Translation invariant diffusions in the space of tempered distributions

In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients σ _ij , b _i 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 σ _ij $\tilde y$ , b _i $\tilde y$ 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.