Directory UMM :Journals:Journal_of_mathematics:GMJ:
Georgian Mathematical Journal
Volume 8 (2001), Number 2, 389–400
ON STOCHASTIC DIFFERENTIAL EQUATIONS IN A
CONFIGURATION SPACE
A. SKOROKHOD
Abstract. Infinite systems of stochastic differential equations for randomly
perturbed particle systems with pairwise interaction are considered. It is
proved that under some reasonable assumption on the potential function
there exists a local weak solution to the system and it is weakly locally
unique for a wide class of initial conditions.
2000 Mathematics Subject Classification: 60H10, 60H30.
Key words and phrases: Configuration space, stochastic differential equation, local weak solution.
1. Introduction
We consider a sequence of Rd -valued stochastic processes
xk (t),
k = 1, 2, . . . ,
satisfying the system of stochastic differential equations of the form
dxk (t) =
X
a(xk (t) − xi (t))dt + σdwk (t),
k = 1, 2, . . . ,
(1)
i6=k
where a(x) = −Ux (x), and U : Rd → R is a smooth function for |x| > 0, and
σ > 0 is a constant, wk (t), k = 1, 2, . . . , is a sequence of independent Wiener
processes in Rd . System (1) describes the evolution of systems of pairwise
interacting particles with the pairwise potential U (x) which is perturbed by
Wiener noises. The problem is to find conditions under which the system has
a solution and this solution is unique.
Unperturbed systems were considered by many authors. We notice the recent
articles of S. Albeverio, Yu. G. Kondratiev, and M. R¨ockner [1], [2] where a
new powerful method for the investigation of unperturbed systems is proposed.
Finite-dimensional perturbed systems were considered in my book [3] and my
article [4]. The first general theorem on the existence and uniqueness of the
solutions to infinite dimensional stochastic differential equations were obtained
by Yu. L. Daletskii in [5]; he considered equations with smooth coefficients in
a Hilbert space. The existence and uniqueness of the solution to system (1) for
locally bounded smooth potentials and d ≤ 2 was proved by J. Fritz in [6]. The
main result of this article was published in [7] without a complete proof.
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
390
A. SKOROKHOD
2. The Space Γ
It is convenient to consider system (1) in the configuration space Γ which is
the set of locally finite counting measures γ on the Borel σ-algebra B(Rd ) of
the space Rd . So a measure γ ∈ Γ satisfies the condition: the support Sγ of the
measure γ is a sequence of different points {xk , k ∈ N } of Rd for which
|xk | → ∞,
γ(A) =
X
1A (xk ).
k
The topology in Γ is generated by the weak convergence of measures: γn → γ0
if
Z
Z
φ(x)γn (dx) → φ(x)γ0 (dx)
for φ ∈ Cf where Cf is the set of continuous functions φ : Rd → R with bounded
supports.
We use the notation
Z
hφ, γi =
and
hΦ, γ × γi =
Z
φ ∈ Cf ,
φ(x)γ(dx),
′
′
Φ(x, x )γ(dx)γ(dx ) −
Z
Φ(x, x)γ(dx),
where Φ : (Rd )2 → R is a continuous function with a bounded support.
We rewrite system (1) for Γ-valued function γt for which
hφ, γi =
X
φ(xk (t)),
φ ∈ Cf .
k
Using the Itˆo’s formula, and considering the function a as a function of two
variables a(x − x′ ), we obtain the relation
σ2
h∆φ, γt i
2
X
(2)
+
σ(φ′ (xk (t)), dwk (t)), φ ∈ Cf ,
dhφ, γt i = h(φ′ , a), γt × γt idt +
(2)
k
(2)
where ∆φ(x) = Trφ′′ (x), and Cf is the set of φ ∈ Cf for which φ′ (x) and φ′′ (x)
are continuous bounded functions.
A weak solution to equation (1). A Γ-valued stochastic process γt (ω) is
(2)
called a weak solution to system (2) if, for all φ ∈ Cf , the stochastic process
µφ (ω, t) = hφ, γt i −
Zt
[h(φ′ , a), γs × γs i +
0
σ2
h∆φ, γs i]ds
2
(3)
is a martingale with respect to the filtration (Ft )t≥0 , where Ft = σ(γs , s ≤ t),
and the square characteristic of the martingale is
hµφ , µφ it = σ
2
Zt
0
h(φ′ , φ′ ), γs ids.
(4)
ON SDES IN A CONFIGURATION SPACE
391
If γt (ω) is a weak solution to system (2) and
hφ, γt (ω)i =
X
φ(xk (t))
for all φ ∈ Cf , then the sequence {xk (t), k ∈ N } is a weak solution to system
(1).
Weak uniqueness. Let γ0 ∈ Γ. System (2) has a unique weak solution with
the initial value γ0 if for any pair of weak solutions to system (2) γt1 (ω) and
γt2 (ω) satisfying the condition γ01 = γ02 = γ0 , the following relations are fulfilled:
1
1
1
1
2
2
2
2
EΦ(ξ11
, . . . , ξ1m
, . . . , ξl1
, . . . , ξlm
) = EΦ(ξ11
, . . . , ξ1m
, . . . , ξl1
, . . . , ξlm
), (5)
where Φ(y11 , . . . , ylm ) is a continuous bounded function on Rlm and,
ξijk = hφi , γtkj (ω)i, k = 1, 2, i = 1, . . . , l,
φ ∈ Cf , t1 , . . . , tm ∈ R+ .
j = 1, . . . , m,
This means that the distribution of the stochastic process γtk does not depend
on k, i.e., the distribution of the weak solution to system (2) is unique if it
exists.
Local weak solutions. Let γt (ω) be a continuous Γ-valued stochastic process, and (Ft )t≥0 be the filtration generated by it. γt (ω). is called a local weak
solution to system (2) if there exists a stopping time τ with respect to the
filtration (Ft )t≥0 for which P {τ > 0} = 1 and the stochastic process µφ (ω, t)
which is determined by relation (3) is a martingale for t < τ with the square
characteristic given by equality (4).
