A BSDE approach to regularization by noise of semilinear stochastic evolution equations

Abstract:

We prove strong well-posedness for a class of stochastic evolution equations in Hilbert spaces H when the drift term is Hölder continuous. This class includes examples of semilinear stochastic damped equations which describe elastic systems with structural damping 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 Itô formula and the associated Kolmogorov equation (the so-called Zvonkin transformation or Itô-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.
The talk is based on a joint work with D. Addona and E. Priola.