getdocf7c4. 271KB Jun 04 2011 12:05:13 AM
Elect. Comm. in Probab. 3 (1998) 65–74
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
ESTIMATES FOR THE DERIVATIVE
OF DIFFUSION SEMIGROUPS
L.A. RINCON1
Department of Mathematics
University of Wales Swansea
Singleton Park
Swansea SA2 8PP, UK
e-mail: [email protected]
submitted April 15, 1998; revised August 18, 1998
AMS 1991 Subject classification: 47D07, 60J55, 60H10.
Keywords and phrases: Diffusion Semigroups, Diffusion Processes, Stochastic Differential
Equations.
Abstract
Let {Pt }t≥0 be the transition semigroup of a diffusion process. It is known that Pt sends
continuous functions into differentiable functions so we can write DPt f. But what happens
with this derivative when t → 0 and P0 f = f is only continuous ?. We give estimates for
the supremum norm of the Fr´echet derivative of the semigroups associated with the operators
A + V and A + Z · ∇ where A is the generator of a diffusion process, V is a potential and Z
is a vector field.
1
Introduction
Consider the following stochastic differential equation on Rn
dXt
X0
= X(Xt ) dBt + A(Xt ) dt ,
= x ∈ Rn ,
for t ≥ 0, where the first integral is an Itˆ
o stochastic integral and the second is a Riemann
integral. Here {Bt }t≥0 is Brownian motion on Rm and the equality holds almost everywhere.
The coefficients of this equation are the mapping X : Rn → L(Rm ; Rn ) and the vector field
A : Rn → Rn . Assume standard regularity conditions on these coefficients so that there exists
a strong solution {Xt }t≥0 to our equation. We write Xtx for Xt when we want to make clear
its dependence on the initial value x. It is known that under further assumptions on the
coefficients of our equation, the mapping x 7→ Xtx is differentiable ( see for instance [1] ).
1 Supported
by a grant from CONACYT (M´exico).
65
66
Electronic Communications in Probability
Assume coefficients X and A are smooth enough and consider the associated derivative equation
dVt
V0
= DX(Xt )(Vt ) dBt + DA(Xt )(Vt ) dt ,
= v ∈ Rn ,
whose solution {Vt }t≥0 is the derivative of the mapping x 7→ Xtx at x in the direction v. We
will assume that there exists a smooth map Y : Rn → L(Rn ; Rm ) such that Y(x) is the right
inverse of X(x). That is, X(x)Y(x) = IRn for all x in Rn . We shall also assume that the process
Rt
2
{Y(Xt )(Vt )}t≥0 belongs to L2 ([0, t]) for each t > 0, that is, 0 |Y(Xs )(Vs )| ds < ∞ and thus
Rt
we can write 0 Y(Xs )(Vs ) dBs .
With all above assumptions, we have from [6], the following result (see Appendix for a proof)
Theorem 1 For every t > 0 there exist positive constants k and a such that
E|
Z
t
0
√
Y(Xs )(Vs ) dBs | ≤ k eat − 1 .
(1)
√
√
at
We are interested in small values for t. Thus,
√ since e −
√ 1 = O( t) as t → 0, we have that
there exist a positive constant N such that eat − 1 ≤ N t for small t. Hence for sufficiently
small t, we have the following estimate
E|
Z
0
t
√
Y(Xs )(Vs ) dBs | ≤ c t ,
(2)
where c is a positive constant.
Let now BC r (Rn ) be the Banach space of bounded measurable functions on Rn which are
r-times continuously differentiable with bounded derivatives. The norm of this space is given
by the supremum norm of the function plus the supremum norm of each of its r derivatives. In
particular B(Rn ) is the Banach space of bounded measurable functions on Rn with supremum
norm kfk∞ = supx∈Rn |f(x)|. Suppose our diffusion process {Xt }t≥0 has transition probabilities P (t, x, Γ). Then this induces a semigroup of operators {Pt }t≥0 as follows. For every t ≥ 0
we define on B(Rn ) the bounded linear operator
Z
f(y) P (t, x, dy) = E(f(Xtx )) .
(3)
(Pt f)(x) =
Rn
The semigroup {Pt }t≥0 is a strongly continuous semigroup on BC 0 (Rn ). Denote by A its
infinitesimal generator.
It is known that {Pt }t≥0 is a strong Feller semigroup, that is, Pt sends continuous functions
into differentiable functions. In fact, under above assumptions, a formula for the derivative of
Pt f is known ( see [4] or [5] ).
Theorem 2 If f ∈ BC 2 (Rn ) then the derivative of Pt f : Rn → R is given by
D(Pt f)(x)(v) =
1
E{f(Xt )
t
Z
0
t
Y(Xs )(Vs ) dBs } .
(4)
Diffusion Semigroups
67
Higher derivatives in a more general setting are given in [4]. See also [2] for a general formula
of this derivative
in the context of a stochastic control system. Observe the mapping f 7→
Rt
1
2
n
Y(X
E{f(X
)
s )(Vs ) dBs } defines a bounded linear functional on BC (R ). Hence there
t
t
0
0
n
exists a unique extension on BC (R ). Since the expression of this linear functional does not
depend on the derivatives of f, it has the same expression for any f in BC 0 (Rn ).