Local weak uniqueness. Let γ0 ∈ Γ. System (2) has a locally unique
weak solution with the initial value γ0 if, for any pair of locally weak solutions
to system (2) γt1 and γt2 satisfying the condition γ01 = γ02 = γ0 , relation (5) is
fulfilled for
ξijk = hφi , γtkj (ω)iI{tj 0 there exists a continuous
decreasing function Φ(t) : [0, ∞) → R+ with Φ(+∞) = 0 such that
ZZ
Φ(|x|)Φ(|x′ |)λ(|x − x′ |)1{x6=x′ } γ(dx)γ(dx′ ) < ∞.
(6)
Denote
Φλ (x, x′ ) = Φ(|x|)Φ(|x′ |)λ(|x − x′ |).
(7)
For any compact set K from Γ and any fanction λ satisfying the conditions
mentioned before there exists a function of the form given by relation (7) for
which
suphΦλ , γ × γi < ∞.
γ∈K
392
A. SKOROKHOD
Note that the set
{γ : hΦλ , γ × γi ≤ c}
is a compact in Γ for any Φλ of form (7) and c > 0. Denote by ΓΦ,λ the set of
those γ ∈ Γ for which relation (6) is fulfilled. Set
dΦ,λ (γ1 , γ2 ) = sup{|hφΦ, γ1 i−hφΦ, γ2 i : φ ∈ Lip1 |}+|hΦλ , γ1 ×γ1 i−hΦλ , γ2 ×γ2 i|,
where
1
(
Lip = φ ∈ Cf : sup |φ(x)| ≤ 1,
x
|φ(x) − φ(x′ )|
sup
≤1 .
|x − x′ |
x,x′
)
ΓΦ,λ with the distance dΦ,λ is a separable locally compact space.
3. An Extension of Girsanov’s Formula
Assume that the potential function U (x) satisfies the condition
(PC) U (x) = u(|x|) where the function u : (0, ∞) → R is continuous, it has
continuous derivatives u′ , u′′ , there exists a constant r > 0 for which u(t) = 0
for t > r, and
Z
td−1 |u(t)|dt < ∞.
Free particle processes. Let a measure γ0 satisfy the condition (IC)
hφδ , γ0 i < ∞,
φδ (x) = exp {−δ|x|2 }.
Introduce Γ-valued stochastic processes by the relation
hφ, γt∗ (γ0 , ω)i =
X
φ(x0k + σwk (t)),
k
where
X
φ(x0k ) = hφ, γ0 i.
k
It is easy to check that γt∗ (γ0 , ω) is a continuous Γ-valued stochastic process if
γ0 ∈ Γ0 , where Γ0 is the set of finite measures from Γ. There exist functions
Φ, λ for which P {γt∗ (γ0 , ω) ∈ ΓΦ,λ } = 1 for all t > 0. The stochastic process
γt∗ (γ0 , ω) is continuous in the space ΓΦ,λ , and for any t0 > 0 the function
EF (γ ∗ (γ0 , ω))
is a continuous function in γ0 ∈ ΓΦ,λ if F is a bounded continuous function
on C[0,t0 ] (ΓΦ,λ ) which is the space of continuous ΓΦ,λ -valued functions on the
interval [0, t0 ].
Girsanov’s formula for finite systems. Let γ0n be a sequence of finite
measures from Γ satisfying the condition:
γ0n → γ0 , γ0n ≤ γ0n+1 .
It was proved in [4] that under the condition (PC), for any n, there exists a
unique strong solution to system (2) with the initial value γ0n . Denote it by
γt (γ0n , ω).
ON SDES IN A CONFIGURATION SPACE
393
Lemma 1. Set a(x, x′ ) = a(x − x′ ),
Gn1 (t)
=σ
t
X Z
−1
(ha(xi + σwi (s), .), γs∗ (γ0n , ω)i, dwi (s)),
xi ∈Sγ n 0
0
Gn2 (t)
=σ
−2
Zt Z
|ha(x, .), γs∗ (γ0n , ω)i|2 γs∗ (γ0n , ω, dx)ds,
0
n
o
ρn (t) = exp Gn1 (t) − 21 Gn2 (t) .
Then
EΦ(hφ1 , γt1 (γ0n , ω)i, . . . , hφk , γtk (γ0n , ωi)
= Eρn (t)Φ(hφ1 , γt∗1 (γ0n , ω)i, . . . , hφk , γt∗k (γ0n , ω)i)
for all k = 1, 2, . . . , bounded continuous functions Φ : Rk → R and φ1 , . . . , φk ∈
Cf , t1 , . . . , tk ∈ [0, t].
The proof of the lemma can be obtained using the approximation of the
function a(x, x′ ) by smooth functions since for smooth a(x, x′ ) the proof is a
consequence of Girsanov’s formula [8].
Introduce the stochastic processes
wkc (t)
=
Zt
1{|xk +wk (s)|≤c} dwk (s),
xk ∈ Sγ0n ,
0
where c > 0 is a constant. Let Ftn,c be the σ -algebra generated by
{wkc (s), s ≤ t, xk ∈ Sγ0n }.
Lemma 2.
E(ρn /Ftn,c ) = ρn (c, t),
where
n
o
ρn (c, t) = exp Gn1 (c, t) − 21 Gn2 (c, t) ,
and
Gn1 (c, t)
=σ
t
X Z
−1
(E(ha(xi + σwi (s), .), γs∗ (γ0n , ω)i/Fsn,c ), dwic (s)),
xi ∈Sγ n 0
0
Gn2 (c, t)
=σ
−2
Zt Z
|E(ha(x, .), γs∗ (γ0n , ω)i/Fsn,c )|2 1{|x|≤c} γs∗ (γ0n , ω, dx)ds.
