→∞ x∈K

n→∞ x∈K

≤ 2εkf k∞ .

Let Rλn denote the resolvent of the finite-dimensional diffusion (Xi )i∈Sn under {Px0 }.

Then Rλn is a Feller resolvent (i.e., it maps Cb (RS+n ) to itself) and so if fen ((xi )i∈Sn ) =

f (Πn (x)) for f ∈ Cb (Mν (S)), then fen ∈ Cb (RS+n ) and Rλ fn (x) = Rλn fen ((xi )i∈Sn ) is continuous on Mν (S). (Πn is defined in (2.2).) The convergence in Corollary 4.6 therefore

shows

Rλ : Cb (Mν (S)) → Cb (Mν (S)).

(4.13)

Our immediate goal is to extend the continuity on Mν (S) to (Rλ f )i and xi (Rλ f )ii for

f ∈ Cα . As explained earlier, this is more delicate in our infinite-dimensional setting.

Lemma 4.7. There is a κ4.7 ≥ 1 such that

(a) if 0 < r ≤ R/2, then

Z ∞

z r−1 |z − r|

κ4.7

e−z

dz ≤

;

Γ(r) r

R+1

R

(b) for any r > 0

Z

0

∞

e−z

z r−1 |z − r|

dz ≤ κ4.7 (r + 1)−1/2 .

Γ(r) r

Proof. (a) In view of (b) we may assume R ≥ 1. The integral in (a) is bounded by

Z ∞

r−1

(z − r)2

−z z

dz,

e

Γ(r) rR/2

R

which is bounded by cR−1 r−1 (r + 1); see, e.g., Lemma 3.2(a) of [BP03]. This gives the

required bound if r ≥ 1 (recall R ≥ 1). Assume now that 0 < r < 1. The integral in (a) is

at most

Z ∞

Z ∞

zr

z r+1

−z

e

dz ≤

dz

e−z

Γ(r + 1)

Γ(r + 1)R

R

0

h Γ(r + 2) i

= R−1

= R−1 (r + 1) ≤ 2R−1 ≤ 4(R + 1)−1 .

Γ(r + 1)

(b) If r ≥ 1 this is immediate from Lemma 3.2(a) of [BP03]. If 0 < r < 1, then the

required integral is at most

Z ∞

Z ∞

zr

z r−1

−z

dz +

dz = 2.

e−z

e

Γ(r + 1)

Γ(r)

0

0

652

Proposition 4.8. Let f ∈ Cα and fn = f ◦ πn . If i ∈ S and λ > 0, then for any compact

set K ⊂ Mν (S)

£

lim sup |Rλ fn (x) − Rλ f (x)| + |(Rλ fn )i (x) − (Rλ f )i (x)|

n→∞ x∈K

¤

+ |xi (Rλ fn )ii (x) − xi (Rλ f )ii (x)| = 0.

Proof. Note first that f ∈ Cα implies fn ∈ Cα and so the existence of the above partial

derivatives follows from Lemma 4.4. We focus on the convergence of the second order

derivatives as the first order derivatives are handled in a similar and slightly simpler way,

while the resolvents themselves were handled in Corollary 4.6. Fix f ∈ C α .

If y ∈ Mν (S), write ybi = y|S−{i} and define Y (v, i) ∈ Mν (S) by setting Y (v, i)(j) =

y(j) if j 6= i and Y (v, i)(i) = v; in other words, Y (v, i) is the point which has the same

coordinates as y except that the ith coordinate is equal to v instead of yi . We may then

define dn (v; ybi ) = f (πn Y (v, i)) − f (Y (v, i)) and

Z

∂2

hn (b

yi ; t, xi ) = xi 2

dn (yi ; ybi )pit (xi , dyi ).

∂xi

If |dn (·; ybi )|α denotes the | · |α,i norm of dn (·; ybi ) with S = {i}, then |dn (·; ybi )|α ≤ 2|f |α,i ,

and so the above derivative exists and satisfies (see Lemmas 4.1, 4.3, 4.5, and 4.6 of [BP03])

³x

´

α

i

khn (·; t, xi )k∞ ≤ cα |f |α,i (γi0 t) 2 −1 0 ∧ 1 .

(4.14)

γi t

Note that

xi (Pt fn )ii (x) − xi (Pt f )ii (x) =

Z

hn (b

yi ; t, xi )

Y

pjt (xj , dyj ),

(4.15)

j6=i

where differentiation through the integrals is justified by the above bound and dominated

convergence.

If g : R+ → R, let kgkR = sup{|g(y)| : y ≤ R}. Assume first b0i > 0 and use (4.14)

in [BP03] to conclude that if w = xi /γi0 t and ri = b0i /γi0 , then

Z ∞

∞

k

X

z k+ri −1 |z − (k + ri )| dz

−w w

|hn (b

yi ; t, xi )| ≤

e

. (4.16)

|w−k|

|dn (zγi0 t; ybi )|e−z

k!

Γ(k + ri )

k + ri

γi0 t

0

k=0

By Lemma 4.7, if R > 0,

Z ∞

z k+ri −1 |z − (k + ri )|

dz

e−z

Γ(k + ri )

k + ri

R/γi0 t

κ4.7

κ4.7

+ 1(k+ri >R/2γi0 t) √

≤ 1(k+ri ≤R/2γi0 t)

0

(R/γi t) + 1

ri + k + 1

p

i

h

γ0t

κ4.7

1(k+ri ≤R/2γi0 t) √ i + 1(k+ri >R/2γi0 t) .

≤√

ri + k + 1

R

653

Use this and Lemma 4.7(b) in (4.16) to see that

|hn (b

yi ; t, xi )|

≤

(γi0 t)−1

∞

X

e

−w w

k!

k=0

h

k

|w − k| kdn (·; ybi )kR

Z

R/γi0 t

e−z

0

z k+ri −1 |z − (k + ri )|

dz

Γ(k + ri )

k + ri

p

ii

γ0t

+ 2kf k∞ κ4.7 (ri + k + 1)−1/2 1(k+ri ≤R/2γi0 t) √ i + 1(k+r1 >R/2γi0 t)

R

∞

h

³ γ 0 t ´1/2

i

X

|w − k|

wk

i

kd

(·;

y

b

)k

+

2kf

k

≤ (γi0 t)−1 κ4.7

e−w

n

i

R

∞

k! (ri + k + 1)1/2

R

h

k=0

+

2kf k∞ (γi0 t)−1 κ4.7

∞

X

1(k+ri >R/2γi0 t) e−w

k=0

|w − k|

wk

k! (ri + k + 1)1/2

= I1 + I2 .

(4.17)

A simple calculation (see Lemma 3.3(b) of [BP03]) shows that

³x

´h

³ γ 0 t ´1/2

i

i

i

∧

1

kd

(·;

y

b

)k

+

kf

k

n

i R

∞

γi0 t

R

h

i

≤ c2 ((γi0 t)−1 + 1) kdn (·; ybi )kR + R−1/2 kf k∞ .

I ≤ c1 (γi0 t)−1

If N is a Poisson random variable with mean w, the series in I2 is

i

h

−1/2

E |N − w|(ri + N + 1)

1(N +ri >R/2γi0 t) .

Assume now

(4.19)

R

> max(1, x2i , γi0 t).

8

(4.20)

If ri < R/(4γi0 t), recalling xi < R/8 by (4.20), the expectation in (4.19) is at most

i

h

−1/2

E |N − w|(ri + N + 1)

1(N −w> R−4xi )

4γ 0 t

i

h

≤ E |N − w|(N + 1)

−1/2

1(N −w>R/8γi0 t)

i

E (N − w)2

(R/8γi0 t)3/2

xi

c4

= c3 3/2 (γi0 t)1/2 ≤ (γi0 t)1/2 ≤ c4 R−1/2 .

R

R

≤

If ri ≥ R/(4γi0 t), then (4.19) is bounded by

√

√

c6 xi

c5 w

E |N − w|

≤p

≤ √

≤ c7 R−1/4 ,

√

0

ri

R

R/γi t

654

(4.18)

the last by (4.20). Therefore under (4.20)

I2 ≤ kf k∞ c8 ((γi0 t)−1 + 1)R−1/4 .

Now use the above bounds in (4.17) to see that for R ≥ 8 max(1, γi0 t),

sup

√

xi ≤ R/4

|h(b

yi ; t, xi )| ≤ c9 ((γi0 t)−1 + 1)[kdn (·; ybi )kR + kf k∞ R−1/4 ].

(4.21)

If b0i = 0, a slightly simpler argument starting with (4.6) in [BP03] will lead to the same

bound.

Now choose a compact set K in Mν (S), T > 1 and ε > 0. Assume R is large enough

so that R ≥ 8 max(1, γi0 t),

√

(4.22)

if x ∈ K, then xi ≤ R/4,

and kf k∞ R−1/4 < ε. Let ηn be such that Kη is a compact set satisfying the conclusion

of Lemma 4.1. Let π

bi (y) = (y(j), j ∈ S − {i}) be the projection of y ∈ Mν (S) onto

Mνi (S − {i}), νi = ν|S−{i} , and let

b η = {y ∈ Mν (S) : π

K

bi (y) ∈ π

bi (Kη ), y(i) ≤ R}.

b η is compact, and so by Lemma 4.5

Then it is easy to use Lemma 3.1 to check that K

lim sup |fn (y) − f (y)| = 0.

bη

y∈K

n→∞

This implies

lim

n→∞

sup kdn (·; ybi )kR = 0.

b

yi ∈b

πi (Kη )

Choose N such that n ≥ N implies

sup kdn (·; ybi )kR < ε.

b

yi ∈b

πi (Kη )

Use this with (4.14), (4.21), and (4.22) in (4.15) and conclude that for n ≥ N and t ∈ [0, T ]

sup |xi (Pt fn )ii (x) − xi (Pt f )ii (x)|

x∈K

α

≤ cα |f |α,i (γi0 t) 2 −1 sup P0x (b

πi (Xt ) ∈

/π

bi (Kη ))

x∈K

≤

≤

+ c9 ((γi0 t)−1 + 1)[ε + kf k∞ R−1/4 ]

α

cα |f |α,i (γi0 t) 2 −1 ε + c9 ((γi0 t)−1 + 1)2ε

c10 (|f |α,i + 1)((γi0 t)−1 + 1)ε.