From last theorem we obtain
|DPt f(x)(v)|
≤
≤
Z t
1
kfk∞ E|
Y(Xs )(Vs ) dBs |
t
0
√
1
kfk∞ k eat − 1 .
t
And hence for small t
kDPt fk∞ ≤
ckfk∞
√ .
t
(5)
Observe that, as expected, our estimate goes to infinity as t approaches 0 since P0 f = f is not
1
necessarily differentiable. Also the rate at which it goes to infinity is not faster than √ does.
t
2
Potential
Let V : Rn → R be a bounded measurable function. We shall perturb the generator A by
adding to it the function V . We define the linear operator
AV = A + V ,
with the same domain as for A. A semigroup
{PtV }t≥0 having AV as generator is given by
Rt
V
V
(X
u ) du
the Feynman-Kac formula Pt f = E{f(Xt )e 0
}. We will find a similar estimate as (5)
V
for kDPt fk∞ . We first derive a recursive formula that will help us calculate the derivative of
PtV f. We have
PtV f
s=t
= Pt f + Pt−s PsV f s=0
Z t
∂
(Pt−s PsV f) ds
= Pt f +
∂s
0
Z t
= Pt f +
[−A(Pt−s PsV f) + Pt−s((A + V )PsV f)] ds .
0
Hence
PtV f = Pt f +
Z
0
t
Pt−s (V PsV f) ds .
68
Electronic Communications in Probability
Now we use our formula for differentiation (4) to calculate the derivative of this semigroup.
We have
Z t
DPtV f(x)(v) = DPt f(x)(v) +
DPt−s (V PsV f)(x)(v) ds
0
Z t
1
E{f(Xt )
Y(Xs )(Vs ) dBs }
=
t
0
Z t−s
Z t
1
E{V (Xt−s )PsV f(Xt−s )
Y(Xu )(Vu ) dBu }ds .
+
0
0 t−s
Then, by the Feynman-Kac formula and the Markov property we have
Z t
1
V
DPt f(x)(v) =
E{f(Xt )
Y(Xs )(Vs ) dBs }
t
0
Z t−s
Z t
Rt
1
E{V (Xt−s )E{f(Xt )e t−s V (Xu )du }
Y(Xu )(Vu ) dBu } ds ,
+
0
0 t−s
from which we obtain
kDPtV fk∞
≤
1
kfk∞ E |
t
Z
t
0
+kV k∞ kfk∞
Y(Xs )(Vs ) dBs |
Z
t
0
eskV k∞
E|
t−s
Z
0
t−s
Y(Xu )(Vu ) dBu | ds .
And hence for small t
kDPtV fk∞
≤
=
Z t
ckfk∞
c
√
√
+ kV k∞ kfk∞ etkV k∞
ds
t−s
t
0
√
ckfk∞
√
+ 2c t kV k∞ kfk∞ etkV k∞ .
t
Observe again that our estimate goes to infinity as t → 0.
3
Bounded Smooth Drift
Let Z : Rn → Rn be a bounded smooth vector field. We shall consider another perturbation
to the generator A. This time we define the linear operator
AZ = A + Z · ∇ .
The existence of a semigroup {PtZ }t≥0 having AZ as infinitesimal generator is guaranteed by
the regularity of Z. Indeed, if we write Z(x) = (Z 1 (x), . . . , Z n (x)), then the operator AZ can
be written as
AZ =
n
n
X
∂2
∂
1X
(X(x)X(x)∗ )ij i j +
(Ai (x) + Z i (x)) i ,
2 i,j=1
∂x ∂x
∂x
i=1
and this operator is the infinitesimal generator associated with the equation
dXt
X0
= X(Xt )dBt + [A(Xt ) + Z(Xt )]dt ,
= x ∈ Rn .
Diffusion Semigroups
69
Thanks to the smoothness of Z, this equation yields a diffusion process (Xtx,Z )t∈T and hence
the semigroup PtZ f(x) = E(f(Xtx,Z )). Then previous estimate applies also to kDPtZ fk∞ .
But we can do better because we can find the explicit dependence of the estimate upon Z as
follows. As before, we first find a recursive formula for this semigroup.
PtZ f
s=t
= Pt f + Pt−s PsZ f s=0
Z t
∂
(Pt−s PsZ f) ds
= Pt f +
0 ∂s
Z t
[−A(Pt−s PsZ f) + Pt−s ((A + Z · ∇)PsZ f) ] ds .
= Pt f +
0
Hence
PtZ f
= Pt f +
Z
0
t
Pt−s (Z · ∇PsZ f) ds .
(6)
We can now calculate its derivative as follows
Z t
DPtZ f(x)(v) = DPt f(x)(v) +
DPt−s (Z · ∇PsZ f)(x)(v) ds
0
Z t
1
Y(Xs )(Vs ) dBs }
E{f(Xt )
=
t
0
Z t−s
Z t
1
E{Z(Xt−s ) · ∇PsZ f(Xt−s )
Y(Xu )(Vu ) dBu } ds .
+
0
0 t−s
We now find an estimate for the supremum norm of this derivative. Taking modulus we obtain
Z t
1
|DPtZ f(x)(v)| ≤
kfk∞ E |
Y(Xs )(Vs ) dBs |
t
0
Z t−s
Z t
1
E |DPsZ f(Xt−s )(Z(Xt−s ))| |
+
Y(Xu )(Vu ) dBu | ds .
0
0 t−s
Hence for small t
kDPtZ fk∞
c
≤ √ kfk∞ + ckZk∞
t
Z
0
t
kDPsZ fk∞
√
ds .
t−s
We now solve this inequality. If we iterate once we obtain
kDPtZ fk∞
Z t
c
ds
2
p
≤ √ kfk∞ + c kZk∞ kfk∞
t
s(t − s)
0
Z tZ s
Z
kDPu fk∞
p
+ c2 kZk2∞
du ds .