0
The proof rests on the statement below.
Statement. Let Ft , t ∈ R+ be a continuous filtration, and its subfiltration
Ft∗ , t ∈ R+ , satisfy the condition
∗
(RE)E(ξ/Ft ) is a Ft∗ -measurable random variable if ξ is a bounded F∞
measurable random variable.
394
A. SKOROKHOD
Let µt be an Ft -martingale with the square characteristic hµit for which the
stochastic process
o
n
ρ(t) = exp µt − 12 hµit
is a martingale. Then
n
o
E(ρ(t)/Ft∗ ) = exp µ∗t − 21 hµ∗ it ,
where µ∗t = E(µt /Ft∗ ) is a martingale, and hµ∗ it is its square characteristic.
Proof. Set µ
˜t = µt − µ∗t . Condition (RE) implies the relation
E
à Zt
∗
g(s)d˜
µs /F∞
0
!
=0
for all Ft -adapted functions g(t) for which
¯
¯ Zt
¯
¯
¯
¯
µs ¯¯ < ∞.
E ¯¯ g(s)d˜
0
Set
n
o
ρ∗ (t) = exp µ∗t − 12 hµ∗ it ,
n
o
ρ˜(t) = exp µ
˜t − 12 h˜
µit ,
where h˜
µit is the square characteristic of the martingale µ
˜t . The proof follows
from the relations ρ(t) = ρ∗ (t)˜
ρ(t) and
E(˜
ρ(t) −
∗
1/F∞
)
=E
à Zt
∗
ρ˜(s)d˜
µ(s)/F∞
0
!
= 0.
Remark 1. Assume that φi (x) = 0 for |x| ≥ c, i = 1, 2, . . . , k, φ1 , . . . , φk ∈
Cf , t1 , . . . , tk ∈ [0, t]. Then for Φ satisfying the conditions of Lemma 1 we have
the relation
EΦ(hφ1 , γt1 (γ0n , ω)i, . . . , hφk , γtk (γ0n , ω)i)
= Eρn (c, t)Φ(hφ1 , γt∗1 (γ0n , ω)i, . . . , hφk , γt∗k (γ0n , ω)i).
Remark 2. Denote by Ftc the σ-algebra generated by {wkc (s), s ≤ t, xk ∈ Sγ0 }.
Then there exist the limits in probability
lim Gn1 (c, t) = G1 (c, t),
n→∞
lim Gn2 (c, t) = G2 (c, t),
n→∞
1
lim ρn (t) = ρ(c, t) = exp G1 (c, t) − G2 (c, t) ,
n→∞
2
½
¾
where
G1 (c, t) = σ
t
X Z
−1
G2 (c, t) = σ −2
xi ∈Sγ0 0
Zt Z
(E(ha(xi + σwi (s), .), γs∗ (γ0 , ω)i/Fsc ), dwic (s))),
|E(ha(x, .), γs∗ (γ0 , ω)i/Fsc )|2 1{|x|≤c} γs∗ (γ0 , ω, dx)ds.
0
ON SDES IN A CONFIGURATION SPACE
395
Let w(t) be a standard Wiener process . Introduce the functions
Qc (s, x, B) = P {x + σw(s) ∈ B, inf |x + σw(u)| > c},
u≤s
x ∈ Vc , B ∈ B(Vc ), Vc = {x ∈ Rd : |x| > c},
Q∗c (s, x, B) = lim
λ↓1
Qc (s, x, B)
, |x| = c, B ∈ B(Vc ).
Qc (s, λx, Vc )
Set
θi (c, s) = inf ∆i (c, s), ζi (c, s) = sup ∆i (c, s),
where
∆i (c, s) = {u ≤ s : |x∗i (u)| ≤ c}, x∗i (u) = xi + wi (u).
Then the following statement holds.
Lemma 3.
E(a(x, x∗i (s))1{|x∗i (s)|≤c} /Fsc ) = a(x, x∗i (s))1{|x∗i (s)|≤c}
+1{θi (c,s) 0.
Proof. Choose ak satisfying the inequality
P (G2 (ck , k) > (k 2 ak )−1 ) < k −2 .
Then for any t > 0 we have the relation
X
P (ak G2 (ck , t) > k −2 ) < t +
k
X
k −2 < ∞.
k≥t
This completes the proof.
Corollary 3. Let a sequence {ak } satisfy the statement of Lemma 5. Set
G(t) =
X
ak G2 (ck , t).
k
With probability 1 G(t) is an increasing continuous function satisfying the relations G(0) = 0, limt→∞ G(t) = ∞. Set
τ ∗ = inf{t : G(t) > b},
where b is a positive number. Then
Eρ(ck , τ ∗ ) = 1, E(ρ(ck , τ ∗ ))2 ≤ exp {ba−1
k }.
(9)
398
A. SKOROKHOD
4. A Theorem on Existence of a Local Weak Solution
Theorem 1. Let conditions (PC) and (IC) be fulfillrd,and let τ ∗ be a stopping time introduced by relation (9). Then the following statements hold:
(i) there exists a probability measure P ∗ on (Ω, F) for which
P ∗ (A) = lim E1A ρ(c, τ ∗), A ∈
c→∞
_
Fτc∗k , P ∗ (A) = P (A), A ∈ F ∗ ,
k
∗
where the σ-algebra F is generated by the processes
{x∗k (t ∨ τ ∗ ) − x∗k (t), t ≥ 0, k = 1, 2, . . . }.
(ii) the stochastic processes given by the formula
σwk∗ (t)
=
x∗k (t)
−
t∧τ
Z ∗
0
X
a(x∗k (s), x∗i (s))ds, k = 1, 2, . . . ,
i6=k
are independent Wiener processes with respect to the filtration (Ft )(t≥0) on the
probability space (Ω, F, P ∗ ).
