Differentiability of transition semigroup of generalized Ornstein-Uhlenbeck process: a probabilistic approach

Ben Goldys and Szymon Peszat

Abstract

Let \(P_s\phi(x)=\mathbb{E}\, \phi(X^x(s))\), be the transition semigroup on the space \(B_b(E)\) of bounded measurable functions on a Banach space \(E\), of the Markov family defined by the linear equation with additive noise \[ d X(s)= \left(AX(s) + a\right)d s + B\mathrm{d}W(s), \qquad X(0)=x\in E. \] We give a simple probabilistic proof of the fact that null-controlla\-bility of the corresponding deterministic system \[ d Y(s)= \left(AY(s)+ B\mathcal{U}(t)x)(s)\right)d s, \qquad Y(0)=x, \] implies that for any \(\phi\in B_b(E)\), \(P_t\phi\) is infinitely many times Fréchet differentiable and that \[ D^nP_t\phi(x)[y_1,\ldots ,y_n]= \mathbb{E}\, \phi(X^x(t))(-1)^nI^n_t(y_1,\ldots, y_n), \] where \(I^n_t(y_1,\ldots,y_n)\) is the symmetric n-fold Itô integral of the controls \(\mathcal{U}(t)y_1,\ldots \mathcal{U}(t)y_n\).

Keywords: Gradient estimates, Ornstein–Uhlenbeck processes, strong Feller property, hypoelliptic diffusions, noise on boundary.

AMS Subject Classification: Primary 60H10; secondary 60H15 60H17, 35B30, 35G15.