(t − s)(s − u)
0
0
By Fubini’s theorem, the double integral becomes
Z t
Z t
ds
Z
p
du ,
kDPu fk∞
(t − s)(s − u)
0
u
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Electronic Communications in Probability
and then we observe that
Z t
u
ds
p
= 2 tan−1
(t − s)(s − u)
r
t
s − u
= π.
t − s u
The case u = 0 solves also the first integral. Hence our inequality reduces to
c
2
kDPtZ fk∞ ≤ √ kfk∞ + c2 πkZk∞ kfk∞ + c2 πkZk∞
t
Z
t
0
kDPuZ fk∞ du .
We now apply Gronwall’s inequality. After some simplifications ( extending the integral up to
infinity ) we finally obtain the estimate
2
c
kDPtZ fk∞ ≤ √ kfk∞ + 2c2 πkZk∞ kfk∞ et(c πkZk∞ )
t
(7)
As expected, our estimate goes to infinity as t → 0 since P0Z f = f is not necessarily differentiable.
4
Bounded Uniformly Continuous Drift
We now find a similar estimate when Z : Rn → Rn is only bounded and uniformly continuous.
We look again at the operator AZ = A + Z · ∇. The problem here is that in this case we do
not have the semigroup {PtZ f}t≥0 since the stochastic equation with the added nonsmooth
drift Z might not have a strong solution. So we cannot even talk about its derivative. To solve
this problem we proceed by approximation.
4.1
Existence of Semigroup
Since Z ∈ BC 0 (Rn ; Rn ) is uniformly continuous, and BC ∞ (Rn ; Rn ) is dense in BC 0 (Rn ; Rn ),
∞
n
n
there exists a sequence {Zi}∞
i=1 in BC (R ; R ) such that Zi converges to Z uniformly. Thus,
for every i ∈ N, we have the semigroup {PtZi }t≥0 since our stochastic equation with the added
smooth drift Zi has a strong solution.
∞
For every t ≥ 0 and f ∈ BC 0 (Rn ) fixed, the sequence of functions {PtZi f}i=1 is a Cauchy
sequence in the Banach space BC 0 (Rn ). We will prove this fact later. Let us denote its limit
by PtZ f. All properties required for PtZ f are inherited from those of the semigroup PtZi f.
Indeed by simply writing PtZ f = limi→∞ PtZi f and using an interchange of limits we can prove
1. f 7→ PtZ f is a bounded linear operator.
2. t 7→ PtZ f is a contraction semigroup of operators.
3. {PtZ }t≥0 has generator AZ = A + Z · ∇.
∞
Now, suppose for a moment that the sequence of derivatives {DPtZi f}i=1 converges uniformly.
Then we would have D(limi→∞ PtZi f) = limi→∞ DPtZi f. This proves that PtZ f is differentiable. Then, for any ǫ > 0 there exists N ∈ N such that if i ≥ N , kDPtZ fk∞ ≤ kDPtZi fk∞ +ǫ
and therefore our estimate also applies to kDPtZ fk∞ .
Diffusion Semigroups
4.2
71
Uniform Convergence of Derivatives
∞
We now prove that the sequence {DPtZi f}i=1 converges uniformly. We use again our recursive
formula (6) to obtain
Z t
Z
PtZi f − Pt j f =
Pt−s (Zi · ∇PsZi f − Zj · ∇PsZj f) ds .
0
We now use our formula for differentiation (4). After differentiating and taking modulus we
obtain
Z
|D(PtZi f − Pt j f)(x)(v)|
Z t
1
E{|DPsZi f(Xt−s )(Zi (Xt−s )) − DPsZj f(Xt−s )(Zj (Xt−s ))|
≤
t
−
s
0
Z t−s
|
Y(Xu )(Vu ) dBu |} ds .
0
Thus
kD(PtZi f
Z
− Pt j f)k∞
≤ kZi − Zj k∞
Z t
+ kZj k∞
0
And hence for small t
t
k p a(t−s)
e
− 1kDPsZi fk∞ ds
0 t−s
k p a(t−s)
e
− 1 kDPsZi f − DPsZj fk∞ ds .
t−s
Z
Z t
k
√
≤ kZi − Zj k∞
kDPsZi fk∞ ds
t
−
s
0
Z t
k
√
kDPsZi f − DPsZj fk∞ ds .
+ kZj k∞
t
−s
0
Z
kD(PtZi f − Pt j f)k∞
Now write estimate (7) as
A
kDPtZi fk∞ ≤ √ + B eC t ,
t
for some positive constants A, B and C depending on kfk∞ and kZi k∞ . Thus, substituting
this in our last estimate gives
Z t
A
k
Z
√
√ ds
kD(PtZi f − Pt j f)k∞ ≤ kZi − Zj k∞
t−s s
0
Z t
k
√
B eC s ds
+ kZi − Zj k∞
t−s
0
Z t
k
√
kDPsZi f − DPsZj fk∞ ds .