Proof. Since τ ∗ is a stopping time, for A ∈ Fτc∗k we can write, using Remark 5,
the relations
E1A ρ(c, τ ∗ ) = E1A ρ(ck , τ ∗ )
if c > ck . This implies the existence of limits
lim 1A×B ρ(c, τ ∗ ), A ∈
c→∞
_
Fτc∗k , B ∈ F ∗ .
k
It follows from the last formula that there exists a limit
lim Φ(ξ1 , . . . , ξm )ρ(c, τ ∗ ),
(10)
c→∞
where Φ ∈ C(Rm ) and
ξi = hfi , γt∗i (γ0 , ω)i, i = 1, . . . , m,
where fi ∈ C(Rd ), fi (x) = 0 for |x| large enough.
Now we prove that there exists a probability measure P ∗ on (Ω, F) for which
a limit in the formula (10) is represented in the form E ∗ Φ(ξ1 , . . . , ξm ), where
E ∗ is the expectation with respect to the probability P ∗ . Set
µ
fn (x, x′ ) = 1 −
|x|
cn
¶µ
1−
|x′ |
cn
¶µ
1+
1
∨ 0.
|x − x′ |
¶
Using Corollary 3 we can write the inequality
Ehfn , γt∗ (γ0 , ω) × γt∗ (γ0 , ω)iρ(c, τ ∗ )
1
≤ (Ehfn , γt∗ (γ0 , ω) × γt∗ (γ0 , ω)i2 ) 2 exp { 12 ba−1
n }.
Denote
An (t) = Ehfn , γt∗ (γ0 , ω) × γt∗ (γ0 , ω)i2 .
(11)
ON SDES IN A CONFIGURATION SPACE
399
The functions An (t) are non-negative and continuous. There exists a sequence
of positive numbers {bn } for which
X
n
bn (1 + An (t)) exp { 12 ba−1
n } < ∞, t > 0.
(12)
Formulas (11) and (12) imply the relation
lim sup E
c→∞
X
bn hfn , γt∗ (γ0 , ω) × γt∗ (γ0 , ω)iρ(c, τ ∗ ) < ∞.
n
This formula implies that the set of measures {Pc∗ , c > 0} for which
dPc∗
(ω) = ρ(c, τ ∗ )
dP
is weakly compact, so there exists a sequence c′k , c′k → ∞ for which Pc∗′ converges
k
weakly to a measure P ∗ . The statement (i) is proved.
Introduce the stopping times
½
τn∗ = inf t :
X
¾
ak G2 (ck , t) ≥ b ,
k≤n
and set
Pn∗ (A) = E1A ρ(cn , τn∗ ), A ∈ F.
Let Φ(ξ1 , . . . , ξm ) be the same as before. Then
lim En∗ Φ(ξ1 , . . . , ξm ) = E ∗ Φ(ξ1 , . . . , ξm );
n→∞
here En∗ is the expectation with respect to probability Pn∗ .This formula is a
consequence of the relations
τn∗ > τ ∗ , τn∗ → τ ∗
in probability and
E(ρ(cn , τn∗ )/Fτc∗n ) = ρ(cn , τ ∗).
Let wi∗ (c, t) be given by formula (8). Then for fixed n the sequence
{wi∗ (cn , t), i = 1, 2, . . . }
represents independent Wiener processes on the probability space {Ω, F, Pn∗ }.
Introduce stopping times
ζin = inf{t : |xi (t)| > cn − r}.
Since ac (x, ω) = 0 for |x| < cn − r,we have
wi∗ (cn , t ∨ ζin ) = wi∗ (t ∨ ζin ).
Using Remark 3 we can prove that the relation
lim lim sup Pn∗ {sup |x∗i (s)| > c} = 0
c→∞
n→∞
s≤t
(13)
400
A. SKOROKHOD
is fulfilled for any i and t > 0. Let Φ, h1 , . . . , hm be the same as before . Set
ξi∗
=
Zt
hi (s)dwi∗ (s),
ξin
=
Zt
hi (s)dwi∗ (cn , s), i = 1, . . . .
0
0
Note that
n
En∗ Φ(ξ1n , . . . , ξm
) = EΦ
à Zt
h1 (s)dw1 (s), . . . ,
0
Zt
!
hm (s)dwm (s)
0
(14)
¯
for all n. Denote the expression in the right hand side of formula (14) by Φ.
Then
∗
¯ + En∗ O
En∗ Φ(ξ1∗ , . . . , ξm
)=Φ
µX
¶
1{sups≤t |x∗i (s)|>cn −r} .
i≤m
(15)
Formulas (13) and (15) imply the relation
∗
∗
¯
E ∗ Φ(ξ1∗ , . . . , ξm
) = lim En∗ Φ(ξ1∗ , . . . , ξm
) = Φ.
n→∞
The statement (ii) is proved.
References
¨ ckner, Analysis and geometry on config1. S. Albeverio, Yu. G. Kondratiev, and Ro
uration spaces. J. Funct. Anal. 154(1998), 444–500.
¨ ckner, Analysis and geometry on config2. S. Albeverio, Yu. G. Kondratiev, and Ro
uration spaces: the Gibbsian case. J. Funct. Anal. 157(1998), 242–291.
3. A. V. Skorokhod, Stochastic equations for a complex system. D. Reidel Publishing Co.,
Dordrecht-Boston, Mass., 1988.
4. A. V. Skorokhod, On the regularity of many-particle dynamical systems perturbed by
white noise. J. Appl. Math. Stoch. Anal. 9(1996), 427–437.
5. Yu. L. Daletskii, Infinite-dimensional elliptic operators and related parabolic equations.
(Russian) Uspekhi. Mat. Nauk 22(1967), 3–54.