655

(4.23)

Use the above for t ∈ [T −1 , T ] and (4.14) for t ∈ [T −1 , T ]c to see that for x ∈ K and n ∈ N

|xi (Rλ fn )ii (x)−xi (Rλ f )ii (x)|

α

≤ cα |f |α,i (γi0 t) 2 −1

hZ

1/T

α

t 2 −1 dt +

0

+ c11 (|f |α,i + 1)((γi0 )−1 T + 1)ε

≤ cα |f |α,i (γi0 t)

α

2 −1

Z

Z

(4.24)

∞

T

T

e−λt dt

i

e−λt dt

1/T

[T −α/2 + e−λT λ−1 ]

+ c12 (|f |α,i + 1)((γi0 )−1 T + 1)ελ−1 .

As T > 1 and ε > 0 are arbitrary, this gives

lim sup |xi (Rλ fn )ii (x) − xi (Rλ f )ii (x)| = 0.

n→∞ x∈K

The differentiation through the time integral in (4.24) does require x i > 0 as in the proof

of Lemma 4.4, but that result shows the left-hand side is 0 if xi = 0.

Corollary 4.9. If f ∈ Cα , then for any i ∈ S and λ > 0, Rλ f, (Rλ f )i , and xi (Rλ f )ii are

all continuous bounded functions on Mν (S).

Proof. Fix i and consider n large enough so that i ∈ Sn . Recall Rλn is the resolvent of

(Xi )i∈Sn and fen (x) = f ◦Πn (x) on RS+n . Then Rλ fn (x) = Rλn fen (πn x) and so by Proposition

5.3 of [BP03] (Rλ fn )i (x) = (Rλn )i fen (πn x) and xi (Rλ fn )ii (x) = xi (Rλn fen )ii (πn (x)) are

bounded continuous functions on Mν (S). The uniform convergence in Proposition 4.8 and

the bound in Lemma 4.4 show that (Rλ f )i and xi (Rλ f )ii are in Cb (Mν (S)). (4.13) already

gave the result for Rλ f .

Corollary 4.10. There is a κ4.10 > 0 depending only on α such that for all f ∈ Cα , λ > 0,

and i, j ∈ S,

1−α

0 −1 0

|(Rλ f )i |α,j + |xi (Rλ f )ii |α,j ≤ κ4.10 |f |α,i

|f |α

(γi /γj0 )(1−α)α/2 .

α,j (γi )

Proof. The proof is almost the same as that of Proposition 5.3 of [BP03] for the finitedimensional case – again the constants given there are independent of dimension; cf. Remark 4.3. The only change is that once the bounds on the increments of x i (Rλ f )ii are

established for xi > 0, they follow for xi = 0 by the continuity established in Corollary

4.9; this is in place of the use of Lemma 2.2 in [BP03].

5. Local uniqueness.

We make the following assumption.

656

Assumption 5.1. Assume γi : Mν (S) → (0, ∞), bi : Mν (S) → R are continuous and

(γi0 )i∈S , (b0i )i∈S satisfy (4.1) and (4.2). Assume also

(a)

X · |γi (x) − γ 0 | |bi (x) − b0 | ¸

i

i

sup

+

≡ ρ < ∞.

(5.1)

0

0

γ

γ

x∈Mν (S)

i

i

i∈S

(b) For all x ∈ Mν (S)

X · |γi (x′ ) − γ 0 | |bi (x′ ) − b0 | ¸ X · |γi (x) − γ 0 | |bi (x) − b0 | ¸

i

i

i

i

+

=

+

. (5.2)

lim

x′ →x

γi0

γi0

γi0

γi0

i∈S

i∈S

(c) For all j ∈ S, 0 < h, x ∈ Mν (S)

X · |γi (x + hej ) − γi (x)| |bi (x + hej ) − bi (x)| ¸

−α/2 0 −α/2

(γj )

. (5.3)

+

≤ κ5.1 hα xj

γi0

γi0

i∈S

The following result uses only Assumption 5.1(a)-(b).

Lemma 5.2. For any λ > 0, (L − L0 )Rλ : Cα → Cb (Mν (S)), and there is a κ5.2 = κ5.2 (α)

such that

k(L − L0 )Rλ f k∞ ≤ κ5.2 ρ|f |α λ−α/2 ,

f ∈ Cα , λ > 0.

Note that L − L0 can be well-defined for a larger class of functions than the domain of L

and L0 thanks to Assumption 5.1 and the results of Section 4. In particular, this lemma

shows that (L − L0 )Rλ f is well-defined for f ∈ Cα .

Proof. Lemma 4.4 shows that for f ∈ Cα

X

|γi (x) − γi0 | |xi (Rλ f )ii (x)| + |bi (x) − b0i | |(Rλ f )i (x)|

i∈S

≤

X h |γi (x) − γ 0 |

i

i∈S

γi0

+

|bi (x) − b0i | i

cα (γi0 )α/2 |f |α,i λ−α/2

γi0

≤ cα ρ|f |α λ−α/2 ,

by Assumption 5.1(a). This gives the required bound on k(L − L0 )Rλ f k∞ .

Note that Lemma 4.4 implies

|γi (x) − γi0 | |x(i)(Rλ f )ii (x)| + |bi (x) − b0i | |(Rλ f )i (x)|

h |γ (x) − γ 0 | |b (x) − b0 | i

i

i

i

i

≤ cα

+

|f |α λ−α/2 .

0

γi

γi0

Let xn → x in Mν (S). Assumptions 5.1(a),(b) imply that if

fn (i) =

|γi (xn ) − γi0 | |bi (xn ) − b0i |

+

,

γi0

γi0

657

(5.4)

then {fn } is uniformly integrable with respect to counting measure on S and hence by

(5.4) so is

gn (i) = |γi (xn ) − γi0 | |xn (i)(Rλ f )ii (xn )| + |bi (xn ) − b0i | |(Rλ f )i (xn )|.

This allows us to take the limit as n → ∞ through the summation and conclude by the

continuity of

(γi (x) − γi0 ) (x(i)(Rλ f )ii (x)) + (bi (x) − b0i ) ((Rλ f )i (x))

(see Corollary 4.9) that

lim (L − L0 )Rλ f (xn ) = (L − L0 )Rλ f (x).

n→∞

This proves (L − L0 )Rλ f is continuous.

We let Bλ denote the operator

Bλ = (L − L0 )Rλ =

X£

¤

(γi − γi0 )x(i)(Rλ )ii + (bi − b0i )(Rλ )i .

i

Proposition 5.3. Assume Assumption 5.1(a)-(c). For any λ > 0, Bλ : Cα → Cα is a

bounded operator. Moreover there exist λ0 = λ0 (α) and ρ0 = ρ0 (α) > 0 such that if

λ ≥ λ0 and ρ ≤ ρ0 , then

kBλ kCα < 1/2.

Proof. Use Lemma 4.4 and Corollary 4.10 to see that for f ∈ Cα , j ∈ S, and h > 0,

|Bλ f (x + hej ) − Bλ f (x)|

X

≤

|γi (x + hej ) − γi (x)| |(x + hej )(i)(Rλ f )ii (x + hej )|

i

+

X

i

≤

X

i

+ |bi (x + hej ) − bi (x)| |(Rλ f )i (x + hej )|

|γi (x) − γi0 | |(x + hej )(i)(Rλ f )ii (x + hej ) − x(i)(Rλ f )ii (x)|

+ |bi (x) − b0i | |(Rλ f )i (x + hej ) − (Rλ f )i (x)|

[|γi (x + hej ) − γi (x)| + |bi (x + hej ) − bi (x)|](γi0 )−1 λ−α/2 κ4.4 sup |f |α,i′ (γi0′ )α/2

+

i′

X

i

−α/2

1−α

0

0 (1−α)α/2 α

[|γi (x) − γi0 | + |bi (x) − b0i |](γi0 )−1 κ4.10 |f |α,i

|f |α

h xj

α,j (γi /γj )

The first summation is bounded by (use Assumption 5.1(c))

−α/2

κ4.4 λ−α/2 |f |α (γj0 )−α/2 κ5.1 hα xj

658

.

.

The second summation is bounded by (use Assumption 5.1(a))

κ4.10 ρ sup(|f |α,i (γi0 )α/2 )1−α sup(|f |α,j ′ (γj0′ )α/2 )α (γj0 )−α/2 hα x(j)−α/2

i

j′

−α/2

≤ κ4.10 ρ|f |α (γj0 )−α/2 hα xj

.

We may therefore conclude

|Bλ f |α,j (γj0 )α/2 ≤ [κ4.4 κ5.1 λ−α/2 + κ4.10 ρ] |f |α

and so

|Bλ f |α ≤ [κ4.4 κ5.1 λ−α/2 + κ4.10 ρ]|f |α .

Combine this with Lemma 5.2 to see that

kBλ f kα ≤ [κ4.4 κ5.1 λ−α/2 + (κ4.10 + κ5.2 λ−α/2 )ρ] |f |α .

This, together with Lemma 5.2, shows Bλ : Cα → Cα is a bounded operator with

kBλ k ≤ 1/2 for ρ ≤ ρ0 (α), λ ≥ λ0 (α).

Let Pµ be a solution of MP(L, µ) for some law µ on Mν (S) and for λ > 0, let

R∞

Sλ f = E µ ( 0 e−λt f (Xt )dt).

The following result uses only (2.5), (2.7), and Assumption 5.1(a) (it only requires

the bound in Lemma 5.2 and so does not require Assumption 5.2(b)). Recall the constant

bp

κ2.3a in (2.5). We use −→ to denote bounded pointwise convergence.

Proposition 5.4. Assume (2.5), (2.7). If f ∈ Cα , then

Z

Sλ f = Rλ f (x)µ(dx) + Sλ Bλ f,

Proof. Assume first that

Z

|x|ν µ(dx) < ∞.

λ > κ2.3a .

(5.5)

Let f (x) = f0 (πn x) for some f0 : RS+n → R, f ∈ Cα . Write xn = πn (x) for x ∈ Mν (S) and

define

Z ∞

Z ∞

−λt

gδ (y) =

e Pt f (y)dt =

e−λt Ptn f0 (y n )dt ≡ geδ (y n ).

δ

δ

Ptn

Here

is the semigroup of {Xi : i ∈ Sn } under P0x . The finite dimensional analysis in

the proof of Lemma 6.1 of [BP03] shows that geδ ∈ Cb2 (RS+n ).

Use (2.4), (5.5), (2.5), a stopping time argument, and a Gronwall argument (cf. the

proof of Theorem 2

ni

o

r

Elect

o

u

a

rn l

o

f

P

c

r

ob

abil

ity

Vol. 9 (2004), Paper no. 22, pages 634-673.

