Translation Invariant Diffusions and Stochastic Partial Differential Equations in ${\cal S}^{\prime}

Abstract

In this article we show that the ordinary stochastic differential equations of K.It\^{o} maybe considered as part of a larger class of second order stochastic PDE's that are quasi linear and have the property of translation invariance. We show using the `monotonicity inequality' and the Lipshitz continuity of the coefficients $\sigma_{ij}$ and $b_i$, existence and uniqueness of strong solutions for these stochastic PDE's. Using pathwise uniqueness, we prove the strong Markov property.