6. J. Fritz, Gradient dynamics of infinite point systems. Ann. Probab. 15(1987), 478–514.
7. A. V. Skorokhod, On infinite systems of stochastic differential equations. Methods
Funct. Anal. Topology (1999), No. 4, 54–61.
8. I. V. Girsanov, On a transformation of some class of stochastic processes by absolute
continuous changing of measures. Theory Probab. Appl. 5(1960), 314–330.
(Received 3.07.2000)
Author’s addresses:
Institute of Mathematics
National Academy of Sciences of Ukraine
3, Tereshchenkivska St., Kyiv 01601, Ukraine
Department of Statistics and Probability
Michigan State University
A 415 Wells Hall, East Lansing, MI 48824, U.S.A.
E-mail: skorokhod@stt.msu.edu
Volume 8 (2001), Number 2, 389–400
ON STOCHASTIC DIFFERENTIAL EQUATIONS IN A
CONFIGURATION SPACE
A. SKOROKHOD
Abstract. Infinite systems of stochastic differential equations for randomly
perturbed particle systems with pairwise interaction are considered. It is
proved that under some reasonable assumption on the potential function
there exists a local weak solution to the system and it is weakly locally
unique for a wide class of initial conditions.
2000 Mathematics Subject Classification: 60H10, 60H30.
Key words and phrases: Configuration space, stochastic differential equation, local weak solution.
1. Introduction
We consider a sequence of Rd -valued stochastic processes
xk (t),
k = 1, 2, . . . ,
satisfying the system of stochastic differential equations of the form
dxk (t) =
X
a(xk (t) − xi (t))dt + σdwk (t),
k = 1, 2, . . . ,
(1)
i6=k
where a(x) = −Ux (x), and U : Rd → R is a smooth function for |x| > 0, and
σ > 0 is a constant, wk (t), k = 1, 2, . . . , is a sequence of independent Wiener
processes in Rd . System (1) describes the evolution of systems of pairwise
interacting particles with the pairwise potential U (x) which is perturbed by
Wiener noises. The problem is to find conditions under which the system has
a solution and this solution is unique.
Unperturbed systems were considered by many authors. We notice the recent
articles of S. Albeverio, Yu. G. Kondratiev, and M. R¨ockner [1], [2] where a
new powerful method for the investigation of unperturbed systems is proposed.
Finite-dimensional perturbed systems were considered in my book [3] and my
article [4]. The first general theorem on the existence and uniqueness of the
solutions to infinite dimensional stochastic differential equations were obtained
by Yu. L. Daletskii in [5]; he considered equations with smooth coefficients in
a Hilbert space. The existence and uniqueness of the solution to system (1) for
locally bounded smooth potentials and d ≤ 2 was proved by J. Fritz in [6]. The
main result of this article was published in [7] without a complete proof.
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
390
A. SKOROKHOD
2. The Space Γ
It is convenient to consider system (1) in the configuration space Γ which is
the set of locally finite counting measures γ on the Borel σ-algebra B(Rd ) of
the space Rd . So a measure γ ∈ Γ satisfies the condition: the support Sγ of the
measure γ is a sequence of different points {xk , k ∈ N } of Rd for which
|xk | → ∞,
γ(A) =
X
1A (xk ).
k
The topology in Γ is generated by the weak convergence of measures: γn → γ0
if
Z
Z
φ(x)γn (dx) → φ(x)γ0 (dx)
for φ ∈ Cf where Cf is the set of continuous functions φ : Rd → R with bounded
supports.
We use the notation
Z
hφ, γi =
and
hΦ, γ × γi =
Z
φ ∈ Cf ,
φ(x)γ(dx),
′
′
Φ(x, x )γ(dx)γ(dx ) −
Z
Φ(x, x)γ(dx),
where Φ : (Rd )2 → R is a continuous function with a bounded support.
We rewrite system (1) for Γ-valued function γt for which
hφ, γi =
X
φ(xk (t)),
φ ∈ Cf .
k
Using the Itˆo’s formula, and considering the function a as a function of two
variables a(x − x′ ), we obtain the relation
σ2
h∆φ, γt i
2
X
(2)
+
σ(φ′ (xk (t)), dwk (t)), φ ∈ Cf ,
dhφ, γt i = h(φ′ , a), γt × γt idt +
(2)
k
(2)
where ∆φ(x) = Trφ′′ (x), and Cf is the set of φ ∈ Cf for which φ′ (x) and φ′′ (x)
are continuous bounded functions.
A weak solution to equation (1). A Γ-valued stochastic process γt (ω) is
(2)
called a weak solution to system (2) if, for all φ ∈ Cf , the stochastic process
µφ (ω, t) = hφ, γt i −
Zt
[h(φ′ , a), γs × γs i +
0
σ2
h∆φ, γs i]ds
2
(3)
is a martingale with respect to the filtration (Ft )t≥0 , where Ft = σ(γs , s ≤ t),
and the square characteristic of the martingale is
hµφ , µφ it = σ
2
Zt
0
h(φ′ , φ′ ), γs ids.
(4)
ON SDES IN A CONFIGURATION SPACE
391
If γt (ω) is a weak solution to system (2) and
hφ, γt (ω)i =
X
φ(xk (t))
for all φ ∈ Cf , then the sequence {xk (t), k ∈ N } is a weak solution to system
(1).
Weak uniqueness. Let γ0 ∈ Γ. System (2) has a unique weak solution with
the initial value γ0 if for any pair of weak solutions to system (2) γt1 (ω) and
γt2 (ω) satisfying the condition γ01 = γ02 = γ0 , the following relations are fulfilled:
1
1
1
1
2
2
2
2
EΦ(ξ11
, . . . , ξ1m
, . . . , ξl1
, . . . , ξlm
) = EΦ(ξ11
, . . . , ξ1m
, . . . , ξl1
, . . . , ξlm
), (5)
where Φ(y11 , . . . , ylm ) is a continuous bounded function on Rlm and,
ξijk = hφi , γtkj (ω)i, k = 1, 2, i = 1, . . . , l,
φ ∈ Cf , t1 , . . . , tm ∈ R+ .
j = 1, . . . , m,
This means that the distribution of the stochastic process γtk does not depend
on k, i.e., the distribution of the weak solution to system (2) is unique if it
exists.