Journal URL

http://www.math.washington.edu/˜ejpecp/

Countable Systems of Degenerate Stochastic Differential Equations

with Applications to Super-Markov Chains

Richard F. Bass1

and

Edwin A. Perkins2

Abstract

We prove well-posedness of the martingale problem for an infinite-dimensional degenerate

elliptic operator under appropriate H¨older continuity conditions on the coefficients. These

martingale problems include large population limits of branching particle systems on a

countable state space in which the particle dynamics and branching rates may depend on

the entire population in a H¨older fashion. This extends an approach originally used by the

authors in finite dimensions.

1. Research supported in part by an NSF grant DMS0244737.

2. Research supported in part by an NSERC Research grant.

Keywords: Martingale problem, infinite dimensional operators, elliptic operators, degenerate, Bessel, stochastic

differential equations, superprocesses

Classification: Primary 60H10

Running Head: Degenerate SDE’s .

Submitted to EJP on January 19, 2004. Final version accepted on August 23, 2004.

634

1. Introduction.

We prove existence and uniqueness of the martingale problem for the infinite-dimensional degenerate operator

Lf (x) =

X

[xi γi (x)fii (x) + bi (x)fi (x)]

i∈S

under suitable H¨older continuity assumptions on the coefficients γi and bi . Here S is a

countably infinite discrete set, we write x = (xi )i∈S with xi ≥ 0 for each i, L operates

on the class of finite-dimensional cylindrical functions, and f i and fii denote the first and

second partials of f in the direction xi .

In the last ten years there has been considerable interest in infinite-dimensional

operators whose coefficients are only H¨older continuous rather than Lipschitz continuous.

See [CD96], [D96], [L96], [Z00], and [DZ02], for example, which consider operators that

are perturbations of either the infinite-dimensional Laplacian or of the infinite-dimensional

Ornstein-Uhlenbeck operator. The operator L given above is not only infinite-dimensional,

but also degenerate, due to the xi factor in the second order term. This degeneracy also

means that the diffusion coefficient will not have a Lipschitz square root even for smooth

γi , invalidating the standard fixed point approaches.

The principal motivation for this work is the question of uniqueness for measurevalued diffusions which behave locally like a superprocess. In general assume S is a Polish

space and let MF (S) denote the space of finite measures on S with the weak topology.

R

Write m(f ) = f dm for m ∈ MF (S) and an appropriate R-valued f on S. Assume

{Ax : x ∈ MF (S)} is a collection of generators, all defined on an appropriate domain D 0 of

bounded continuous functions on S, and γ : S × MF (S) → R+ . Let Ω be C(R+ , MF (S)),

equipped with the topology of uniform convergence on bounded intervals, its Borel σ-field

F, canonical right-continuous filtration Ft , and coordinate maps Xt (ω) = ωt .

For each law µ on MF (S), a probability P on (Ω, F) is a solution of the martingale

problem associated with A, γ and initial law µ, written M P (A, γ, µ), if for each f ∈ D 0 ,

Xt (f ) = X0 (f ) +

Mtf

+

Z

t

Xs (AXs f ) ds,

0

where M f is a continuous Ft -martingale such that

f

hM it =

Z

t

Xs (2γ(·, Xs )f 2 ) ds.

0

Under appropriate continuity conditions on (Ax , γ) one can usually construct solutions to M P (A, γ, µ) as the weak limit points of large population (N ), small mass (N −1 )

systems of branching particle systems. In these approximating systems a particle at x

635

in population Xt branches into a mean 1 number of offspring with rate N γ(x, Xt ), and

between branch times particles evolve like a Markov process with generator A Xt (see e.g.

[MR92]). The main difficulty lies in questions of uniqueness of solutions to M P (A, γ, µ).

A case of particular interest is S = Rd and

X

X

AX f (y) =

aij (y, X)fij (y) +

bi (y, X)fi (y),

i,j≤d

i≤d

in which particles evolve according to a state dependent Itˆ

o equation between branch

times. For γ = γ0 constant, uniqueness is proved in [DK98] under appropriate Lipschitz

conditions on a, b, using methods in [P95]. The latter also effectively handles the case

γ(y, X) = γ(y) (and some other special cases of X-dependence) by proving uniqueness for

an associated strong equation and historical martingale problem.

Even in the case where S is finite, the problem of handling general γ was only

recently solved in [ABBP02] and [BP03]. If S = {1, . . . , d}, then M F (S) = Rd+ and

Pd

Ax f (i) = j=1 qij (x)f (j), where for each x ∈ Rd+ , (qij (x)) is a Q-matrix of a Markov

P

chain on S, that is, qij ≥ 0 for i 6= j and qii = − j6=i qij . Then X solves M P (A, γ, µ) if

and only if Xt ∈ Rd+ solves the degenerate stochastic differential equation

Xti

=

X0i

+

Z tX

d

0 j=1

Xsj qji (Xs ) ds

+

Z

0

t

(2γi (Xs )Xsi )1/2 dBsi ,

i = 1 . . . d.

(1.1)

Here γi : Rd+ → [0, ∞), i = 1, . . . , d, B 1 , . . . , B d are independent one-dimensional Brownian

motions, and X0 has law µ for a given probability measure µ on Rd+ . More generally,

consider the generator

Lf (x) =

d

X

[xi γi (x)fii (x) + bi (x)fi (x)]

i=1

for f ∈ Cb2 (Rd+ ), the space of bounded continuous functions on Rd+ whose first and second

P

partials are also bounded and continuous; bi (x) = j xj qji (x) would correspond to (1.1).

If µ is a law on Rd+ , a probability P on C(R+ , Rd+ ) solves M P (L, µ) if and only if for all

f ∈ Cb2 (Rd+ ),

Z t

f

Mt = f (Xt ) − f (X0 ) −

Lf (Xs ) ds

0

is an Ft -martingale under P and X0 has law µ.

Theorem A (Corollary 1.3 of [BP03]). Assume γi : Rd+ → (0, ∞), bi : Rd+ → R are

α-H¨older continuous on compact sets and satisfy

bi (x) ≥ 0 on {xi = 0},

636

(1.2)

|bi (x)| ≤ c(1 + |x|).

(1.3)

Then there is a unique solution to M P (L, µ) for each law µ on Rd+ .

A similar existence and uniqueness theorem was proved in [ABBP02] (see Theorem

A of [BP03]) assuming only continuity of γi and bi but with (1.2) strengthened to

bi (x) > 0 on {xi = 0}.

(1.4)

A simple one-dimensional example shows these results are sharp in the sense that uniqueness fails if only continuity and (1.2) are assumed (see Section 8 of [ABBP03]). Clearly

(1.2) is needed to ensure solutions remain in the positive orthant.

In this work we extend the method of [BP03] to the case where S is a countably

infinite discrete set and hence take a step towards resolving the general uniqueness problem

described above. Both [ABBP02] and [BP03] adapt the perturbation approach of [SV79]

P

to this setting by considering L as a perturbation of L0 f = i xi γi0 fii + b0i fi for some

constants γi0 > 0 and b0i ≥ 0. If Rλ is the resolvent associated with L0 , the key step is to

show that on a suitable Banach space (B, k · k), one has

k(Rλ f )i k + kxi (Rλ f )ii k ≤ Ckf k for all i.

(1.5)

Qd b0 /γ 0 −1

In [ABBP02] the space B is L2 (Rd+ , 1 xi i i dxi ) (here b0i > 0), while in [BP] the space

B is a weighted H¨older space ((4.9) below gives the precise norm). In both cases the constant C in (1.5) is independent of d. The L2 setting in [ABBP02], however, does not appear

to extend readily to infinite dimensions. There is the question of an appropriate measure

on R∞

+ , there are problems extending the Krylov-Safonov type theorems on regularity of

the resolvents which are required to handle all starting points as opposed to almost all

starting points, and, as in the finite-dimensional setting, (1.4) will not hold for the most

natural Q-matrices such as nearest random walk on the discrete circle. We therefore will

extend the weighted H¨older approach in [BP03]. This approach has also been effective in

other (non-singular) infinite-dimensional settings ([ABP04]).

Our main result (Theorem 2.7) states that the natural infinite dimensional analogue

of M P (L, ν) has a unique solution when S is a discrete countably infinite set and X

takes on values in an appropriate space of measures. To understand the nature of the

assumptions made on the coefficients bi and γi , consider the Corollaries 2.10–2.12 when

S = Zd . Basically, we require bi and γi to be H¨older continuous in the j th variable, where

the H¨older constant approaches 0 at a certain polynomial rate as |i − j| approaches ∞.

P

The state space of X will be measures x(·) satisfying i |i|q x(i) < ∞ where q > 0 may

approach zero for α close to 1 but becomes large as α gets small. There are cases where

infinite measures are allowed but they require stronger H¨older conditions on the coefficients

as the mass gets large (see Remark 2.13).

637

The main result and a number of corollaries are stated in Section 2. In Section 3 we

prove a more general existence theorem (Theorem 2.4) by truncating to a finite-dimensional

system and taking weak limits. Although these type of arguments are well-known (see

[SS80]), we could not find the particular result we needed in the literature and have included

the proof for completeness: in addition, there is an unexpected mild condition needed; see

Theorem 2.4 and Remark 2.5. The weighted H¨older spaces are introduced in Section 4

where the infinite dimensional analogues of (1.5) are derived. Since the constants in [BP03]

are independent of dimension this should be easy, but some complications arise in infinite

dimensions since boundedness of the weighted H¨older norms does not imply continuity, in

contrast to the case of finite dimensions. We must establish uniform convergence of the

appropriate derivatives of the resolvent by the corresponding quantities for a sequence of

approximating finite-dimensional functions to carry over the finite-dimensional estimates

from [BP03] and obtain continuity (Proposition 4.8). The key bounds on the weighted

H¨older norm then follow from the finite-dimensional result in [BP] (Corollary 4.10). This

approximation is also used to derive the perturbation equation for the resolvent of strong

Markov solutions of M P (L, µ) in terms of Rλ = (λ − L0 )−1 (Proposition 5.4).

In Section 5 local uniqueness is established (Theorem 5.5), i.e., if γ i and bi are

sufficiently close to constant functions, uniqueness is shown. In this setting our state space

may include counting measure, but as the coefficients become asymptotically constant this

is not surprising. In Section 6 we use the local uniqueness and a localization argument

to prove Theorem 2.7. Localization in infinite dimensions still seems to be an awkward

process and our arguments here are surely not optimal–we believe some of the additional

continuity conditions in Assumption 2.6 may be weakened. Still it is important to note that

the weighted H¨older spaces at least allow for localization. In their ground-breaking paper

[DM95], Dawson and March establish a quite general uniqueness result in the FlemingViot setting but were unable to carry out the localization step. Nonetheless [DM95] still

represents the best available uniqueness result in general infinite dimensional settings albeit

in the Fleming-Viot setting and for close to constant coefficients. Finally in Section 7 we

prove the various corollaries to Theorem 2.7.

