Preprint

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

Ben Goldys and Szymon Peszat


Abstract

Let Psϕ(x)=Eϕ(Xx(s)), be the transition semigroup on the space Bb(E) of bounded measurable functions on a Banach space E, of the Markov family defined by the linear equation with additive noise dX(s)=(AX(s)+a)ds+BdW(s),X(0)=xE. We give a simple probabilistic proof of the fact that null-controlla\-bility of the corresponding deterministic system dY(s)=(AY(s)+BU(t)x)(s))ds,Y(0)=x, implies that for any ϕBb(E), Ptϕ is infinitely many times Fréchet differentiable and that DnPtϕ(x)[y1,,yn]=Eϕ(Xx(t))(1)nItn(y1,,yn), where Itn(y1,,yn) is the symmetric n-fold Itô integral of the controls U(t)y1,U(t)yn.

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

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

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Saturday, October 26, 2024