Local weak solutions. Let γt (ω) be a continuous Γ-valued stochastic process, and (Ft )t≥0 be the filtration generated by it. γt (ω). is called a local weak
solution to system (2) if there exists a stopping time τ with respect to the
filtration (Ft )t≥0 for which P {τ > 0} = 1 and the stochastic process µφ (ω, t)
which is determined by relation (3) is a martingale for t < τ with the square
characteristic given by equality (4).
Local weak uniqueness. Let γ0 ∈ Γ. System (2) has a locally unique
weak solution with the initial value γ0 if, for any pair of locally weak solutions
to system (2) γt1 and γt2 satisfying the condition γ01 = γ02 = γ0 , relation (5) is
fulfilled for
ξijk = hφi , γtkj (ω)iI{tj 0 there exists a continuous
decreasing function Φ(t) : [0, ∞) → R+ with Φ(+∞) = 0 such that
ZZ
Φ(|x|)Φ(|x′ |)λ(|x − x′ |)1{x6=x′ } γ(dx)γ(dx′ ) < ∞.
(6)
Denote
Φλ (x, x′ ) = Φ(|x|)Φ(|x′ |)λ(|x − x′ |).
(7)
For any compact set K from Γ and any fanction λ satisfying the conditions
mentioned before there exists a function of the form given by relation (7) for
which
suphΦλ , γ × γi < ∞.
γ∈K
392
A. SKOROKHOD
Note that the set
{γ : hΦλ , γ × γi ≤ c}
is a compact in Γ for any Φλ of form (7) and c > 0. Denote by ΓΦ,λ the set of
those γ ∈ Γ for which relation (6) is fulfilled. Set
dΦ,λ (γ1 , γ2 ) = sup{|hφΦ, γ1 i−hφΦ, γ2 i : φ ∈ Lip1 |}+|hΦλ , γ1 ×γ1 i−hΦλ , γ2 ×γ2 i|,
where
1
(
Lip = φ ∈ Cf : sup |φ(x)| ≤ 1,
x
|φ(x) − φ(x′ )|
sup
≤1 .
|x − x′ |
x,x′
)
ΓΦ,λ with the distance dΦ,λ is a separable locally compact space.
3. An Extension of Girsanov’s Formula
Assume that the potential function U (x) satisfies the condition
(PC) U (x) = u(|x|) where the function u : (0, ∞) → R is continuous, it has
continuous derivatives u′ , u′′ , there exists a constant r > 0 for which u(t) = 0
for t > r, and
Z
td−1 |u(t)|dt < ∞.
Free particle processes. Let a measure γ0 satisfy the condition (IC)
hφδ , γ0 i < ∞,
φδ (x) = exp {−δ|x|2 }.
Introduce Γ-valued stochastic processes by the relation
hφ, γt∗ (γ0 , ω)i =
X
φ(x0k + σwk (t)),
k
where
X
φ(x0k ) = hφ, γ0 i.
k
It is easy to check that γt∗ (γ0 , ω) is a continuous Γ-valued stochastic process if
γ0 ∈ Γ0 , where Γ0 is the set of finite measures from Γ. There exist functions
Φ, λ for which P {γt∗ (γ0 , ω) ∈ ΓΦ,λ } = 1 for all t > 0. The stochastic process
γt∗ (γ0 , ω) is continuous in the space ΓΦ,λ , and for any t0 > 0 the function
EF (γ ∗ (γ0 , ω))
is a continuous function in γ0 ∈ ΓΦ,λ if F is a bounded continuous function
on C[0,t0 ] (ΓΦ,λ ) which is the space of continuous ΓΦ,λ -valued functions on the
interval [0, t0 ].
Girsanov’s formula for finite systems. Let γ0n be a sequence of finite
measures from Γ satisfying the condition:
γ0n → γ0 , γ0n ≤ γ0n+1 .
It was proved in [4] that under the condition (PC), for any n, there exists a
unique strong solution to system (2) with the initial value γ0n . Denote it by
γt (γ0n , ω).
ON SDES IN A CONFIGURATION SPACE
393
Lemma 1. Set a(x, x′ ) = a(x − x′ ),
Gn1 (t)
=σ
t
X Z
−1
(ha(xi + σwi (s), .), γs∗ (γ0n , ω)i, dwi (s)),
xi ∈Sγ n 0
0
Gn2 (t)
=σ
−2
Zt Z
|ha(x, .), γs∗ (γ0n , ω)i|2 γs∗ (γ0n , ω, dx)ds,
0
n
o
ρn (t) = exp Gn1 (t) − 21 Gn2 (t) .
Then
EΦ(hφ1 , γt1 (γ0n , ω)i, . . . , hφk , γtk (γ0n , ωi)
= Eρn (t)Φ(hφ1 , γt∗1 (γ0n , ω)i, . . . , hφk , γt∗k (γ0n , ω)i)
for all k = 1, 2, . . . , bounded continuous functions Φ : Rk → R and φ1 , . . . , φk ∈
Cf , t1 , . . . , tk ∈ [0, t].
The proof of the lemma can be obtained using the approximation of the
function a(x, x′ ) by smooth functions since for smooth a(x, x′ ) the proof is a
consequence of Girsanov’s formula [8].
Introduce the stochastic processes
wkc (t)
=
Zt
1{|xk +wk (s)|≤c} dwk (s),
xk ∈ Sγ0n ,
0
where c > 0 is a constant. Let Ftn,c be the σ -algebra generated by
{wkc (s), s ≤ t, xk ∈ Sγ0n }.
