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

Ben Goldys, Szymon Peszat

Abstract

Let $P_sφ(x)=\mathbb{E}\, φ(X^x(s))$, 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 $$ d X(s)= \left(AX(s) + a\right)ds + BdW(s), \qquad X(0)=x\in E. $$ We give a simple probabilistic proof of the fact that null-controlla\-bility of the corresponding deterministic system $$ d Y(s)= \left(AY(s)+ B\mathcal{U}(t)x)(s)\right)ds, \qquad Y(0)=x, $$ implies that for any $φ\in B_b(E)$, $P_tφ$ is infinitely many times Fréchet differentiable and that $$ D^nP_tφ(x)[y_1,\ldots ,y_n]= \mathbb{E}\, φ(X^x(t))(-1)^nI^n_t(y_1,\ldots, y_n), $$ where $I^n_t(y_1,\ldots,y_n)$ is the symmetric n-fold Itô integral of the controls $\mathcal{U}(t)y_1,\ldots \mathcal{U}(t)y_n$.

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

Abstract

Let , be the transition semigroup on the space of bounded measurable functions on a Banach space , of the Markov family defined by the linear equation with additive noise We give a simple probabilistic proof of the fact that null-controlla\-bility of the corresponding deterministic system implies that for any , is infinitely many times Fréchet differentiable and that where is the symmetric n-fold Itô integral of the controls .

Paper Structure

This paper contains 17 sections, 19 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Formulation and main results
  3. Some definitions
  4. $n$-fold symmetric Itô's integral
  5. General result
  6. Gradient estimate implies null-controllability
  7. Conditions for null-controllability
  8. The case of invertible $B$
  9. Analytic semigroup on a Hilbert space and degenerate noise
  10. Examples
  11. Finite dimensional degenerate linear system
  12. Kolmogorov's diffusion
  13. The case of boundary noise
  14. Well posedness
  15. The case of white noise
  16. ...and 2 more sections

Key Result

Theorem 2.6

Assume that the deterministic system E12 is null-controllable at time $T$. Then for $\phi\in B_b(E)$, $t\ge T$ and $x\in E$ we have Moreover, for any $n$ and $\phi\in B_b(E)$, $P_t\phi$ is n-times differentiable, and Finally, we have

Theorems & Definitions (40)

  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • Corollary 2.8
  • proof
  • Corollary 2.9
  • ...and 30 more