A BSDEs approach to pathwise uniqueness for stochastic evolution equations

TL;DR

This paper establishes strong well-posedness for certain stochastic evolution equations in Hilbert spaces with Holder continuous drifts, using a novel approach involving infinite dimensional FBSDEs, and demonstrates the role of additive noise in restoring pathwise uniqueness.

Contribution

It introduces a new method based on infinite dimensional FBSDEs to prove pathwise uniqueness for SPDEs with Holder continuous drifts, differing from traditional Kolmogorov equation techniques.

Findings

Strong well-posedness for a class of SPDEs with Holder continuous drifts.

Pathwise uniqueness is achieved through additive Wiener noise.

Solutions depend Lipschitz continuously on initial conditions.

Abstract

We prove strong well-posedness for a class of stochastic evolution equations in Hilbert spaces H when the drift term is Holder continuous. This class includes examples of semilinear stochastic damped wave equations which describe elastic systems with structural damping (for such equations even existence of solutions in the linear case is a delicate issue) and semilinear stochastic 3D heat equations. In the deterministic case, there are examples of non-uniqueness in our framework. Strong (or pathwise) uniqueness is restored by means of a suitable additive Wiener noise. The proof of uniqueness relies on the study of related systems of infinite dimensional forward-backward SDEs (FBSDEs). This is a different approach with respect to the well-known method based on the Ito formula and the associated Kolmogorov equation (the so-called Zvonkin transformation or Ito-Tanaka trick). We deal with approximating FBSDEs in which the linear part generates a group of bounded linear operators in H; such approximations depend on the type of SPDEs we are considering. We also prove Lipschitz dependence of solutions from their initial conditions.