Path-dependent infinite-dimensional SDE with non-regular drift: an existence result

TL;DR

This paper proves the existence of weak solutions for a class of infinite-dimensional stochastic differential equations with very general, non-regular drift functions, using entropy-based tightness and variational methods.

Contribution

It introduces a novel approach employing specific entropy as a tightness criterion and describes solutions as variational problems, extending previous results to more general drifts.

Findings

Established existence of weak solutions for non-regular drifts

Used entropy as a tightness tool in infinite dimensions

Extended previous bounded drift results to more general cases

Abstract

We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither bounded or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy and a finite second moment. The originality of our method leads in the use of the specific entropy as a tightness tool and in the description of such infinite-dimensional stochastic process as solution of a variational problem on the path space. Our result clearly improves previous ones obtained for free dynamics with bounded drift.