Lemma 2.
E(ρn /Ftn,c ) = ρn (c, t),
where
n
o
ρn (c, t) = exp Gn1 (c, t) − 21 Gn2 (c, t) ,
and
Gn1 (c, t)
=σ
t
X Z
−1
(E(ha(xi + σwi (s), .), γs∗ (γ0n , ω)i/Fsn,c ), dwic (s)),
xi ∈Sγ n 0
0
Gn2 (c, t)
=σ
−2
Zt Z
|E(ha(x, .), γs∗ (γ0n , ω)i/Fsn,c )|2 1{|x|≤c} γs∗ (γ0n , ω, dx)ds.
0
The proof rests on the statement below.
Statement. Let Ft , t ∈ R+ be a continuous filtration, and its subfiltration
Ft∗ , t ∈ R+ , satisfy the condition
∗
(RE)E(ξ/Ft ) is a Ft∗ -measurable random variable if ξ is a bounded F∞
measurable random variable.
394
A. SKOROKHOD
Let µt be an Ft -martingale with the square characteristic hµit for which the
stochastic process
o
n
ρ(t) = exp µt − 12 hµit
is a martingale. Then
n
o
E(ρ(t)/Ft∗ ) = exp µ∗t − 21 hµ∗ it ,
where µ∗t = E(µt /Ft∗ ) is a martingale, and hµ∗ it is its square characteristic.
Proof. Set µ
˜t = µt − µ∗t . Condition (RE) implies the relation
E
à Zt
∗
g(s)d˜
µs /F∞
0
!
=0
for all Ft -adapted functions g(t) for which
¯
¯ Zt
¯
¯
¯
¯
µs ¯¯ < ∞.
E ¯¯ g(s)d˜
0
Set
n
o
ρ∗ (t) = exp µ∗t − 12 hµ∗ it ,
n
o
ρ˜(t) = exp µ
˜t − 12 h˜
µit ,
where h˜
µit is the square characteristic of the martingale µ
˜t . The proof follows
from the relations ρ(t) = ρ∗ (t)˜
ρ(t) and
E(˜
ρ(t) −
∗
1/F∞
)
=E
à Zt
∗
ρ˜(s)d˜
µ(s)/F∞
0
!
= 0.
Remark 1. Assume that φi (x) = 0 for |x| ≥ c, i = 1, 2, . . . , k, φ1 , . . . , φk ∈
Cf , t1 , . . . , tk ∈ [0, t]. Then for Φ satisfying the conditions of Lemma 1 we have
the relation
EΦ(hφ1 , γt1 (γ0n , ω)i, . . . , hφk , γtk (γ0n , ω)i)
= Eρn (c, t)Φ(hφ1 , γt∗1 (γ0n , ω)i, . . . , hφk , γt∗k (γ0n , ω)i).
Remark 2. Denote by Ftc the σ-algebra generated by {wkc (s), s ≤ t, xk ∈ Sγ0 }.
Then there exist the limits in probability
lim Gn1 (c, t) = G1 (c, t),
n→∞
lim Gn2 (c, t) = G2 (c, t),
n→∞
1
lim ρn (t) = ρ(c, t) = exp G1 (c, t) − G2 (c, t) ,
n→∞
2
½
¾
where
G1 (c, t) = σ
t
X Z
−1
G2 (c, t) = σ −2
xi ∈Sγ0 0
Zt Z
(E(ha(xi + σwi (s), .), γs∗ (γ0 , ω)i/Fsc ), dwic (s))),
|E(ha(x, .), γs∗ (γ0 , ω)i/Fsc )|2 1{|x|≤c} γs∗ (γ0 , ω, dx)ds.
0
ON SDES IN A CONFIGURATION SPACE
395
Let w(t) be a standard Wiener process . Introduce the functions
Qc (s, x, B) = P {x + σw(s) ∈ B, inf |x + σw(u)| > c},
u≤s
x ∈ Vc , B ∈ B(Vc ), Vc = {x ∈ Rd : |x| > c},
Q∗c (s, x, B) = lim
λ↓1
Qc (s, x, B)
, |x| = c, B ∈ B(Vc ).
Qc (s, λx, Vc )
Set
θi (c, s) = inf ∆i (c, s), ζi (c, s) = sup ∆i (c, s),
where
∆i (c, s) = {u ≤ s : |x∗i (u)| ≤ c}, x∗i (u) = xi + wi (u).
Then the following statement holds.
Lemma 3.
E(a(x, x∗i (s))1{|x∗i (s)|≤c} /Fsc ) = a(x, x∗i (s))1{|x∗i (s)|≤c}
+1{θi (c,s) 0.
Proof. Choose ak satisfying the inequality
P (G2 (ck , k) > (k 2 ak )−1 ) < k −2 .
Then for any t > 0 we have the relation
X
P (ak G2 (ck , t) > k −2 ) < t +
k
X
k −2 < ∞.
k≥t
This completes the proof.
Corollary 3. Let a sequence {ak } satisfy the statement of Lemma 5. Set
G(t) =
X
ak G2 (ck , t).
k
With probability 1 G(t) is an increasing continuous function satisfying the relations G(0) = 0, limt→∞ G(t) = ∞. Set
τ ∗ = inf{t : G(t) > b},
where b is a positive number. Then
Eρ(ck , τ ∗ ) = 1, E(ρ(ck , τ ∗ ))2 ≤ exp {ba−1
k }.
(9)
398
A. SKOROKHOD
4. A Theorem on Existence of a Local Weak Solution
Theorem 1. Let conditions (PC) and (IC) be fulfillrd,and let τ ∗ be a stopping time introduced by relation (9). Then the following statements hold:
(i) there exists a probability measure P ∗ on (Ω, F) for which
P ∗ (A) = lim E1A ρ(c, τ ∗), A ∈
c→∞
_
Fτc∗k , P ∗ (A) = P (A), A ∈ F ∗ ,
k
∗
where the σ-algebra F is generated by the processes
{x∗k (t ∨ τ ∗ ) − x∗k (t), t ≥ 0, k = 1, 2, . . . }.