+ kZj k∞
t
−
s
0
The first two integrals are bounded if we allow t to move within a finite interval (0, T ]. Thus
there exist positive constants M1 and M2 such that
Z
kD(PtZi f − Pt j f)k∞
≤ M1 kZi − Zj k∞
Z t
1
√
+ M2
kDPsZi f − DPsZj fk∞ ds .
t−s
0
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Electronic Communications in Probability
Now iterate this to obtain
Z
kD(PtZi f − Pt j f)k∞
≤ M1 kZi − Zj k∞
+ M2 M1 kZi − Zj k∞
+ M22
Z tZ
0
s
Z
kD(PuZi f
p
0
t
0
√
1
ds
t−s
Z
− Pu j f)k∞
(t − s)(s − u)
du ds .
As before the double integral reduces to
Z t
2
πM2
kD(PuZi f − PuZj f)k∞ du .
0
Collecting constants into new constants M and N we arrive at
Z t
Z
kD(PtZi f − Pt j f)k∞ ≤ M kZi − Zj k∞ + N
kD(PuZi f − PuZj f)k∞ du .
0
Now we apply again Gronwall’s inequality to obtain
Z
kD(PtZi f − Pt j f)k∞ ≤ M kZi − Zj k∞ + N M kZi − Zj k∞
Z
t
eN(t−u) du .
0
∞
The right hand side goes to zero as kZi − Zj k∞ → 0. This proves the sequence {DPtZi }i=1 is
uniformly convergent.
4.3
Uniform Convergence of Semigroups
∞
We finally prove that the sequence of functions {PtZi f}i=1 is a Cauchy sequence. From our
recursive formula (6) we obtain
Z t
Z
kPtZi f − Pt j fk∞ ≤
kZi · ∇PsZi f − Zj · ∇PsZj fk∞ ds
0
Z t
≤
kZi − Zj k∞ kDPsZi fk∞ ds
0
Z t
+
kZj k∞ kDPsZi f − DPsZj fk∞ ds
0
Z t
√
≤ kZi − Zj k∞
k eas − 1 ds
0
+kZj k∞
Z
0
t
kDPsZi f − DPsZj fk∞ ds .
The first integral is bounded so the first part goes to zero as kZi − Zj k∞ approaches 0. The
second part also goes to zero since we just proved the sequence of derivatives is a Cauchy
∞
sequence. Hence {PtZi f}i=1 is uniformly convergent.
Thus, with this approximating procedure we found a semigroup for the operator A + Z · ∇ for
Z bounded and uniformly continuous and we proved it is differentiable and that our estimate
also applies to its derivative.
Diffusion Semigroups
73
Observe that the boundedness and uniform continuity of Z are required in order to ensure
the existence of a sequence of smooth vector fields Zi uniformly convergent to Z. Alternative
assumptions on Z that guarantee the existence of such approximating sequence may be used.
Appendix
We here give a proof of inequality (1) which is taken from [6]. By using the Cauchy-Schwarz
inequality and then the isometric property, we have
E|
t
Z
0
Z
t
Y(xs )(vs ) dBs | ≤ ( E|
Y(xs )(vs ) dBs |2 )1/2
0
Z t
2
= (E
kY(xs )(vs )k ds )1/2
0
Z t
2
Ekvs k ds )1/2 .
≤ kYk2∞(
(8)
0
We will estimate the right-hand side of the last inequality. Itˆ
o’s formula applied to the function
2
f(·) = k · k : Rn → R and the semimartingale (vt )t≥0 yields
kvs k2 = kv0 k2 + 2
Z
s
vu dvu +
0
n Z
X
i=1
0
s
dhvi iu .
Therefore
E kvs k
2
2
= kv0 k + Ehvis
Z s
= kv0 k2 + E
[DX(xu )(vu )] [DX(xu )(vu )]∗ du
0
Z s
= kv0 k2 + E
kDX(xu )(vu )k2 du
0
Z s
2
2
≤ kv0 k + kDXk∞
E kvu k2 du .
0
We now use Gronwall’s inequality to obtain
E kvs k2 ≤ kv0 k2 ekDXk∞ s .
Integrating from 0 to t yields
Z
0
t
E kvs k2 ds ≤
kv0 k2
(ekDXk∞ t − 1) ,
kDXk∞
and thus, substituting in (8) we obtain
E|
Z
0
t
Y(xs )(vs ) dBs | ≤ kYk2∞
This proves inequality (1).
kv0 k
p
et kDXk∞ − 1 .
1/2
kDXk∞
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Electronic Communications in Probability
Acknowledgments
I would like to thank Prof. K. D. Elworthy at Warwick from whom I have enormously benefitted. Also thanks to Dr. H. Zhao and Prof. A. Truman at Swansea for very helpful suggestions. CONACYT has financially supported me for my postgraduate studies at Warwick and
Swansea.
References
[1] Da Prato, G. and Zabczyk, J. (1992) Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications 44. Cambridge University Press.
[2] Da Prato, G. and Zabczyk, J. (1997) Differentiability of the Feynman-Kac Semigroup and
a Control Application. Rend. Mat. Acc. Lincei s. 9, v. 8, 183–188.
[3] Elworthy, K. D. (1982) Stochastic Differential Equations on Manifolds. Cambridge University Press.
[4] Elworthy, K. D. and Li, X. M. (1994) Formulae for the Derivative of Heat Semigroups.
Journal of Functional Analysis 125, 252–286.
[5] Elworthy, K. D. and Li, X. M. (1993) Differentiation of Heat Semigroups and applications.
Warwick Preprints 77/1993.
[6] Li, X.-M. (1992) Stochastic Flows on Noncompact Manifolds. Ph.D. Thesis. Warwick
University.
[7] Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential
Equations. Springer-Verlag. New York.
