Translation Invariant Diffusions in the space of tempered distributions

Abstract

In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients $\sigma_{ij},b_i$ and initial condition $y$ in the space of tempered distributions) that maybe viewed as a generalisation of Ito's original equations with smooth coefficients . The solutions are characterized as the translates of a finite dimensional diffusion whose coefficients $\sigma_{ij}\star \tilde{y},b_i\star \tilde{y}$ are assumed to be locally Lipshitz.Here $\star$ denotes convolution and $\tilde{y}$ is the distribution which on functions, is realised by the formula $\tilde{y}(r) := y(-r)$ . The expected value of the solution satisfies a non linear evolution equation which is related to the forward Kolmogorov equation associated with the above finite dimensional diffusion.