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: L.Rincon@swansea.ac.uk
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 DPtf. But what happens
with this derivative when t → 0and P0f = 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· ∇whereA is the generator of a diffusion process,V is a potential andZ
is a vector field.
1
Introduction
Consider the following stochastic differential equation onRn
dXt = X(Xt)dBt+A(Xt)dt ,
X0 = x∈Rn,
for t ≥ 0, where the first integral is an Itˆo stochastic integral and the second is a Riemann integral. Here {Bt}t≥0 is Brownian motion onRmand 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 writeXtx forXtwhen 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 mappingx7→Xx
t is differentiable ( see for instance [1] ).
1Supported by a grant from CONACYT (M´exico).
Assume coefficientsXandAare smooth enough and consider the associated derivative equation
dVt = DX(Xt)(Vt)dBt+DA(Xt)(Vt)dt ,
V0 = v∈Rn,
whose solution {Vt}t≥0is the derivative of the mappingx7→Xtx at xin the direction v. We
will assume that there exists a smooth map Y:Rn →L(Rn;Rm) such thatY(x) is the right inverse ofX(x). That is,X(x)Y(x) =IRnfor allxinRn. We shall also assume that the process {Y(Xt)(Vt)}t≥0 belongs toL2([0, t]) for eacht >0, that is, R0t|Y(Xs)(Vs)|2ds <∞ and thus
we can writeRt
0Y(Xs)(Vs)dBs.
With all above assumptions, we have from [6], the following result (see Appendix for a proof)
Theorem 1 For every t >0there exist positive constants kand asuch that
E|
Z t
0 Y
(Xs)(Vs)dBs| ≤k
√
eat−1. (1)
We are interested in small values fort. Thus, since√eat−1 =O(√t) as t→0, we have that
there exist a positive constantN such that√eat−1≤N√tfor smallt. Hence for sufficiently
small t, we have the following estimate
E|
Z t
0 Y
(Xs)(Vs)dBs| ≤c
√
t , (2)
where cis a positive constant.
Let now BCr(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 itsrderivatives. In particularB(Rn) is the Banach space of bounded measurable functions onRn with supremum normkfk∞= supx∈Rn|f(x)|. Suppose our diffusion process{Xt}t≥0has transition
probabili-tiesP(t, x,Γ). Then this induces a semigroup of operators{Pt}t≥0as follows. For everyt≥0
we define onB(Rn) the bounded linear operator
(Ptf)(x) =
Z
Rn
f(y)P(t, x, dy) =E(f(Xtx)). (3)
The semigroup {Pt}t≥0 is a strongly continuous semigroup on BC0(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 Ptf is known ( see [4] or [5] ).
Theorem 2 If f ∈BC2(Rn)then the derivative ofP
tf:Rn →Ris given by
D(Ptf)(x)(v) =
1
tE{f(Xt)
Z t
0 Y
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→
1
tE{f(Xt)
Rt
0Y(Xs)(Vs)dBs} defines a bounded linear functional on BC2(Rn). Hence there
exists a unique extension onBC0(Rn). Since the expression of this linear functional does not depend on the derivatives off, it has the same expression for anyf inBC0(Rn).
From last theorem we obtain
|DPtf(x)(v)| ≤
1
tkfk∞E|
Z t
0 Y
(Xs)(Vs)dBs|
≤ 1
tkfk∞k √
eat−1.
And hence for smallt
kDPtfk∞≤
ckfk∞ √
t . (5)
Observe that, as expected, our estimate goes to infinity astapproaches 0 sinceP0f =f is not
necessarily differentiable. Also the rate at which it goes to infinity is not faster than √1 t does.
2
Potential
Let V:Rn → R be a bounded measurable function. We shall perturb the generator A by adding to it the functionV. We define the linear operator
AV =A+V ,
with the same domain as for A. A semigroup {PtV}t≥0 havingAV as generator is given by
the Feynman-Kac formulaPV
t f =E{f(Xt)e
Rt
0V(Xu)du}. We will find a similar estimate as (5) forkDPV
t fk∞. We first derive a recursive formula that will help us calculate the derivative of
PV
t f. We have
PtVf = Ptf+Pt−sPsVf
s=t
s=0
= Ptf+
Z t
0
∂
∂s(Pt−sP
V s f)ds
= Ptf+
Z t
0
[−A(Pt−sPsVf) +Pt−s((A+V)PsVf)]ds .
Hence
PtVf =Ptf +
Z t
0
Now we use our formula for differentiation (4) to calculate the derivative of this semigroup.
Then, by the Feynman-Kac formula and the Markov property we have
DPtVf(x)(v) = 1
from which we obtain
kDPtVfk∞ ≤
And hence for smallt
kDPtVfk∞ ≤ ck√fk∞
Observe again that our estimate goes to infinity ast→0.
3
Bounded Smooth Drift
LetZ :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 {PZ
t }t≥0 havingAZ as infinitesimal generator is guaranteed by
the regularity ofZ. Indeed, if we writeZ(x) = (Z1(x), . . . , Zn(x)), then the operatorAZ can
and this operator is the infinitesimal generator associated with the equation
dXt = X(Xt)dBt+ [A(Xt) +Z(Xt)]dt ,
Thanks to the smoothness ofZ, this equation yields a diffusion process (Xtx,Z)t∈T and hence
the semigroup PtZf(x) = E(f(Xtx,Z)). Then previous estimate applies also to kDPtZfk∞. But we can do better because we can find the explicit dependence of the estimate uponZ as follows. As before, we first find a recursive formula for this semigroup.