(ii) the stochastic processes given by the formula
σwk∗ (t)
=
x∗k (t)
−
t∧τ
Z ∗
0
X
a(x∗k (s), x∗i (s))ds, k = 1, 2, . . . ,
i6=k
are independent Wiener processes with respect to the filtration (Ft )(t≥0) on the
probability space (Ω, F, P ∗ ).
Proof. Since τ ∗ is a stopping time, for A ∈ Fτc∗k we can write, using Remark 5,
the relations
E1A ρ(c, τ ∗ ) = E1A ρ(ck , τ ∗ )
if c > ck . This implies the existence of limits
lim 1A×B ρ(c, τ ∗ ), A ∈
c→∞
_
Fτc∗k , B ∈ F ∗ .
k
It follows from the last formula that there exists a limit
lim Φ(ξ1 , . . . , ξm )ρ(c, τ ∗ ),
(10)
c→∞
where Φ ∈ C(Rm ) and
ξi = hfi , γt∗i (γ0 , ω)i, i = 1, . . . , m,
where fi ∈ C(Rd ), fi (x) = 0 for |x| large enough.
Now we prove that there exists a probability measure P ∗ on (Ω, F) for which
a limit in the formula (10) is represented in the form E ∗ Φ(ξ1 , . . . , ξm ), where
E ∗ is the expectation with respect to the probability P ∗ . Set
µ
fn (x, x′ ) = 1 −
|x|
cn
¶µ
1−
|x′ |
cn
¶µ
1+
1
∨ 0.
|x − x′ |
¶
Using Corollary 3 we can write the inequality
Ehfn , γt∗ (γ0 , ω) × γt∗ (γ0 , ω)iρ(c, τ ∗ )
1
≤ (Ehfn , γt∗ (γ0 , ω) × γt∗ (γ0 , ω)i2 ) 2 exp { 12 ba−1
n }.
Denote
An (t) = Ehfn , γt∗ (γ0 , ω) × γt∗ (γ0 , ω)i2 .
(11)
ON SDES IN A CONFIGURATION SPACE
399
The functions An (t) are non-negative and continuous. There exists a sequence
of positive numbers {bn } for which
X
n
bn (1 + An (t)) exp { 12 ba−1
n } < ∞, t > 0.
(12)
Formulas (11) and (12) imply the relation
lim sup E
c→∞
X
bn hfn , γt∗ (γ0 , ω) × γt∗ (γ0 , ω)iρ(c, τ ∗ ) < ∞.
n
This formula implies that the set of measures {Pc∗ , c > 0} for which
dPc∗
(ω) = ρ(c, τ ∗ )
dP
is weakly compact, so there exists a sequence c′k , c′k → ∞ for which Pc∗′ converges
k
weakly to a measure P ∗ . The statement (i) is proved.
Introduce the stopping times
½
τn∗ = inf t :
X
¾
ak G2 (ck , t) ≥ b ,
k≤n
and set
Pn∗ (A) = E1A ρ(cn , τn∗ ), A ∈ F.
Let Φ(ξ1 , . . . , ξm ) be the same as before. Then
lim En∗ Φ(ξ1 , . . . , ξm ) = E ∗ Φ(ξ1 , . . . , ξm );
n→∞
here En∗ is the expectation with respect to probability Pn∗ .This formula is a
consequence of the relations
τn∗ > τ ∗ , τn∗ → τ ∗
in probability and
E(ρ(cn , τn∗ )/Fτc∗n ) = ρ(cn , τ ∗).
Let wi∗ (c, t) be given by formula (8). Then for fixed n the sequence
{wi∗ (cn , t), i = 1, 2, . . . }
represents independent Wiener processes on the probability space {Ω, F, Pn∗ }.
Introduce stopping times
ζin = inf{t : |xi (t)| > cn − r}.
Since ac (x, ω) = 0 for |x| < cn − r,we have
wi∗ (cn , t ∨ ζin ) = wi∗ (t ∨ ζin ).
Using Remark 3 we can prove that the relation
lim lim sup Pn∗ {sup |x∗i (s)| > c} = 0
c→∞
n→∞
s≤t
(13)
400
A. SKOROKHOD
is fulfilled for any i and t > 0. Let Φ, h1 , . . . , hm be the same as before . Set
ξi∗
=
Zt
hi (s)dwi∗ (s),
ξin
=
Zt
hi (s)dwi∗ (cn , s), i = 1, . . . .
0
0
Note that
n
En∗ Φ(ξ1n , . . . , ξm
) = EΦ
à Zt
h1 (s)dw1 (s), . . . ,
0
Zt
!
hm (s)dwm (s)
0
(14)
¯
for all n. Denote the expression in the right hand side of formula (14) by Φ.
Then
∗
¯ + En∗ O
En∗ Φ(ξ1∗ , . . . , ξm
)=Φ
µX
¶
1{sups≤t |x∗i (s)|>cn −r} .
i≤m
(15)
Formulas (13) and (15) imply the relation
∗
∗
¯
E ∗ Φ(ξ1∗ , . . . , ξm
) = lim En∗ Φ(ξ1∗ , . . . , ξm
) = Φ.
n→∞
The statement (ii) is proved.
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(Received 3.07.2000)
Author’s addresses:
Institute of Mathematics
National Academy of Sciences of Ukraine
3, Tereshchenkivska St., Kyiv 01601, Ukraine
Department of Statistics and Probability
Michigan State University
A 415 Wells Hall, East Lansing, MI 48824, U.S.A.
E-mail: skorokhod@stt.msu.edu