2. Notation and statement of results.

We will use the letter c with or without subscripts to denote positive finite constants

whose exact value does not matter and which may change from line to line. We use κ with

subscripts to denote positive finite constants whose value does matter.

Let S be a countable set equipped with a map | · | : S → [0, ∞) such that S n = {i ∈

S : |i| < n} is finite for all n ∈ Z+ . (S0 = ∅). Our prototype is of course S = Zd with | · |

equal to the usual Euclidean length of i. Let ν : S → (0, ∞) and for x ∈ R S let

X

|x|ν =

|xi |νi ∈ [0, ∞].

i∈S

638

ν will be called a weight functon. We will use both νi and ν(i) for the ith coordinate of ν

and similarly for other maps defined on S.

We let

Mν (S) = {x ∈ RS+ : |x|ν < ∞}.

and consider elements of Mν (S) as measures on S with

hx, ϕi =

X

xi ϕ(i)

i∈S

when

ϕ : S → R,

X

i∈S

xi |ϕ(i)| < ∞.

It is easy to see that Mν (S) is a Polish space when equipped with the distance |x − x′ |ν .

If νi ≡ 1, it is easy to check that Mν (S) = MF (S) is the usual space of finite measures on

S equipped with the topology of weak convergence for the discrete topology on S.

R+ = [0, ∞) and Cb2 (RS+n ) is the set of bounded continuous functions f : RS+n → R

whose first and second partial derivatives fi , fij are bounded and continuous. If xi = 0,

then the partials fi , fij are interpreted as right-hand derivatives.

Define the projection operator πn : RS+ → RS+n by

πn x(i) = x(i),

i ∈ Sn ,

(2.1)

and define the lift operator Πn : RS+n → RS+ by

Πn x(i) = 1(i∈Sn ) x(i).

(2.2)

Let

2

Cb,F

(Mν (S)) = {f : Mν (S) → R : ∃n, fn ∈ Cb2 (RS+n ) such that f (x) = fn (πn x)}.

These are the functions which only depend (in a C 2 way) on the coordinates xi with i ∈ Sn .

2

Clearly Cb,F

(Mν (S)) ⊂ Cb (Mν (S)), the space of bounded continuous maps from Mν (S)

to R.

For i ∈ S assume γi : Mν (S) → [0, ∞), bi : Mν (S) → R, and for f : Mν (S) → R

define

X£

¤

xi γi (x)fii (x) + bi (x)fi (x) ,

x ∈ Mν (S),

(2.3)

Lf (x) =

i∈S

provided these partial derivatives exist and the above series is absolutely convergent. Note

2

that this is the case if f ∈ Cb,F

(Mν (S)). Let Ων equal C(R+ , Mν (S)), equipped with the

topology of uniform convergence on bounded intervals. Let Xt (ω) = ω(t) for ω ∈ Ων , let

Fu0 be the universal completion of σ(Xs : s ≤ u), and set Ft = ∩u>t Fu0 , F = ∪t≥0 Ft0 .

639

Definition 2.1. Let µ be a probability on Mν (S). A probability P on (Ων , F) solves

MP(L, µ), the martingale problem for L in Mν (S) started at µ, if P(X0 ∈ ·) = µ(·) and

2

for any f ∈ Cb,F

(Mν (S))

Mtf

= f (Xt ) − f (X0 ) −

Z

t

0

Lf (Xs )ds

is an (Ft )-local martingale under P. MP(L) is well-posed in Mν (S) if and only if there is

a unique solution to MP(L, µ) in Mν (S) for every initial law µ on Mν (S).

Note that s → Lf (Xs ) and t → f (Xt ) and hence t → Mtf are all necessarily

continuous functions.

Remark 2.2 (a) As in Remark 1.1(d) of [BP03] one could also consider test functions

f (x) = fn (πn x) for some fn ∈ Cb2 (RSn ), (i.e., those which extend in a C 2 manner to all of

RSn instead of RS+n ).

(b) Changing the class of test functions changes the martingale problem. The

smaller the class of test functions for which one establish uniqueness, the stronger the

2

theorem. Cb,F

(Mν (S)) is a reasonably small class.

(c) Let {Bi : i ∈ S} be a sequence of independent one-dimensional adapted Brownian motions on a filtered probability space (Ω, F, F t , P) and consider the stochastic differential equation

Yi (t) = Yi (0) +

Z

t

(2Yi (s))γi (Ys ))

1/2

0

dBsi

+

Z

t

bi (Ys )ds,

0

i ∈ S, t ≥ 0.

(2.4)

Here Y (0) is an F 0 -measurable random vector in Mν (S). If Y is a continuous Mν (S)valued solution to (2.4), a simple application of Itˆ

o’s formula shows that the law P Y of Y

is a solution to MP(L, µ) where µ is the law of Y (0). Conversely, given a solution P of

MP(L, µ), a standard construction allows one to build Brownian motions {B i : i ∈ S}

and a solution Y to (2.4) on some Ω such that P is the law of Y .

We introduce conditions on bi , the first of which we assume throughout this work:

Assumption 2.3. (a) There exists a constant κ2.3a (b) such that

X

i∈S

|bi (x)|νi ≤ κ2.3a (b)(|x|ν + 1),

x ∈ Mν (S),

(2.5)

(b) There exists a constant κ2.3b (b) such that

bi (x) ≥ −κ2.3b (b)xi ,

i ∈ S,

640

x ∈ Mν (S).

(2.6)

Assumption 2.3(a) will avoid explosions in finite time while a condition such as

Assumption 2.3(b) is needed to ensure that our solutions have non-negative components

(although a weaker condition bi (x) ≥ 0 if xi = 0 sufficed in finite dimensions – see [BP03].)

Note that no analogue of (2.3) is needed for the diffusion coefficients.

Our focus is on uniqueness in law of solutions to MP(L, δx0 ), but as our setting is

slightly different from that considered in the literature (e.g., Shiga and Shimizu [SS80]),

we state a general existence result. The proof is given in Section 3.

Theorem 2.4. Assume there exists β : S → (0, ∞) such that lim|i|→∞ β(i) = 0 and

for all i ∈ S, bi , γi have (necessarily unique) continuous extensions bi : Mβν (S) → R,

γi : Mβν (S) → [0, ∞). In addition to (2.5) and (2.6) assume there exists a constant κ 2.4 (γ)

such that

sup kγi k∞ = κ2.4 (γ) < ∞.

(2.7)

i

Then for any x0 ∈ Mν (S) there is a solution to MP(L, δx0 ).

Remark 2.5. Note the above continuity condition is trivially satisfied if γ i , bi are given as

continuous functions on RS+ with the product topology (as in Shiga and Shimizu [SS80]).

This condition is only needed to obtain a compact containment condition in the usual

tightness proof and will be easy to verify in the examples of interest. If γ i , bi on Mν (S)

are uniformly continuous with respect to | · |βν , the above extensions exist.

Here is our key H¨older continuity hypothesis on γi and bi :

Assumption 2.6. For some β : S → (0, ∞) satisfying lim|i|→∞ β(i) = 0 and

β(i) ≤ κ2.6a ν(i)−α/2 , for each i ∈ S, bi and γi have (necessarily unique) continuous

extensions bi : Mβν (S) → R, γi : Mβν (S) → (0, ∞). For any x0 ∈ Mν (S), η > 0, there

exists δ0 > 0 and a κ2.6b > 0 such that if x ∈ Mν (S) and |x − x0 |βν < δ0 , then (a) holds

and either (b) or (c) holds, where

(a)

X |γi (x) − γi (x0 )| |bi (x) − bi (x0 )|

+

0,

X |γi (x + hej ) − γi (x)|

i

γi (x0 )

+

|bi (x + hej ) − bi (x)|

−α/2 α

h

≤ κ2.6b γj (x0 )−α/2 xj

γi (x0 )

or

(c) lim|i|→∞ ν(i) = ∞ and for any j ∈ S, h ∈ (0, 1],

X |γi (x + hej ) − γi (x)| |bi (x + hej ) − bi (x)|

−α/2 α

+

≤ κ2.6b γj (x0 )−α/2 (1 + xj

)h .

γ

(x

)

γ

(x

)

i

0

i

0

i

Our main result is the following.

641

Theorem 2.7. Suppose Assumptions 2.3(a) and 2.6 hold, (2.7) holds, and either

bi (x) ≥ 0 for all i ∈ S and x ∈ Mν (S),

(2.9)

or

lim ν(i) = ∞

bi (x) ≥ −κ2.7 xi γi (x), for all i ∈ S and x ∈ Mν (S).

and

|i|→∞

(2.10)

Then MP(L) is well-posed in Mν (S).

We state some corollaries to Theorem 2.7, the proofs of which are given in Section 7.

Corollary 2.8. Assume bi : Mν (S) → R, γi : Mν (S) → R+ are continuous maps for all

i ∈ S, where ν : S → (0, ∞) satisfies lim|i|→∞ ν(i) = ∞. Assume Assumption 2.3(a)-(b),

(2.7),

inf γi (x) = ε0 > 0,

(2.11)

i,x

there exists non-negative constants {C(i, j) : i, j ∈ S} such that

|γi (x + hej ) − γi (x)| + |bi (x + hej ) − bi (x)| ≤ C(i, j)hα ,

and if C(j) =

h > 0,

P

x ∈ Mν (S), (2.12)

C(i, j), then

i

h

X

α

C(j)1/(1−α) ν(j)−α(1− 2 )/(1−α) < ∞.

sup C(j) + 1(α d(1 − α)(α(1 − α2 ))−1 , and, for some

c0 > 0, p > d, set C(i, j) = c0 (|i − j| + 1)−p for i, j ∈ Zd . Let bi : Mν (Zd ) → R and

γi : Mν (Zd ) → (0, ∞) be continuous maps satisfying (2.5), (2.6), (2.7), (2.11), and (2.12).

Then MP(L) is well-posed in Mν (Zd ).

Corollary 2.11. Let p : Zd − {0} → [0, 1] be a probability on Zd − {0} such that mq =

P q

i |i| p(i) < ∞, where q is as in Corollary 2.10. Let q ij = λp(j −i) (i 6= j) be the Q-matrix

of a random walk which takes steps distributed as p with rate λ > 0. Let ν, (bbi ), and (γi )

P

satisfy the hypotheses of Corollary 2.10, and let bi (x) = bbi (x) + j xj qji . Then MP(L)

is well-posed in Mν (Zd ).