[8] Rinc´
on, L. A. (1994) Some Formulae and Estimates for the Derivative of Diffusion Semigroups. Warwick MSc. Dissertation.
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
ESTIMATES FOR THE DERIVATIVE
OF DIFFUSION SEMIGROUPS
L.A. RINCON1
Department of Mathematics
University of Wales Swansea
Singleton Park
Swansea SA2 8PP, UK
e-mail: [email protected]
submitted April 15, 1998; revised August 18, 1998
AMS 1991 Subject classification: 47D07, 60J55, 60H10.
Keywords and phrases: Diffusion Semigroups, Diffusion Processes, Stochastic Differential
Equations.
Abstract
Let {Pt }t≥0 be the transition semigroup of a diffusion process. It is known that Pt sends
continuous functions into differentiable functions so we can write DPt f. But what happens
with this derivative when t → 0 and P0 f = f is only continuous ?. We give estimates for
the supremum norm of the Fr´echet derivative of the semigroups associated with the operators
A + V and A + Z · ∇ where A is the generator of a diffusion process, V is a potential and Z
is a vector field.
1
Introduction
Consider the following stochastic differential equation on Rn
dXt
X0
= X(Xt ) dBt + A(Xt ) dt ,
= x ∈ Rn ,
for t ≥ 0, where the first integral is an Itˆ
o stochastic integral and the second is a Riemann
integral. Here {Bt }t≥0 is Brownian motion on Rm and the equality holds almost everywhere.
The coefficients of this equation are the mapping X : Rn → L(Rm ; Rn ) and the vector field
A : Rn → Rn . Assume standard regularity conditions on these coefficients so that there exists
a strong solution {Xt }t≥0 to our equation. We write Xtx for Xt when we want to make clear
its dependence on the initial value x. It is known that under further assumptions on the
coefficients of our equation, the mapping x 7→ Xtx is differentiable ( see for instance [1] ).
1 Supported
by a grant from CONACYT (M´exico).
65
66
Electronic Communications in Probability
Assume coefficients X and A are smooth enough and consider the associated derivative equation
dVt
V0
= DX(Xt )(Vt ) dBt + DA(Xt )(Vt ) dt ,
= v ∈ Rn ,
whose solution {Vt }t≥0 is the derivative of the mapping x 7→ Xtx at x in the direction v. We
will assume that there exists a smooth map Y : Rn → L(Rn ; Rm ) such that Y(x) is the right
inverse of X(x). That is, X(x)Y(x) = IRn for all x in Rn . We shall also assume that the process
Rt
2
{Y(Xt )(Vt )}t≥0 belongs to L2 ([0, t]) for each t > 0, that is, 0 |Y(Xs )(Vs )| ds < ∞ and thus
Rt
we can write 0 Y(Xs )(Vs ) dBs .
With all above assumptions, we have from [6], the following result (see Appendix for a proof)
Theorem 1 For every t > 0 there exist positive constants k and a such that
E|
Z
t
0
√
Y(Xs )(Vs ) dBs | ≤ k eat − 1 .
(1)
√
√
at
We are interested in small values for t. Thus,
√ since e −
√ 1 = O( t) as t → 0, we have that
there exist a positive constant N such that eat − 1 ≤ N t for small t. Hence for sufficiently
small t, we have the following estimate
E|
Z
0
t
√
Y(Xs )(Vs ) dBs | ≤ c t ,
(2)
where c is a positive constant.
Let now BC r (Rn ) be the Banach space of bounded measurable functions on Rn which are
r-times continuously differentiable with bounded derivatives. The norm of this space is given
by the supremum norm of the function plus the supremum norm of each of its r derivatives. In
particular B(Rn ) is the Banach space of bounded measurable functions on Rn with supremum
norm kfk∞ = supx∈Rn |f(x)|. Suppose our diffusion process {Xt }t≥0 has transition probabilities P (t, x, Γ). Then this induces a semigroup of operators {Pt }t≥0 as follows. For every t ≥ 0
we define on B(Rn ) the bounded linear operator
Z
f(y) P (t, x, dy) = E(f(Xtx )) .
(3)
(Pt f)(x) =
Rn
The semigroup {Pt }t≥0 is a strongly continuous semigroup on BC 0 (Rn ). Denote by A its
infinitesimal generator.
It is known that {Pt }t≥0 is a strong Feller semigroup, that is, Pt sends continuous functions
into differentiable functions. In fact, under above assumptions, a formula for the derivative of
Pt f is known ( see [4] or [5] ).
Theorem 2 If f ∈ BC 2 (Rn ) then the derivative of Pt f : Rn → R is given by
D(Pt f)(x)(v) =
1
E{f(Xt )
t
Z
0
t
Y(Xs )(Vs ) dBs } .
(4)
Diffusion Semigroups
67
Higher derivatives in a more general setting are given in [4]. See also [2] for a general formula
of this derivative
in the context of a stochastic control system. Observe the mapping f 7→
Rt
1
2
n
Y(X
E{f(X
)
s )(Vs ) dBs } defines a bounded linear functional on BC (R ). Hence there
t
t
0
0
n
exists a unique extension on BC (R ). Since the expression of this linear functional does not
depend on the derivatives of f, it has the same expression for any f in BC 0 (Rn ).
