Backward uniqueness of stochastic parabolic like equations driven by Gaussian multiplicative noise

TL;DR

This paper establishes backward uniqueness for stochastic parabolic equations driven by Gaussian noise and explores applications to controllability, using a logarithmic convexity approach in Hilbert space settings.

Contribution

It introduces a novel backward uniqueness result for stochastic parabolic and Navier-Stokes equations with multiplicative Gaussian noise, employing a logarithmic convexity method.

Findings

Backward uniqueness holds for stochastic parabolic equations.

Applications demonstrated in approximate controllability.

Method extends to tamed Navier-Stokes equations.

Abstract

One proves here the backward uniqueness of solutions to stochastic semilinear parabolic equations and also for the tamed Navier-Stokes equations driven by linearly multiplicative Gaussian noises. Applications to approximate controllability of nonlinear stochastic parabolic equations with initial controllers are given. The method of proof relies on the logarithmic convexity property known to hold for solutions to linear evolution equations in Hilbert spaces with self-adjoint principal part.