Differentiability of transition semigroup of generalized Ornstein-Uhlenbeck process: a probabilistic approach
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arXiv
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| Format: | Preprint |
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2024
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| author | Goldys, Ben Peszat, Szymon |
| author_facet | Goldys, Ben Peszat, Szymon |
| contents | 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 $\mathcal{U}(t)y_1,\ldots \mathcal{U}(t)y_n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_20074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Differentiability of transition semigroup of generalized Ornstein-Uhlenbeck process: a probabilistic approach Goldys, Ben Peszat, Szymon Probability Analysis of PDEs 60H10, 60H15 60H17, 35B30, 35G15 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 $\mathcal{U}(t)y_1,\ldots \mathcal{U}(t)y_n$. |
| title | Differentiability of transition semigroup of generalized Ornstein-Uhlenbeck process: a probabilistic approach |
| topic | Probability Analysis of PDEs 60H10, 60H15 60H17, 35B30, 35G15 |
| url | https://arxiv.org/abs/2410.20074 |