−1

Corollary 2.12. Let R > 0, N = {i ∈ Zd : |i| ≤ R} and γ : RN

] be H¨older

+ → [ε, ε

α −1

continuous of index α ∈ (0, 1]. Assume q > d(1 − α)(α(1 − 2 )) and set ν(i) = (|i| + 1)q ,

i ∈ Zd . Let qij = λp(j − i), (i 6= j) where p is a probability on Zd − {0} with finite q th

P

moment. If bi (x) = j xj qji and γi (x) = γ((xi+j : j ∈ N )), then MP(L) is well-posed in

Mν (Zd ).

Corollary 2.8 is a version of Theorem 2.7 where the hypotheses are given in terms

of the H¨older constants of the γi and bi ; Corollary 2.9 applies Corollary 2.8 to the case of

super-Markov chains. Corollaries 2.10–2.12 are the application of Corollaries 2.8 and 2.9

to the case where S = Zd and an explicit bound is given for the H¨older constants of the

γi and bi .

Remark 2.13. If we assume Assumptions 2.6 (a),(b) and (2.9) we may take ν i → 0 so

P

that Mν (S) will contain some infinite measures, that is, points x such that i∈S xi = ∞.

In this case the H¨older condition Assumption 2.6 (b) becomes rather strong if x j gets large.

3. Existence.

If ε = {εn } is a sequence in (0, ∞) decreasing to 0 and S ′ = {Sn′ } is a sequence of

finite subsets of S which increases to S with S0′ = ∅, let

Kε,S ′ = {x ∈ Mν (S) :

X

′

i∈S

/ n

xi νi ≤ εn for all n ∈ Z+ }.

Write Kε for the above in the case when Sn′ = Sn for all n ∈ Z+ .

Lemma 3.1. (a) For any ε, S ′ as above, Kε,S ′ is a compact subset of Mν (S).

(b) If K is a compact subset of Mν (S), there is a sequence εn decreasing to 0 such that

K ⊂ Kε .

Proof. The standard proof is left for the reader.

643

Remark 3.2. If σN : Mν (S) → Mν (S) is defined by σN (x)(i) = 1(i∈SN ) x(i), then for any

sequence εn decreasing to 0, it is easy to check that σN (Kε ) ⊂ Kε .

Proof of Theorem 2.4. First, let Xtn be the solution to

Z t³

hZ t

i

´1/2

n,i

n +

i

bi ((Xs ) )ds +

dBsi .

Xt = x0 + 1(i∈Sn )

γi ((Xsn )+ )|Xsn,i |

0

(3.2)

0

Here (Xsn )+ = ((Xsn,i )+ : i ∈ S) and {B i } is a sequence of independent one-dimensional

Brownian motions on some filtered probability space. Note that

bbi ((xj )j∈S ) = bi ((x+ )j∈S , (xj )j∈S c )

n

n

0

j

n

is a continuous function on RSn with linear growth (by (2.5) and the continuity assumptions

on bi ). The same is true of

´

³

j

+

σ

bi ((xj )j∈Sn ) = γi ((xj )j∈Sn , (x0 )j∈Snc )|xi |

and so the existence of X n follows from Skorokhod’s existence theorem for finite-dimensional SDEs. Using L0t (X n,i ) = 0 and bi (x) ≥ 0 if xi = 0 (by (2.6)), one can use Tanaka’s

formula to see that Xtn,i ≥ 0 for all t ≥ 0 and for all i almost surely, and one may therefore

remove the superscript + ’s in (3.2). Let Tkn = inf{t : |Xtn |ν ≥ k}. For each n, Tkn ↑ ∞ as

Rt

P

k → ∞, since x0 ∈ Mν (S). Define Mtn = i∈Sn νi 0 (γi (Xsn )Xsn,i )1/2 dBsi , and note that

if ν n = maxi∈Sn νi , then

Z tX

n

hMt i =

νi2 γi (Xsn )Xsn,i ds

0 i∈S

n

≤ κ2.4 ν n

Z

t

0

|Xsn |ν ds

≤ κ2.4 ν n kt

for t ≤ Tkn . Let

n

Xt

Z

|Xtn |ν

(3.3)

t

+ κ2.3b

|Xsn |ν ds

0

Z tX

= |x0 |ν +

νi bi (Xsn ) + κ2.3b |Xsn |ν ds + Mtn .

=

0 i∈S

n

n

Mt∧T

n is a martingale by (3.3) and so (2.5) implies

k

n

E X t∧T n

k

Z

t

n

n

E (κ2.3a (|Xs∧T

n |ν + 1) + κ2.3b |Xs∧T n |ν )ds

k

k

0

Z t

n

≤ |x0 |ν + κ2.3a t + c1

E (|X s∧T n |ν )ds.

≤ |x0 |ν +

0

644

k

(3.4)

The left hand side is clearly finite by the definition of Tkn , Gronwall’s lemma implies

n

E (X t∧T n ) ≤ [|x0 |ν + κ2.3a t]ec1 t ,

k

t ≥ 0,

and so Fatou’s lemma gives

n

E (X t ) ≤ [|x0 |ν + κ2.3a t]ec1 t ,

t ≥ 0.

(3.5)

n

Therefore, using (2.6) in (3.4), we see that X t is a submartingale. The weak L1 inequality

and (3.5) imply

n

P(sup |Xtn |ν ≥ k) ≤ P(sup X t ≥ k) ≤ k −1 [|x0 |ν + κ2.3a T ]ec1 T ,

t≤T

(3.6)

t≤T

which implies

lim sup P(Tkn ≤ T ) = 0,

k→∞ n

Define Xtn (ϕ) =

(recall S0c = S)

P

i∈S

T > 0.

(3.7)

Xtn,i ϕ(i) for ϕ : S → R+ and β m = supi∈Sm

c β(i) ↓ 0. Then

c νβ) > ε0 ) ≤ P(sup

P(sup Xtn (1Sm

t≤T

t≤T

≤

X

ν(i)Xtn,i > ε0 /β m )

c

i∈Sm

βm

[|x0 |ν + κ2.3a T ]ec1 T .

ε0

(3.8)

Let ε, T > 0. Choose mk ↑ ∞ and K0 > 0 such that

∞

X

¡

¢1/2

β mk

[|x0 |ν + κ2.3a T ]ec1 T + K0−1 [|x0 |ν + κ2.3a T ]ec1 T < ε,

(3.9)

k=1

and define

K = {x ∈ Mνβ (S) :

X

c

i∈Sm

xi νi βi ≤ (β mk )1/2 for all k ∈ N, |x|ν ≤ K0 }.

k

¡

¢1/2

Then K is compact in Mνβ (S) by Lemma 3.1. By (3.8) with ε0 = β mk

and m = mk

and (3.6) with k = K0 we get that for each n

P(Xtn ∈ K for all t ≤ T )

(3.10)

∞

i

hX

(β mk )1/2 [|x0 |ν + κ2.3a T ]ec1 T + K0−1 [|x0 |ν + κ2.3a T ]ec1 T

≥1−

k=1

≥ 1 − ε,

645

by (3.9). This will give us the compact containment required for the tightness of

{P(X·n ∈ ·) : n ∈ N}.

We claim next that if i ∈ S is fixed, then

{X n,i : n ∈ N} is a tight sequence of processes in C([0, ∞), R).

(3.11)

By (3.7) it suffices to show

{X n,i (· ∧ Tkn ) : n ∈ N} is tight in C([0, ∞), R) for each k ∈ N.

(3.12)

Let Mtn,i denote the stochastic integral on the right hand side of (3.2). Then for s ≤ t,

E

³³ Z

´4 ´

t

s

1(r≤Tkn ) dMrn,i

≤ c2 E

³³ Z

s

2

t

1(r≤Tkn ) Xrn,i γi (Xrn )dr

≤ c3 (κ2.4 ) (k/νi )2 (t − s)2 .

´2 ´

(3.13)

In addition by (2.5),

Z

s

t

1

(r≤Tkn )

|bi (Xrn )|dr

Z

κ2.3a t

≤

1(r≤Tkn ) (|Xrn |ν + 1)dr

νi

s

κ2.3a

≤

(k + 1)(t − s).

νi

This, (3.13), (3.2), and standard arguments now imply (3.12).

(3.10) and (3.11) imply {X·n : n ∈ N} is a tight sequence in C([0, ∞), Mνβ (S)). If

νβ(i) ≡ 1, this is standard, as then Mνβ (S) = MF (S) (see, e.g., Theorem II.4.1 in [Pe02]).

In general, define Φ : Mνβ (S) → MF (S) by

Φ(x)(i) = x(i)νi βi .

Then Φ is an isometry and the above result for MF (S) gives the required implication.

By Skorokhod’s theorem we may first extract a weakly convergent subsequence

nm

{X } and then assume X nm → X a.s. in C([0, ∞), Mνβ (S)). It is easy to use the

continuity of bi , γi on Mνβ (S) to let n = nm → ∞ in (3.2) and conclude

Xti

=

xi0

+

Z

t

0

Z tp

γi (Xs )Xsi dBsi

bi (Xs )ds +

0

for all t ≥ 0 and all i ∈ S, a.s. Fatou’s lemma implies for any t > 0

|Xt |ν =

X

i∈S

lim inf Xtnm ,i νi ≤ lim inf |Xtnm |ν ,

m→∞

m→∞

646

(3.14)

and so an elementary argument implies

P(sup |Xt |ν > k) ≤ P(lim inf sup |Xtnm |ν > k)

m→∞ t≤T

t≤T

≤ lim inf P(sup |Xtnm |ν > k)

m→∞

≤k

−1

t≤T

[|x0 |ν + κ2.3a T ]ec1 T ,

the last by (3.6). This proves

sup |Xt |ν < ∞,

T > 0,

a.s.,

(3.15)

t≤T

and so X· has Mν (S) valued paths a.s.

To show that X· has continuous Mν (S)-valued paths a.s. we use the following

lemma, whose elementary proof is left to the reader.

Lemma 3.3. Suppose x : [0, ∞) → Mν (S) is such that xt (i), i ∈ S, and |xt |ν are all

continuous. Then x is continuous.

Clearly Xt (i) is continuous for all i ∈ S a.s. since it is continuous in Mνβ (S). From

Rtp

P

(3.14) we see that if Mn (t) = i∈Sn νi 0 γi (Xs )Xsi dBsi , then

X

X

Xt (i)νi =

x0 (i)νi +

t

X

νi bi (Xs )ds + Mn (t).