From last theorem we obtain
|DPt f(x)(v)|
≤
≤
Z t
1
kfk∞ E|
Y(Xs )(Vs ) dBs |
t
0
√
1
kfk∞ k eat − 1 .
t
And hence for small t
kDPt fk∞ ≤
ckfk∞
√ .
t
(5)
Observe that, as expected, our estimate goes to infinity as t approaches 0 since P0 f = f is not
1
necessarily differentiable. Also the rate at which it goes to infinity is not faster than √ does.
t
2
Potential
Let V : Rn → R be a bounded measurable function. We shall perturb the generator A by
adding to it the function V . We define the linear operator
AV = A + V ,
with the same domain as for A. A semigroup
{PtV }t≥0 having AV as generator is given by
Rt
V
V
(X
u ) du
the Feynman-Kac formula Pt f = E{f(Xt )e 0
}. We will find a similar estimate as (5)
V
for kDPt fk∞ . We first derive a recursive formula that will help us calculate the derivative of
PtV f. We have
PtV f
s=t
= Pt f + Pt−s PsV f s=0
Z t
∂
(Pt−s PsV f) ds
= Pt f +
∂s
0
Z t
= Pt f +
[−A(Pt−s PsV f) + Pt−s((A + V )PsV f)] ds .
0
Hence
PtV f = Pt f +
Z
0
t
Pt−s (V PsV f) ds .
68
Electronic Communications in Probability
Now we use our formula for differentiation (4) to calculate the derivative of this semigroup.
We have
Z t
DPtV f(x)(v) = DPt f(x)(v) +
DPt−s (V PsV f)(x)(v) ds
0
Z t
1
E{f(Xt )
Y(Xs )(Vs ) dBs }
=
t
0
Z t−s
Z t
1
E{V (Xt−s )PsV f(Xt−s )
Y(Xu )(Vu ) dBu }ds .
+
0
0 t−s
Then, by the Feynman-Kac formula and the Markov property we have
Z t
1
V
DPt f(x)(v) =
E{f(Xt )
Y(Xs )(Vs ) dBs }
t
0
Z t−s
Z t
Rt
1
E{V (Xt−s )E{f(Xt )e t−s V (Xu )du }
Y(Xu )(Vu ) dBu } ds ,
+
0
0 t−s
from which we obtain
kDPtV fk∞
≤
1
kfk∞ E |
t
Z
t
0
+kV k∞ kfk∞
Y(Xs )(Vs ) dBs |
Z
t
0
eskV k∞
E|
t−s
Z
0
t−s
Y(Xu )(Vu ) dBu | ds .
And hence for small t
kDPtV fk∞
≤
=
Z t
ckfk∞
c
√
√
+ kV k∞ kfk∞ etkV k∞
ds
t−s
t
0
√
ckfk∞
√
+ 2c t kV k∞ kfk∞ etkV k∞ .
t
Observe again that our estimate goes to infinity as t → 0.
3
Bounded Smooth Drift
Let Z : Rn → Rn be a bounded smooth vector field. We shall consider another perturbation
to the generator A. This time we define the linear operator
AZ = A + Z · ∇ .
The existence of a semigroup {PtZ }t≥0 having AZ as infinitesimal generator is guaranteed by
the regularity of Z. Indeed, if we write Z(x) = (Z 1 (x), . . . , Z n (x)), then the operator AZ can
be written as
AZ =
n
n
X
∂2
∂
1X
(X(x)X(x)∗ )ij i j +
(Ai (x) + Z i (x)) i ,
2 i,j=1
∂x ∂x
∂x
i=1
and this operator is the infinitesimal generator associated with the equation
dXt
X0
= X(Xt )dBt + [A(Xt ) + Z(Xt )]dt ,
= x ∈ Rn .
Diffusion Semigroups
69
Thanks to the smoothness of Z, this equation yields a diffusion process (Xtx,Z )t∈T and hence
the semigroup PtZ f(x) = E(f(Xtx,Z )). Then previous estimate applies also to kDPtZ fk∞ .
But we can do better because we can find the explicit dependence of the estimate upon Z as
follows. As before, we first find a recursive formula for this semigroup.
PtZ f
s=t
= Pt f + Pt−s PsZ f s=0
Z t
∂
(Pt−s PsZ f) ds
= Pt f +
0 ∂s
Z t
[−A(Pt−s PsZ f) + Pt−s ((A + Z · ∇)PsZ f) ] ds .
= Pt f +
0
Hence
PtZ f
= Pt f +
Z
0
t
Pt−s (Z · ∇PsZ f) ds .
(6)
We can now calculate its derivative as follows
Z t
DPtZ f(x)(v) = DPt f(x)(v) +
DPt−s (Z · ∇PsZ f)(x)(v) ds
0
Z t
1
Y(Xs )(Vs ) dBs }
E{f(Xt )
=
t
0
Z t−s
Z t
1
E{Z(Xt−s ) · ∇PsZ f(Xt−s )
Y(Xu )(Vu ) dBu } ds .
+
0
0 t−s
We now find an estimate for the supremum norm of this derivative. Taking modulus we obtain
Z t
1
|DPtZ f(x)(v)| ≤
kfk∞ E |
Y(Xs )(Vs ) dBs |
t
0
Z t−s
Z t
1
E |DPsZ f(Xt−s )(Z(Xt−s ))| |
+
Y(Xu )(Vu ) dBu | ds .
0
0 t−s
Hence for small t
kDPtZ fk∞
c
≤ √ kfk∞ + ckZk∞
t
Z
0
t
kDPsZ fk∞
√
ds .
t−s
We now solve this inequality. If we iterate once we obtain
kDPtZ fk∞
Z t
c
ds
2
p
≤ √ kfk∞ + c kZk∞ kfk∞
t
s(t − s)
0
Z tZ s
Z
kDPu fk∞
p
+ c2 kZk2∞
du ds .
