A mild Girsanov formula

Abstract

We consider a well posed SPDE$\colon dZ=(AZ+b(Z)) dt+dW(t),\,Z_0=x, $ on a separable Hilbert space $H$, where $A\colon H\to H$ is self-adjoint, negative and such that $A^{-1+\beta}$ is of trace class for some $\beta>0$, $b\colon H\to H$ is Lipschitz continuous and $W$ is a cylindrical Wiener process on $H$. We denote by $W_A(t)=\int_0^te^{(t-s)A}\,dW(s),\,t\in[0,T],$ the stochastic convolution. We prove, with the help of a formula for nonlinear transformations of Gaussian integrals due to R. Ramer, the following identity $$(P\circ Z_x^{-1})(\Phi) =\int_X\Phi(h+e^{\cdot A}x)\, \exp\left\{ -\tfrac12|\gamma_x(h)|^2_{ H_{Q_T}} + I(\gamma_x)(h)\right\} N_{Q_T}(dh), $$ where $ N_{Q_T}$ is the law of $W_A$ in $C([0,T],H)$, $ H_{Q_T}$ its Cameron--Martin space, $$ [\gamma_x(k)](t)=\int_0^t e^{(t-s)A}b(k(s)+e^{sA}x) ds,\quad t\in[0,T], \; k \in C([0,T],H) $$ and $I(\gamma_x) $ is the It\^o integral of $\gamma_x$. Some applications are discussed; in particular, when $b$ is dissipative we provide an explicit formula for the law of the stationary process and the invariant measure $\nu$ of the Markov semigroup $(P_t)$. Some concluding remarks are devoted to a similar problem with colored noise.