PtZf = Ptf+Pt−sPsZf
We can now calculate its derivative as follows
DPtZf(x)(v) = DPtf(x)(v) +
We now find an estimate for the supremum norm of this derivative. Taking modulus we obtain
|DPtZf(x)(v)| ≤ 1
Hence for smallt
kDPtZfk∞≤√c
We now solve this inequality. If we iterate once we obtain
kDPtZfk∞ ≤
By Fubini’s theorem, the double integral becomes
and then we observe that
Z t
u
ds
p
(t−s)(s−u)= 2 tan
−1
r
s−u t−s
t
u
=π .
The case u= 0 solves also the first integral. Hence our inequality reduces to
kDPtZfk∞≤√c
tkfk∞+c
2πkZk
∞kfk∞+c2πkZk 2 ∞
Z t
0 k
DPuZfk∞du .
We now apply Gronwall’s inequality. After some simplifications ( extending the integral up to infinity ) we finally obtain the estimate
kDPtZfk∞≤
c √
tkfk∞+ 2c
2πkZk
∞kfk∞et(c
2
πkZk∞) (7)
As expected, our estimate goes to infinity ast→0 since P0Zf =f is not necessarily differen-tiable.
4
Bounded Uniformly Continuous Drift
We now find a similar estimate whenZ: Rn→Rnis only bounded and uniformly continuous. We look again at the operatorAZ =A+Z· ∇. The problem here is that in this case we do
not have the semigroup {PtZf}t≥0 since the stochastic equation with the added nonsmooth
driftZ 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
SinceZ∈BC0(Rn;Rn) is uniformly continuous, andBC∞(Rn;Rn) is dense inBC0(Rn;Rn), there exists a sequence{Zi}∞i=1inBC∞(Rn;Rn) such thatZiconverges toZ uniformly. Thus,
for everyi∈N, we have the semigroup{PZi
t }t≥0since our stochastic equation with the added
smooth driftZi has a strong solution.
For every t ≥0 and f ∈ BC0(Rn) fixed, the sequence of functions{PZi
t f}
∞
i=1 is a Cauchy
sequence in the Banach space BC0(Rn). We will prove this fact later. Let us denote its limit by PtZf. All properties required for PtZf are inherited from those of the semigroup PZi
t f.
Indeed by simply writingPtZf = limi→∞PtZif and using an interchange of limits we can prove
1. f 7→PtZf is a bounded linear operator.
2. t7→PZ
t f is a contraction semigroup of operators.
3. {PZ
t }t≥0has generator AZ =A+Z· ∇.
Now, suppose for a moment that the sequence of derivatives{DPZi
t f}
∞
i=1converges uniformly.
Then we would have D(limi→∞PtZif) = limi→∞DPtZif. This proves that PtZf is
differen-tiable. Then, for anyǫ >0 there existsN ∈Nsuch that ifi≥N,kDPtZfk∞≤ kDPtZifk∞+ǫ
4.2
Uniform Convergence of Derivatives
We now prove that the sequence{DPZi
t f}
∞
i=1converges uniformly. We use again our recursive
formula (6) to obtain
PZi
We now use our formula for differentiation (4). After differentiating and taking modulus we obtain
And hence for smallt
kD(PZi
Now write estimate (7) as
kDPZi
this in our last estimate gives
kD(PZi
The first two integrals are bounded if we allowt to move within a finite interval (0, T]. Thus there exist positive constantsM1andM2such that
Now iterate this to obtain
As before the double integral reduces to
πM22
Z t
0k
D(PZi
u f−PuZjf)k∞du .
Collecting constants into new constantsM andN we arrive at
kD(PZi
Now we apply again Gronwall’s inequality to obtain
kD(PZi
4.3
Uniform Convergence of Semigroups
We finally prove that the sequence of functions {PZi
t f}
∞
i=1 is a Cauchy sequence. From our
recursive formula (6) we obtain
kPZi
second part also goes to zero since we just proved the sequence of derivatives is a Cauchy sequence. Hence {PZi
t f}
∞
i=1 is uniformly convergent.
Observe that the boundedness and uniform continuity of Z are required in order to ensure the existence of a sequence of smooth vector fieldsZi uniformly convergent toZ. Alternative
assumptions onZ 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|
Z t
0 Y
(xs)(vs)dBs| ≤ (E|
Z t
0 Y
(xs)(vs)dBs|2)1/2
= (E
Z t
0 k
Y(xs)(vs)k2ds)1/2
≤ kYk2 ∞(
Z t
0
Ekvsk2ds)1/2. (8)
We will estimate the right-hand side of the last inequality. Itˆo’s formula applied to the function f(·) =k · k2:Rn→Rand the semimartingale (vt)t≥0yields
kvsk2=kv0k2+ 2
Z s
0
vudvu+ n
X
i=1
Z s
0
dhviiu.
Therefore
Ekvsk2 = kv0k2+Ehvis
= kv0k2+E
Z s
0
[DX(xu)(vu)] [DX(xu)(vu)]∗du
= kv0k2+E
Z s
0 k
DX(xu)(vu)k2du
≤ kv0k2+kDXk2∞
Z s
0 Ek
vuk2du .
We now use Gronwall’s inequality to obtain
Ekvsk2≤ kv0k2ekDXk∞s.
Integrating from 0 tot yields
Z t
0 Ek
vsk2ds≤ k
v0k2
kDXk∞
(ekDXk∞t
−1),
and thus, substituting in (8) we obtain
E|
Z t
0
Y(xs)(vs)dBs| ≤ kYk2∞ k
v0k
kDXk1∞/2
p
etkDXk∞−1.
Acknowledgments
I would like to thank Prof. K. D. Elworthy at Warwick from whom I have enormously bene-fitted. Also thanks to Dr. H. Zhao and Prof. A. Truman at Swansea for very helpful sugges-tions. 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. Ency-clopedia 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 Uni-versity 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.
