A mild Girsanov formula
Giuseppe Da Prato, Enrico Priola, Luciano Tubaro
TL;DR
The paper advances a mild Girsanov framework for stochastic evolution equations on a Hilbert space by combining a nonlinear Gaussian transformation (Ramer identity) with Malliavin calculus. It derives a density representation of the law of the mild solution $Z_x$ in terms of a Gaussian reference measure for the stochastic convolution, expressed through the adapted shift $\gamma_x$ and its Itô integral $I(\gamma_x)$. This yields a concrete semigroup representation and, in the dissipative drift case, explicit descriptions of the stationary law and invariant measures, including a density formula for $\nu$ with respect to the Gaussian baseline $\mu$. The work also extends to colored-noise settings and constructs a stationary two-sided framework on $\mathbb R$, enabling analysis of long-time behavior via the mild Girsanov transform.
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 $A^{-1+β}$ is of trace class for some $β>0$, $b\colon H\to H$ 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 $$(P\circ Z_x^{-1})(Φ) =\int_XΦ(h+e^{\cdot A}x)\, \exp\left\{ -\tfrac12|γ_x(h)|^2_{ H_{Q_T}} + I(γ_x)(h)\right\} N_{Q_T}(dh), $$ where $ N_{Q_T}$ is the law of $W_A$ in $C([0,T],H)$, $ H_{Q_T}$ its Cameron--Martin space, $$ [γ_x(k)](t)=\int_0^t e^{(t-s)A}b(k(s)+e^{sA}x) ds,\quad t\in[0,T], \; k \in C([0,T],H) $$ and $I(γ_x) $ is the Itô integral of $γ_x$. 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 $ν$ of the Markov semigroup $(P_t)$. Some concluding remarks are devoted to a similar problem with colored noise.