(3.16)

0 i∈S

n

i∈Sn

i∈Sn

Z

(3.15) and (2.5) show that

X

i∈Sn

νi |bi (Xs )| →

X

i∈S

νi |bi (Xs )|,

which is bounded uniformly on compact time intervals a.s. as n → ∞, and so

¯Z

¯

sup¯

t≤T

t

X

0 i∈S

n

≤

Z

T

0

νi bi (Xs )ds −

X

i∈S−Sn

Z tX

0 i∈S

¯

¯

νi bi (Xs )ds¯

νi |bi (Xs )|ds

→0

(3.17)

as n → ∞ by dominated convergence. By (3.16) and (2.5),

sup |Mn (t)| ≤ sup |Xt |ν + |x0 |ν +

t≤T

t≤T

647

Z

0

T

κ2.3a (|Xs |ν + 1)ds.

By (3.15) and the Dubins-Schwartz theorem, this means {hM n iT } remains bounded in

probability as n → ∞. Therefore

Z TX

νi2 γi (Xs )Xs (i)ds = lim hM n iT < ∞,

a.s., T > 0.

0

n→∞

i∈S

A standard square function inequality now implies

sup |Mn (t) − Mm (t)| → 0

t≤T

in probability as m, n → 0 for all T > 0, and so we may take a subsequence such that

Mnk converges uniformly on compact time intervals a.s. Let n = nk → ∞ in (3.16). The

above and (3.17) show that the right hand side of (3.16) converges uniformly on compact

time intervals a.s. to a necessarily continuous process. As the left hand side converges to

|Xt |ν for all t ≥ 0, a.s., it follows that t → |Xt |ν is continuous. Lemma 3.3 therefore shows

t → Xt is a continuous Mν (S)-valued process. By (3.14) and Remark 2.2(c), the law of X

is a solution of the martingale problem for L starting at x0 .

4. Estimates.

We first obtain some key analytic estimates for the special case when γi = γi0 and

bi = b0i are constants. Assume

0 < γi0 ≤ sup γi0′ = κ4.1 (γ 0 ) < ∞,

i′

0 ≤ b0i ,

i ∈ S,

Let

L0 f (x) =

and

X

|b0 |ν =

X

i∈S

i ∈ S,

(4.1)

b0i ν(i) < ∞.

(4.2)

γi0 xi fii (x) + b0i fi (x)

i∈S

for functions f for which the partial derivatives exist and the sum is absolutely convergent.

By Theorem 2.4 there is a solution P0x0 to MP(L, δx0 ) in Mν (S) for each x0 ∈ Mν (S). In

fact it is easy to see that under P0x0 , the processes {Xi : i ∈ S} are independent diffusions

and each X i is a suitably scaled squared Bessel process whose law is that of the pathwise

unique solution to

Z

t

X i (t) = x0 (i) +

0

(X i (s)γi0 )1/2 dBsi + bi0 t,

(4.3)

where the Bsi are independent one dimensional Brownian motions. (Theorem 2.4 is only

needed here to ensure X has paths in Ων .) An explicit formula for the transition kernel

of pit (xi , dyi ) of Xi is given in (2.2) and (2.4) of [BP03]. Let (Pt )t≥0 and (Rλ )λ≥0 be the

semigroup and resolvent, respectively, of the Mν (S)-valued diffusion Xt = (Xti )i∈S .

648

Lemma 4.1. For any compact set K ∈ Mν (S), T > 0, and ε > 0, there is a sequence

η = {ηn } decreasing to zero such that

sup P0x0 (Xt ∈ Kη for t ≤ T ) ≥ 1 − ε.

x0 ∈K

Proof. By Lemma 3.1(b) we may assume K = Kδ for some sequence δn decreasing to

P

0

zero. Set B(N ) = i∈S

/ N ν(i)bi . If

X

Zn,N (t) =

ν(i)Xi (t)

i∈Sn −SN

for n > N ≥ 0 and

ZN (t) =

X

ν(i)Xi (t),

i∈S

/ N

then for x0 ∈ Kδ

E 0x0 [ZN (t)] =

X

i∈S

/ N

x0 (i)ν(i) + b0i ν(i)t ≤ δ(N ) + B(N )t,

where δ(N ), B(N ) ↓ 0 by (4.2). Since Zn,N (t) is a submartingale, the weak L1 inequality

implies that for any x0 ∈ Kδ

P0x0 (sup ZN (t) > A) = lim P0x0 (sup Zn,N (t) > A)

n→∞

t≤T

t≤T

−1

≤ lim A

n→∞

−1

≤A

E 0x0 [Zn,N (t)]

(δ(N ) + B(N )T ).

(4.4)

sup P0x0 (sup ZNk (t) > 2−k ) ≤ 2−k−1 ε

(4.5)

Choose Nk ↑ ∞ such that

x0 ∈Kδ

t≤T

and then η > 1 sufficiently large so that

sup P0x0 (sup |Xt |ν > η) ≤ ε/2.

x0 ∈Kδ

t≤T

The latter is possible by (4.4) with N = 0 since Z0 (t) = |Xt |ν . Now define

ηn =

½

η,

n≤N

−k

2 , Nk ≤ n < Nk+1 .

649

(4.6)

Then for any x0 ∈ Kδ

P0x0 (Xt

∈

Kηc

for some t ≤ T ) ≤

P0x0 (sup |Xt |ν

t≤T

> η) +

∞

X

P0x0 (sup ZNk (t) > 2−k ) < ε

k=1

t≤T

by (4.5) and (4.6).

Define ei ∈ Mν (S) by ei (j) = 1(i=j) . Let α ∈ (0, 1] and for f : Mν (S) → R and

i ∈ S define

α

|f |α,i = sup{|f (x + hei ) − f (x)|x(i) 2 h−α : x ∈ Mν (S), h > 0}.

(4.7)

Set

α

|f |α = sup(γi0 ) 2 |f |α,i ,

kf kα = kf k∞ + |f |α .

i

(4.8)

If

Cα = {f : Mν (S) → R : f continuous , kf kα < ∞},

then it is easy to check that (Cα , k · kα ) is a Banach space.

Remark 4.2. One difference between our infinite dimensional setting and the finite dimensional setting in [BP03] is that supi |f |α,i < ∞ does not imply that f is uniformly

continuous on I = {x ∈ Mν (S) : x(i) > 0 for all i ∈ S} and hence has a continuous

extension to Mν (S). This is true on Rd+ (see Lemma 2.2 of [BP03]). Suppose we define

f (x) to be 1 if infinitely many of the x(i) 6= 2−i /ν(i) and 0 otherwise. Then |f |α,i = 0

but f is discontinuous on I. This complicates things a bit when checking whether various

operators preserve Cα .

Remark 4.3. A key fact in our argument is that the estimates on (Rλ f )i and xi (Rλ f )ii

from [BP03] are independent of the dimension of the space. Recall that the way we obtained the estimates in [BP03] was to first consider the one-dimensional case. If P ti denotes

the semigroup corresponding to the operator xγi0 f ′′ (x) + b0i f ′ (x), then we obtained bounds

on |(Pti f )′ (x)| and on |(Pti f )′ (x + ∆) − (Pti f )′ (x)| in terms of constants depending only on

γi0 and b0i ; see Lemmas 4.3, 4.4, 4.6, and 4.7 of [BP03]. If Pt denotes the semigroup correPd

sponding to L0 f (x) = i=1 [xi γi0 fii (x) + b0i fi (x)], we then derived bounds on |(Pt f )i (x)|

and on |(Pt f )i (x + ∆ej ) − (Pt f )i (x)| with the same constants (Propositions 5.1 and 5.2 of

[BP03]); hence the constants did not depend on the dimension d of the underlying space.

We then deduced estimates on (Rλ f )i . The same reasoning was applied for xi (Rλ f )ii .

Lemma 4.4. Let f ∈ Cα , λ > 0, and i ∈ S. There is a κ4.4 = κ4.4 (α) independent of

f, i, λ such that the following hold:

(a) The partial derivative (Rλ f )i (x) exists for every x ∈ Mν (S) and satisfies

α

α

k(Rλ f )i k∞ ≤ κ4.4 (γi0 ) 2 −1 λ− 2 |f |α,i .

650

(4.9)

(b) The second order derivative (Rλ f )ii (x) exists on {x ∈ Mν (S) : xi > 0} and satisfies

|xi (Rλ f )ii (x)| ≤

κ4.4 (γi0 )−1 |f |α,i

h³ γ 0 ´ α2

i

λ

α

2

i

∧ xi .

(4.10)

In particular, limxi →0 xi (Rλ f )ii (x) = 0 uniformly on Mν (S) and if xi (Rλ f )ii (x) is set to

be this limit on {x ∈ Mν (S) : xi = 0}, then

α

α

kxi (Rλ f )ii (x)k∞ ≤ κ4.4 (γi0 ) 2 −1 λ− 2 |f |α,i .

(4.11)

Proof. We only prove (b) as (a) is similar but easier. Let t > 0. Then argue as in the

finite-dimensional argument (Proposition 5.1 of [BP03]), noting the constants there are

independent of dimension, to see that (Pt f )ii exists on Mν (S) (in fact on RS+ ) and satisfies

´

³x

α

i

(4.12)

|xi (Pt f )ii (x)| ≤ c1 |f |α,i (γi0 t) 2 −1 0 ∧ 1 .

γi t

If xi > 0, this allows one to differentiate through the time integral (by the dominated

convergence and the mean value theorem) and conclude for xi > 0 that (Rλ f )ii exists and

satisfies

Z ∞

e−λt xi (Pt f )ii (x)dt.

xi (Rλ f )ii (x) =

0

A simple calculation using (4.12) leads to (4.10) for xi > 0. The fact that xi (Rλ f )ii

approaches 0 uniformly as xi ↓ 0 is then immediate, as is (4.11).

Lemma 4.5. If f ∈ Cb (Mν (S)) and fn = f ◦ πn , then for any compact subset K of Mν (S)

lim sup |f (x) − fn (x)| = 0.

n→∞ x∈K

Proof. By Lemma 3.1(b) we may assume K = Kη for some sequence η = {ηn } decreasing

to 0. Then

X

sup |x − πn x|ν = sup

xi νi ≤ ηn → 0.

x∈Kη

x∈Kη

c

i∈Sn

Since πn (Kη ) ⊂ Kη (recall Remark 3.2) and f is uniformly continuous on Kη , the result

follows.

Corollary 4.6. If f ∈ Cb (Mν (S)) and fn = f ◦ πn , then for any λ > 0, Rλ fn → Rλ f

uniformly on compact subsets of Mν (S).

