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

## TL;DR

This paper extends the framework of Itô stochastic differential equations to a class of translation-invariant second-order stochastic PDEs, establishing existence, uniqueness, and the strong Markov property under certain conditions.

## Contribution

It introduces a class of translation-invariant stochastic PDEs that generalize Itô SDEs and proves fundamental properties like existence, uniqueness, and the strong Markov property.

## Key findings

- Existence and uniqueness of strong solutions for the class of stochastic PDEs.
- Application of the monotonicity inequality and Lipschitz conditions.
- Proof of the strong Markov property for these equations.

## 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.