(t − s)(s − u)
0
0
By Fubini’s theorem, the double integral becomes
Z t
Z t
ds
Z
p
du ,
kDPu fk∞
(t − s)(s − u)
0
u
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Electronic Communications in Probability
and then we observe that
Z t
u
ds
p
= 2 tan−1
(t − s)(s − u)
r
t
s − u
= π.
t − s u
The case u = 0 solves also the first integral. Hence our inequality reduces to
c
2
kDPtZ fk∞ ≤ √ kfk∞ + c2 πkZk∞ kfk∞ + c2 πkZk∞
t
Z
t
0
kDPuZ fk∞ du .
We now apply Gronwall’s inequality. After some simplifications ( extending the integral up to
infinity ) we finally obtain the estimate
2
c
kDPtZ fk∞ ≤ √ kfk∞ + 2c2 πkZk∞ kfk∞ et(c πkZk∞ )
t
(7)
As expected, our estimate goes to infinity as t → 0 since P0Z f = f is not necessarily differentiable.
4
Bounded Uniformly Continuous Drift
We now find a similar estimate when Z : Rn → Rn is only bounded and uniformly continuous.
We look again at the operator AZ = A + Z · ∇. The problem here is that in this case we do
not have the semigroup {PtZ f}t≥0 since the stochastic equation with the added nonsmooth
drift Z might not have a strong solution. So we cannot even talk about its derivative. To solve
this problem we proceed by approximation.
4.1
Existence of Semigroup
Since Z ∈ BC 0 (Rn ; Rn ) is uniformly continuous, and BC ∞ (Rn ; Rn ) is dense in BC 0 (Rn ; Rn ),
∞
n
n
there exists a sequence {Zi}∞
i=1 in BC (R ; R ) such that Zi converges to Z uniformly. Thus,
for every i ∈ N, we have the semigroup {PtZi }t≥0 since our stochastic equation with the added
smooth drift Zi has a strong solution.
∞
For every t ≥ 0 and f ∈ BC 0 (Rn ) fixed, the sequence of functions {PtZi f}i=1 is a Cauchy
sequence in the Banach space BC 0 (Rn ). We will prove this fact later. Let us denote its limit
by PtZ f. All properties required for PtZ f are inherited from those of the semigroup PtZi f.
Indeed by simply writing PtZ f = limi→∞ PtZi f and using an interchange of limits we can prove
1. f 7→ PtZ f is a bounded linear operator.
2. t 7→ PtZ f is a contraction semigroup of operators.
3. {PtZ }t≥0 has generator AZ = A + Z · ∇.
∞
Now, suppose for a moment that the sequence of derivatives {DPtZi f}i=1 converges uniformly.
Then we would have D(limi→∞ PtZi f) = limi→∞ DPtZi f. This proves that PtZ f is differentiable. Then, for any ǫ > 0 there exists N ∈ N such that if i ≥ N , kDPtZ fk∞ ≤ kDPtZi fk∞ +ǫ
and therefore our estimate also applies to kDPtZ fk∞ .
Diffusion Semigroups
4.2
71
Uniform Convergence of Derivatives
∞
We now prove that the sequence {DPtZi f}i=1 converges uniformly. We use again our recursive
formula (6) to obtain
Z t
Z
PtZi f − Pt j f =
Pt−s (Zi · ∇PsZi f − Zj · ∇PsZj f) ds .
0
We now use our formula for differentiation (4). After differentiating and taking modulus we
obtain
Z
|D(PtZi f − Pt j f)(x)(v)|
Z t
1
E{|DPsZi f(Xt−s )(Zi (Xt−s )) − DPsZj f(Xt−s )(Zj (Xt−s ))|
≤
t
−
s
0
Z t−s
|
Y(Xu )(Vu ) dBu |} ds .
0
Thus
kD(PtZi f
Z
− Pt j f)k∞
≤ kZi − Zj k∞
Z t
+ kZj k∞
0
And hence for small t
t
k p a(t−s)
e
− 1kDPsZi fk∞ ds
0 t−s
k p a(t−s)
e
− 1 kDPsZi f − DPsZj fk∞ ds .
t−s
Z
Z t
k
√
≤ kZi − Zj k∞
kDPsZi fk∞ ds
t
−
s
0
Z t
k
√
kDPsZi f − DPsZj fk∞ ds .
+ kZj k∞
t
−s
0
Z
kD(PtZi f − Pt j f)k∞
Now write estimate (7) as
A
kDPtZi fk∞ ≤ √ + B eC t ,
t
for some positive constants A, B and C depending on kfk∞ and kZi k∞ . Thus, substituting
this in our last estimate gives
Z t
A
k
Z
√
√ ds
kD(PtZi f − Pt j f)k∞ ≤ kZi − Zj k∞
t−s s
0
Z t
k
√
B eC s ds
+ kZi − Zj k∞
t−s
0
Z t
k
√
kDPsZi f − DPsZj fk∞ ds .
