ESTIMATES FOR THE DERIVATIVE OF DIFFUSION SEMIGROUPS

Let f P t g t (cid:21) 0 be the transition semigroup of a di(cid:11)usion process. It is known that P t sends continuous functions into di(cid:11)erentiable functions so we can write DP t f . But what happens with this derivative when t ! 0 and P 0 f = f is only continuous ?. We give estimates for the supremum norm of the Fr(cid:19)echet derivative of the semigroups associated with the operators A + V and A + Z (cid:1) r where A is the generator of a di(cid:11)usion process, V is a potential and Z is a vector (cid:12)eld.


Introduction
Consider the following stochastic differential equation on R n dX t = X(X t ) dB t + A(X t ) dt , X 0 = x ∈ R n , for t ≥ 0, where the first integral is an Itô stochastic integral and the second is a Riemann integral. Here {B t } t≥0 is Brownian motion on R m and the equality holds almost everywhere. The coefficients of this equation are the mapping X : R n → L(R m ; R n ) and the vector field A : R n → R n . Assume standard regularity conditions on these coefficients so that there exists a strong solution {X t } t≥0 to our equation. We write X x t for X t 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 → X x t is differentiable ( see for instance [1] ).
Assume coefficients X and A are smooth enough and consider the associated derivative equation whose solution {V t } t≥0 is the derivative of the mapping x → X x t at x in the direction v. We will assume that there exists a smooth map Y : R n → L(R n ; R m ) such that Y(x) is the right inverse of X(x). That is, X(x)Y(x) = I R n for all x in R n . We shall also assume that the process 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 We are interested in small values for t. Thus, since Hence for sufficiently small t, we have the following estimate where c is a positive constant. Let now BC r (R n ) be the Banach space of bounded measurable functions on R n 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(R n ) is the Banach space of bounded measurable functions on R n with supremum norm f ∞ = sup x∈R n |f(x)|. Suppose our diffusion process {X t } t≥0 has transition probabilities P (t, x, Γ). Then this induces a semigroup of operators {P t } t≥0 as follows. For every t ≥ 0 we define on B(R n ) the bounded linear operator The semigroup {P t } t≥0 is a strongly continuous semigroup on BC 0 (R n ). Denote by A its infinitesimal generator. It is known that {P t } t≥0 is a strong Feller semigroup, that is, P t sends continuous functions into differentiable functions. In fact, under above assumptions, a formula for the derivative of P t f is known ( see [4] or [5] ).
Theorem 2 If f ∈ BC 2 (R n ) then the derivative of P t f : R n → R is given by 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 Hence there exists a unique extension on BC 0 (R n ). 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 (R n ). From last theorem we obtain And hence for small t Observe that, as expected, our estimate goes to infinity as t approaches 0 since P 0 f = f is not necessarily differentiable. Also the rate at which it goes to infinity is not faster than 1 √ t does.

Potential
Let V : R n → R be a bounded measurable function. We shall perturb the generator A by adding to it the function V . We define the linear operator with the same domain as for A. A semigroup {P V t } t≥0 having A V as generator is given by the Feynman-Kac formula P V t f = E{f(X t )e t 0 V (Xu) du }. We will find a similar estimate as (5) for DP V t f ∞ . We first derive a recursive formula that will help us calculate the derivative of P V t f. We have Hence Now we use our formula for differentiation (4) to calculate the derivative of this semigroup. We have Then, by the Feynman-Kac formula and the Markov property we have from which we obtain And hence for small t Observe again that our estimate goes to infinity as t → 0.

Bounded Smooth Drift
Let Z : R n → R n be a bounded smooth vector field. We shall consider another perturbation to the generator A. This time we define the linear operator The existence of a semigroup {P Z t } t≥0 having A Z 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 A Z can be written as and this operator is the infinitesimal generator associated with the equation Thanks to the smoothness of Z, this equation yields a diffusion process (X x,Z t ) t∈T and hence the semigroup P Z t f(x) = E(f(X x,Z t )). Then previous estimate applies also to DP Z t f ∞ . 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. Hence We can now calculate its derivative as follows We now find an estimate for the supremum norm of this derivative. Taking modulus we obtain Hence for small t We now solve this inequality. If we iterate once we obtain By Fubini's theorem, the double integral becomes and then we observe that The case u = 0 solves also the first integral. Hence our inequality reduces to We now apply Gronwall's inequality. After some simplifications ( extending the integral up to infinity ) we finally obtain the estimate As expected, our estimate goes to infinity as t → 0 since P Z 0 f = f is not necessarily differentiable.

Bounded Uniformly Continuous Drift
We now find a similar estimate when Z : R n → R n is only bounded and uniformly continuous. We look again at the operator A Z = A + Z · ∇. The problem here is that in this case we do not have the semigroup {P Z t 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.

Existence of Semigroup
Since Z ∈ BC 0 (R n ; R n ) is uniformly continuous, and BC ∞ (R n ; R n ) is dense in BC 0 (R n ; R n ), there exists a sequence {Z i } ∞ i=1 in BC ∞ (R n ; R n ) such that Z i converges to Z uniformly. Thus, for every i ∈ N, we have the semigroup {P Zi t } t≥0 since our stochastic equation with the added smooth drift Z i has a strong solution. For every t ≥ 0 and f ∈ BC 0 (R n ) fixed, the sequence of functions {P Zi t f}