The Martingale Problem in Hilbert Spaces

Da Prato, Giuseppe ; Tubaro, Luciano

Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 839-855

Résumé

We consider an SPDE in a Hilbert space $H$ of the form $dX(t) = ( AX(t) + b(X(t)) ) \, dt + \sigma(X(t)) \, dW(t)$, $X(0) = x \in H$ and the corresponding transition semigroup $P_t f (x)= \mathbb{E}[ f(X(t, x)) ]$. We define the infinitesimal generator $\bar L$ of $P_t$ through the Laplace transform of $P_t$ as in [1]. Then we consider the differential operator $L\varphi = \frac{1}{2} \operatorname{Tr}[\sigma(x)\sigma^*(x)D^2\varphi] + \langle b(x), D\varphi \rangle$ defined on a suitable set $V$ of regular functions. Our main result is that if $V$ is a core for $\bar L$, then there exists a unique solution of the martingale problem defined in terms of $L$. Application to the Ornstein-Uhlenbeck equation and to some regular perturbation of it are given.