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 sigma (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 sigma (ij) a similar to... , b (i) a similar to... are assumed to be locally Lipshitz.Here a similar to... denotes convolution and is the distribution which on functions, is realised by the formula . 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.