Proof. Let K be a compact subset of Mν (S) and ε > 0. Lemma 4.1 shows there is a

compact Kη such that

sup (Rλ 1Kηc )(x) < ε.

x∈K

651

Lemma 4.5 implies that kRλ f 1Kη − Rλ fn 1Kη k∞ → 0 as n → ∞ and so

lim sup sup |Rλ fn (x) − Rλ f (x)| ≤ lim sup sup |Rλ fn 1Kηc (x) − Rλ f 1Kηc (x)|

n→∞ x∈K

n→∞ x∈K

≤ 2εkf k∞ .

Let Rλn denote the resolvent of the finite-dimensional diffusion (Xi )i∈Sn under {Px0 }.

Then Rλn is a Feller resolvent (i.e., it maps Cb (RS+n ) to itself) and so if fen ((xi )i∈Sn ) =

f (Πn (x)) for f ∈ Cb (Mν (S)), then fen ∈ Cb (RS+n ) and Rλ fn (x) = Rλn fen ((xi )i∈Sn ) is continuous on Mν (S). (Πn is defined in (2.2).) The convergence in Corollary 4.6 therefore

shows

Rλ : Cb (Mν (S)) → Cb (Mν (S)).

(4.13)

Our immediate goal is to extend the continuity on Mν (S) to (Rλ f )i and xi (Rλ f )ii for

f ∈ Cα . As explained earlier, this is more delicate in our infinite-dimensional setting.

Lemma 4.7. There is a κ4.7 ≥ 1 such that

(a) if 0 < r ≤ R/2, then

Z ∞

z r−1 |z − r|

κ4.7

e−z

dz ≤

;

Γ(r) r

R+1

R

(b) for any r > 0

Z

0

∞

e−z

z r−1 |z − r|

dz ≤ κ4.7 (r + 1)−1/2 .

Γ(r) r

Proof. (a) In view of (b) we may assume R ≥ 1. The integral in (a) is bounded by

Z ∞

r−1

(z − r)2

−z z

dz,

e

Γ(r) rR/2

R

which is bounded by cR−1 r−1 (r + 1); see, e.g., Lemma 3.2(a) of [BP03]. This gives the

required bound if r ≥ 1 (recall R ≥ 1). Assume now that 0 < r < 1. The integral in (a) is

at most

Z ∞

Z ∞

zr

z r+1

−z

e

dz ≤

dz

e−z

Γ(r + 1)

Γ(r + 1)R

R

0

h Γ(r + 2) i

= R−1

= R−1 (r + 1) ≤ 2R−1 ≤ 4(R + 1)−1 .

Γ(r + 1)

(b) If r ≥ 1 this is immediate from Lemma 3.2(a) of [BP03]. If 0 < r < 1, then the

required integral is at most

Z ∞

Z ∞

zr

z r−1

−z

dz +

dz = 2.

e−z

e

Γ(r + 1)

Γ(r)

0

0

652

Proposition 4.8. Let f ∈ Cα and fn = f ◦ πn . If i ∈ S and λ > 0, then for any compact

set K ⊂ Mν (S)

£

lim sup |Rλ fn (x) − Rλ f (x)| + |(Rλ fn )i (x) − (Rλ f )i (x)|

n→∞ x∈K

¤

+ |xi (Rλ fn )ii (x) − xi (Rλ f )ii (x)| = 0.

Proof. Note first that f ∈ Cα implies fn ∈ Cα and so the existence of the above partial

derivatives follows from Lemma 4.4. We focus on the convergence of the second order

derivatives as the first order derivatives are handled in a similar and slightly simpler way,

while the resolvents themselves were handled in Corollary 4.6. Fix f ∈ C α .

If y ∈ Mν (S), write ybi = y|S−{i} and define Y (v, i) ∈ Mν (S) by setting Y (v, i)(j) =

y(j) if j 6= i and Y (v, i)(i) = v; in other words, Y (v, i) is the point which has the same

coordinates as y except that the ith coordinate is equal to v instead of yi . We may then

define dn (v; ybi ) = f (πn Y (v, i)) − f (Y (v, i)) and

Z

∂2

hn (b

yi ; t, xi ) = xi 2

dn (yi ; ybi )pit (xi , dyi ).

∂xi

If |dn (·; ybi )|α denotes the | · |α,i norm of dn (·; ybi ) with S = {i}, then |dn (·; ybi )|α ≤ 2|f |α,i ,

and so the above derivative exists and satisfies (see Lemmas 4.1, 4.3, 4.5, and 4.6 of [BP03])

³x

´

α

i

khn (·; t, xi )k∞ ≤ cα |f |α,i (γi0 t) 2 −1 0 ∧ 1 .

(4.14)

γi t

Note that

xi (Pt fn )ii (x) − xi (Pt f )ii (x) =

Z

hn (b

yi ; t, xi )

Y

pjt (xj , dyj ),

(4.15)

j6=i

where differentiation through the integrals is justified by the above bound and dominated

convergence.

If g : R+ → R, let kgkR = sup{|g(y)| : y ≤ R}. Assume first b0i > 0 and use (4.14)

in [BP03] to conclude that if w = xi /γi0 t and ri = b0i /γi0 , then

Z ∞

∞

k

X

z k+ri −1 |z − (k + ri )| dz

−w w

|hn (b

yi ; t, xi )| ≤

e

. (4.16)

|w−k|

|dn (zγi0 t; ybi )|e−z

k!

Γ(k + ri )

k + ri

γi0 t

0

k=0

By Lemma 4.7, if R > 0,

Z ∞

z k+ri −1 |z − (k + ri )|

dz

e−z

Γ(k + ri )

k + ri

R/γi0 t

κ4.7

κ4.7

+ 1(k+ri >R/2γi0 t) √

≤ 1(k+ri ≤R/2γi0 t)

0

(R/γi t) + 1

ri + k + 1

p

i

h

γ0t

κ4.7

1(k+ri ≤R/2γi0 t) √ i + 1(k+ri >R/2γi0 t) .

≤√

ri + k + 1

R

653

Use this and Lemma 4.7(b) in (4.16) to see that

|hn (b

yi ; t, xi )|

≤

(γi0 t)−1

∞

X

e

−w w

k!

k=0

h

k

|w − k| kdn (·; ybi )kR

Z

R/γi0 t

e−z

0

z k+ri −1 |z − (k + ri )|

dz

Γ(k + ri )

k + ri

p

ii

γ0t

+ 2kf k∞ κ4.7 (ri + k + 1)−1/2 1(k+ri ≤R/2γi0 t) √ i + 1(k+r1 >R/2γi0 t)

R

∞

h

³ γ 0 t ´1/2

i

X

|w − k|

wk

i

kd

(·;

y

b

)k

+

2kf

k

≤ (γi0 t)−1 κ4.7

e−w

n

i

R

∞

k! (ri + k + 1)1/2

R

h

k=0

+

2kf k∞ (γi0 t)−1 κ4.7

∞

X

1(k+ri >R/2γi0 t) e−w

k=0

|w − k|

wk

k! (ri + k + 1)1/2

= I1 + I2 .

(4.17)

A simple calculation (see Lemma 3.3(b) of [BP03]) shows that

³x

´h

³ γ 0 t ´1/2

i

i

i

∧

1

kd

(·;

y

b

)k

+

kf

k

n

i R

∞

γi0 t

R

h

i

≤ c2 ((γi0 t)−1 + 1) kdn (·; ybi )kR + R−1/2 kf k∞ .

I ≤ c1 (γi0 t)−1

If N is a Poisson random variable with mean w, the series in I2 is

i

h

−1/2

E |N − w|(ri + N + 1)

1(N +ri >R/2γi0 t) .

Assume now

(4.19)

R

> max(1, x2i , γi0 t).

8

(4.20)

If ri < R/(4γi0 t), recalling xi < R/8 by (4.20), the expectation in (4.19) is at most

i

h

−1/2

E |N − w|(ri + N + 1)

1(N −w> R−4xi )

4γ 0 t

i

h

≤ E |N − w|(N + 1)

−1/2

1(N −w>R/8γi0 t)

i

E (N − w)2

(R/8γi0 t)3/2

xi

c4

= c3 3/2 (γi0 t)1/2 ≤ (γi0 t)1/2 ≤ c4 R−1/2 .

R

R

≤

If ri ≥ R/(4γi0 t), then (4.19) is bounded by

√

√

c6 xi

c5 w

E |N − w|

≤p

≤ √

≤ c7 R−1/4 ,

√

0

ri

R

R/γi t

654

(4.18)

the last by (4.20). Therefore under (4.20)

I2 ≤ kf k∞ c8 ((γi0 t)−1 + 1)R−1/4 .

Now use the above bounds in (4.17) to see that for R ≥ 8 max(1, γi0 t),

sup

√

xi ≤ R/4

|h(b

yi ; t, xi )| ≤ c9 ((γi0 t)−1 + 1)[kdn (·; ybi )kR + kf k∞ R−1/4 ].

(4.21)

If b0i = 0, a slightly simpler argument starting with (4.6) in [BP03] will lead to the same

bound.

Now choose a compact set K in Mν (S), T > 1 and ε > 0. Assume R is large enough

so that R ≥ 8 max(1, γi0 t),

√

(4.22)

if x ∈ K, then xi ≤ R/4,

and kf k∞ R−1/4 < ε. Let ηn be such that Kη is a compact set satisfying the conclusion

of Lemma 4.1. Let π

bi (y) = (y(j), j ∈ S − {i}) be the projection of y ∈ Mν (S) onto

Mνi (S − {i}), νi = ν|S−{i} , and let

b η = {y ∈ Mν (S) : π

K

bi (y) ∈ π

bi (Kη ), y(i) ≤ R}.

b η is compact, and so by Lemma 4.5

Then it is easy to use Lemma 3.1 to check that K

lim sup |fn (y) − f (y)| = 0.

bη

y∈K

n→∞

This implies

lim

n→∞

sup kdn (·; ybi )kR = 0.

b

yi ∈b

πi (Kη )

Choose N such that n ≥ N implies

sup kdn (·; ybi )kR < ε.

b

yi ∈b

πi (Kη )

Use this with (4.14), (4.21), and (4.22) in (4.15) and conclude that for n ≥ N and t ∈ [0, T ]

sup |xi (Pt fn )ii (x) − xi (Pt f )ii (x)|

x∈K

α

≤ cα |f |α,i (γi0 t) 2 −1 sup P0x (b

πi (Xt ) ∈

/π

bi (Kη ))

x∈K

≤

≤

+ c9 ((γi0 t)−1 + 1)[ε + kf k∞ R−1/4 ]

α

cα |f |α,i (γi0 t) 2 −1 ε + c9 ((γi0 t)−1 + 1)2ε

c10 (|f |α,i + 1)((γi0 t)−1 + 1)ε.