+ kZj k∞
t
−
s
0
The first two integrals are bounded if we allow t to move within a finite interval (0, T ]. Thus
there exist positive constants M1 and M2 such that
Z
kD(PtZi f − Pt j f)k∞
≤ M1 kZi − Zj k∞
Z t
1
√
+ M2
kDPsZi f − DPsZj fk∞ ds .
t−s
0
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Electronic Communications in Probability
Now iterate this to obtain
Z
kD(PtZi f − Pt j f)k∞
≤ M1 kZi − Zj k∞
+ M2 M1 kZi − Zj k∞
+ M22
Z tZ
0
s
Z
kD(PuZi f
p
0
t
0
√
1
ds
t−s
Z
− Pu j f)k∞
(t − s)(s − u)
du ds .
As before the double integral reduces to
Z t
2
πM2
kD(PuZi f − PuZj f)k∞ du .
0
Collecting constants into new constants M and N we arrive at
Z t
Z
kD(PtZi f − Pt j f)k∞ ≤ M kZi − Zj k∞ + N
kD(PuZi f − PuZj f)k∞ du .
0
Now we apply again Gronwall’s inequality to obtain
Z
kD(PtZi f − Pt j f)k∞ ≤ M kZi − Zj k∞ + N M kZi − Zj k∞
Z
t
eN(t−u) du .
0
∞
The right hand side goes to zero as kZi − Zj k∞ → 0. This proves the sequence {DPtZi }i=1 is
uniformly convergent.
4.3
Uniform Convergence of Semigroups
∞
We finally prove that the sequence of functions {PtZi f}i=1 is a Cauchy sequence. From our
recursive formula (6) we obtain
Z t
Z
kPtZi f − Pt j fk∞ ≤
kZi · ∇PsZi f − Zj · ∇PsZj fk∞ ds
0
Z t
≤
kZi − Zj k∞ kDPsZi fk∞ ds
0
Z t
+
kZj k∞ kDPsZi f − DPsZj fk∞ ds
0
Z t
√
≤ kZi − Zj k∞
k eas − 1 ds
0
+kZj k∞
Z
0
t
kDPsZi f − DPsZj fk∞ ds .
The first integral is bounded so the first part goes to zero as kZi − Zj k∞ approaches 0. The
second part also goes to zero since we just proved the sequence of derivatives is a Cauchy
∞
sequence. Hence {PtZi f}i=1 is uniformly convergent.
Thus, with this approximating procedure we found a semigroup for the operator A + Z · ∇ for
Z bounded and uniformly continuous and we proved it is differentiable and that our estimate
also applies to its derivative.
Diffusion Semigroups
73
Observe that the boundedness and uniform continuity of Z are required in order to ensure
the existence of a sequence of smooth vector fields Zi uniformly convergent to Z. Alternative
assumptions on Z that guarantee the existence of such approximating sequence may be used.
Appendix
We here give a proof of inequality (1) which is taken from [6]. By using the Cauchy-Schwarz
inequality and then the isometric property, we have
E|
t
Z
0
Z
t
Y(xs )(vs ) dBs | ≤ ( E|
Y(xs )(vs ) dBs |2 )1/2
0
Z t
2
= (E
kY(xs )(vs )k ds )1/2
0
Z t
2
Ekvs k ds )1/2 .
≤ kYk2∞(
(8)
0
We will estimate the right-hand side of the last inequality. Itˆ
o’s formula applied to the function
2
f(·) = k · k : Rn → R and the semimartingale (vt )t≥0 yields
kvs k2 = kv0 k2 + 2
Z
s
vu dvu +
0
n Z
X
i=1
0
s
dhvi iu .
Therefore
E kvs k
2
2
= kv0 k + Ehvis
Z s
= kv0 k2 + E
[DX(xu )(vu )] [DX(xu )(vu )]∗ du
0
Z s
= kv0 k2 + E
kDX(xu )(vu )k2 du
0
Z s
2
2
≤ kv0 k + kDXk∞
E kvu k2 du .
0
We now use Gronwall’s inequality to obtain
E kvs k2 ≤ kv0 k2 ekDXk∞ s .
Integrating from 0 to t yields
Z
0
t
E kvs k2 ds ≤
kv0 k2
(ekDXk∞ t − 1) ,
kDXk∞
and thus, substituting in (8) we obtain
E|
Z
0
t
Y(xs )(vs ) dBs | ≤ kYk2∞
This proves inequality (1).
kv0 k
p
et kDXk∞ − 1 .
1/2
kDXk∞
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Electronic Communications in Probability
Acknowledgments
I would like to thank Prof. K. D. Elworthy at Warwick from whom I have enormously benefitted. Also thanks to Dr. H. Zhao and Prof. A. Truman at Swansea for very helpful suggestions. CONACYT has financially supported me for my postgraduate studies at Warwick and
Swansea.
References
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[2] Da Prato, G. and Zabczyk, J. (1997) Differentiability of the Feynman-Kac Semigroup and
a Control Application. Rend. Mat. Acc. Lincei s. 9, v. 8, 183–188.
[3] Elworthy, K. D. (1982) Stochastic Differential Equations on Manifolds. Cambridge University Press.
[4] Elworthy, K. D. and Li, X. M. (1994) Formulae for the Derivative of Heat Semigroups.
Journal of Functional Analysis 125, 252–286.
[5] Elworthy, K. D. and Li, X. M. (1993) Differentiation of Heat Semigroups and applications.
Warwick Preprints 77/1993.
[6] Li, X.-M. (1992) Stochastic Flows on Noncompact Manifolds. Ph.D. Thesis. Warwick
University.
[7] Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential
Equations. Springer-Verlag. New York.
[8] Rinc´
on, L. A. (1994) Some Formulae and Estimates for the Derivative of Diffusion Semigroups. Warwick MSc. Dissertation.