Invariant measures of stochastic partial differential equations and conditioned diffusions

This work establishes and exploits a connection between the invariant measure of stochastic partial differential equations (SPDEs) and the law of bridge processes. Namely, it is shown that the invariant measure of $u_t=u_{xx}+f(u)+\sqrt{2?}\eta (x\text{,}t)$, where $\eta (x\text{,}t)$ is a space–time white-noise, is identical to the law of the bridge process associated to $\mathrm{d}U=a(U)\mathrm{d}x+\sqrt{?}\mathrm{d}W(x)$, provided that a and f are related by $?a^{″}(u)+2a^{\prime }(u)a(u)=?2f(u)$, $u\in \mathbb{R}$. Some consequences of this connection are investigated, including the existence and properties of the invariant measure for the SPDE on the line, $x\in \mathbb{R}$. To cite this article: M.G. Reznikoff, E. Vanden-Eijnden, C. R. Acad. Sci. Paris, Ser. I 340 (2005).