655

(4.23)

Use the above for t ∈ [T −1 , T ] and (4.14) for t ∈ [T −1 , T ]c to see that for x ∈ K and n ∈ N

|xi (Rλ fn )ii (x)−xi (Rλ f )ii (x)|

α

≤ cα |f |α,i (γi0 t) 2 −1

hZ

1/T

α

t 2 −1 dt +

0

+ c11 (|f |α,i + 1)((γi0 )−1 T + 1)ε

≤ cα |f |α,i (γi0 t)

α

2 −1

Z

Z

(4.24)

∞

T

T

e−λt dt

i

e−λt dt

1/T

[T −α/2 + e−λT λ−1 ]

+ c12 (|f |α,i + 1)((γi0 )−1 T + 1)ελ−1 .

As T > 1 and ε > 0 are arbitrary, this gives

lim sup |xi (Rλ fn )ii (x) − xi (Rλ f )ii (x)| = 0.

n→∞ x∈K

The differentiation through the time integral in (4.24) does require x i > 0 as in the proof

of Lemma 4.4, but that result shows the left-hand side is 0 if xi = 0.

Corollary 4.9. If f ∈ Cα , then for any i ∈ S and λ > 0, Rλ f, (Rλ f )i , and xi (Rλ f )ii are

all continuous bounded functions on Mν (S).

Proof. Fix i and consider n large enough so that i ∈ Sn . Recall Rλn is the resolvent of

(Xi )i∈Sn and fen (x) = f ◦Πn (x) on RS+n . Then Rλ fn (x) = Rλn fen (πn x) and so by Proposition

5.3 of [BP03] (Rλ fn )i (x) = (Rλn )i fen (πn x) and xi (Rλ fn )ii (x) = xi (Rλn fen )ii (πn (x)) are

bounded continuous functions on Mν (S). The uniform convergence in Proposition 4.8 and

the bound in Lemma 4.4 show that (Rλ f )i and xi (Rλ f )ii are in Cb (Mν (S)). (4.13) already

gave the result for Rλ f .

Corollary 4.10. There is a κ4.10 > 0 depending only on α such that for all f ∈ Cα , λ > 0,

and i, j ∈ S,

1−α

0 −1 0

|(Rλ f )i |α,j + |xi (Rλ f )ii |α,j ≤ κ4.10 |f |α,i

|f |α

(γi /γj0 )(1−α)α/2 .

α,j (γi )

Proof. The proof is almost the same as that of Proposition 5.3 of [BP03] for the finitedimensional case – again the constants given there are independent of dimension; cf. Remark 4.3. The only change is that once the bounds on the increments of x i (Rλ f )ii are

established for xi > 0, they follow for xi = 0 by the continuity established in Corollary

4.9; this is in place of the use of Lemma 2.2 in [BP03].

5. Local uniqueness.

We make the following assumption.

656

Assumption 5.1. Assume γi : Mν (S) → (0, ∞), bi : Mν (S) → R are continuous and

(γi0 )i∈S , (b0i )i∈S satisfy (4.1) and (4.2). Assume also

(a)

X · |γi (x) − γ 0 | |bi (x) − b0 | ¸

i

i

sup

+

≡ ρ < ∞.

(5.1)

0

0

γ

γ

x∈Mν (S)

i

i

i∈S

(b) For all x ∈ Mν (S)

X · |γi (x′ ) − γ 0 | |bi (x′ ) − b0 | ¸ X · |γi (x) − γ 0 | |bi (x) − b0 | ¸

i

i

i

i

+

=

+

. (5.2)

lim

x′ →x

γi0

γi0

γi0

γi0

i∈S

i∈S

(c) For all j ∈ S, 0 < h, x ∈ Mν (S)

X · |γi (x + hej ) − γi (x)| |bi (x + hej ) − bi (x)| ¸

−α/2 0 −α/2

(γj )

. (5.3)

+

≤ κ5.1 hα xj

γi0

γi0

i∈S

The following result uses only Assumption 5.1(a)-(b).

Lemma 5.2. For any λ > 0, (L − L0 )Rλ : Cα → Cb (Mν (S)), and there is a κ5.2 = κ5.2 (α)

such that

k(L − L0 )Rλ f k∞ ≤ κ5.2 ρ|f |α λ−α/2 ,

f ∈ Cα , λ > 0.

Note that L − L0 can be well-defined for a larger class of functions than the domain of L

and L0 thanks to Assumption 5.1 and the results of Section 4. In particular, this lemma

shows that (L − L0 )Rλ f is well-defined for f ∈ Cα .

Proof. Lemma 4.4 shows that for f ∈ Cα

X

|γi (x) − γi0 | |xi (Rλ f )ii (x)| + |bi (x) − b0i | |(Rλ f )i (x)|

i∈S

≤

X h |γi (x) − γ 0 |

i

i∈S

γi0

+

|bi (x) − b0i | i

cα (γi0 )α/2 |f |α,i λ−α/2

γi0

≤ cα ρ|f |α λ−α/2 ,

by Assumption 5.1(a). This gives the required bound on k(L − L0 )Rλ f k∞ .

Note that Lemma 4.4 implies

|γi (x) − γi0 | |x(i)(Rλ f )ii (x)| + |bi (x) − b0i | |(Rλ f )i (x)|

h |γ (x) − γ 0 | |b (x) − b0 | i

i

i

i

i

≤ cα

+

|f |α λ−α/2 .

0

γi

γi0

Let xn → x in Mν (S). Assumptions 5.1(a),(b) imply that if

fn (i) =

|γi (xn ) − γi0 | |bi (xn ) − b0i |

+

,

γi0

γi0

657

(5.4)

then {fn } is uniformly integrable with respect to counting measure on S and hence by

(5.4) so is

gn (i) = |γi (xn ) − γi0 | |xn (i)(Rλ f )ii (xn )| + |bi (xn ) − b0i | |(Rλ f )i (xn )|.

This allows us to take the limit as n → ∞ through the summation and conclude by the

continuity of

(γi (x) − γi0 ) (x(i)(Rλ f )ii (x)) + (bi (x) − b0i ) ((Rλ f )i (x))

(see Corollary 4.9) that

lim (L − L0 )Rλ f (xn ) = (L − L0 )Rλ f (x).

n→∞

This proves (L − L0 )Rλ f is continuous.

We let Bλ denote the operator

Bλ = (L − L0 )Rλ =

X£

¤

(γi − γi0 )x(i)(Rλ )ii + (bi − b0i )(Rλ )i .

i

Proposition 5.3. Assume Assumption 5.1(a)-(c). For any λ > 0, Bλ : Cα → Cα is a

bounded operator. Moreover there exist λ0 = λ0 (α) and ρ0 = ρ0 (α) > 0 such that if

λ ≥ λ0 and ρ ≤ ρ0 , then

kBλ kCα < 1/2.

Proof. Use Lemma 4.4 and Corollary 4.10 to see that for f ∈ Cα , j ∈ S, and h > 0,

|Bλ f (x + hej ) − Bλ f (x)|

X

≤

|γi (x + hej ) − γi (x)| |(x + hej )(i)(Rλ f )ii (x + hej )|

i

+

X

i

≤

X

i

+ |bi (x + hej ) − bi (x)| |(Rλ f )i (x + hej )|

|γi (x) − γi0 | |(x + hej )(i)(Rλ f )ii (x + hej ) − x(i)(Rλ f )ii (x)|

+ |bi (x) − b0i | |(Rλ f )i (x + hej ) − (Rλ f )i (x)|

[|γi (x + hej ) − γi (x)| + |bi (x + hej ) − bi (x)|](γi0 )−1 λ−α/2 κ4.4 sup |f |α,i′ (γi0′ )α/2

+

i′

X

i

−α/2

1−α

0

0 (1−α)α/2 α

[|γi (x) − γi0 | + |bi (x) − b0i |](γi0 )−1 κ4.10 |f |α,i

|f |α

h xj

α,j (γi /γj )

The first summation is bounded by (use Assumption 5.1(c))

−α/2

κ4.4 λ−α/2 |f |α (γj0 )−α/2 κ5.1 hα xj

658

.

.

The second summation is bounded by (use Assumption 5.1(a))

κ4.10 ρ sup(|f |α,i (γi0 )α/2 )1−α sup(|f |α,j ′ (γj0′ )α/2 )α (γj0 )−α/2 hα x(j)−α/2

i

j′

−α/2

≤ κ4.10 ρ|f |α (γj0 )−α/2 hα xj

.

We may therefore conclude

|Bλ f |α,j (γj0 )α/2 ≤ [κ4.4 κ5.1 λ−α/2 + κ4.10 ρ] |f |α

and so

|Bλ f |α ≤ [κ4.4 κ5.1 λ−α/2 + κ4.10 ρ]|f |α .

Combine this with Lemma 5.2 to see that

kBλ f kα ≤ [κ4.4 κ5.1 λ−α/2 + (κ4.10 + κ5.2 λ−α/2 )ρ] |f |α .

This, together with Lemma 5.2, shows Bλ : Cα → Cα is a bounded operator with

kBλ k ≤ 1/2 for ρ ≤ ρ0 (α), λ ≥ λ0 (α).

Let Pµ be a solution of MP(L, µ) for some law µ on Mν (S) and for λ > 0, let

R∞

Sλ f = E µ ( 0 e−λt f (Xt )dt).

The following result uses only (2.5), (2.7), and Assumption 5.1(a) (it only requires

the bound in Lemma 5.2 and so does not require Assumption 5.2(b)). Recall the constant

bp

κ2.3a in (2.5). We use −→ to denote bounded pointwise convergence.

Proposition 5.4. Assume (2.5), (2.7). If f ∈ Cα , then

Z

Sλ f = Rλ f (x)µ(dx) + Sλ Bλ f,

Proof. Assume first that

Z

|x|ν µ(dx) < ∞.

λ > κ2.3a .

(5.5)

Let f (x) = f0 (πn x) for some f0 : RS+n → R, f ∈ Cα . Write xn = πn (x) for x ∈ Mν (S) and

define

Z ∞

Z ∞

−λt

gδ (y) =

e Pt f (y)dt =

e−λt Ptn f0 (y n )dt ≡ geδ (y n ).

δ

δ

Ptn

Here

is the semigroup of {Xi : i ∈ Sn } under P0x . The finite dimensional analysis in

the proof of Lemma 6.1 of [BP03] shows that geδ ∈ Cb2 (RS+n ).

Use (2.4), (5.5), (2.5), a stopping time argument, and a Gronwall argument (cf. the

proof of Theorem 2