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 σij, bi and initial condition y in the space of tempered distributions) that may be 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 σij ★ \(\tilde y\), bi ★ \(\tilde y\) are assumed to be locally Lipshitz.Here ★ denotes convolution and \(\tilde y\) is the distribution which on functions, is realised by the formula \(\tilde y\left( r \right): = y\left( { - r} \right)\). 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.