Regularization by noise for linear SPDEs

We say that a regularization by noise phenomenon occurs for a possibly ill-posed differential equation if this equation becomes well-posed under addition of noise. In this talk we show such a regularization for stochastic linear transport-like equations, namely $${\partial }_{t}v+b\cdot \mathrm{\nabla }v+hv+\mathrm{\nabla }v\circ \dot{W}=0,$$ where $b=b(t,x)$, $h=h(t,x)$ are given deterministic, possibly irregular vector fields, $W$ is a $d$-dimensional Brownian motion, $\circ$ denotes Stratonovich integration and $v=v(t,x,\omega )$ is the solution.

We show, under a certain integrability assumption on $b$ and $h$ (the Ladyzhenskaya-Prodi-Serrin integrability condition), that this equation admits a unique distributional solution (in a suitable integrability class), which is also Sobolev regular for regular initial condition. The result is false in general without noise.

The existence of a (Sobolev) regular solution is obtained by a priori estimates: using the renormalization property of the transport equation, we get a system of transport-like SPDEs for the powers of the derivative of the solution $v$; then we use parabolic estimates to bound the expectation of such powers. The uniqueness we get is of pathwise type (actually even stronger than pathwise) and is obtained through a duality method, using the regular solution to the dual SPDE.

These results can be applied to get well-posedness for the associated SDE $$\text{rmd}X=b(X)\text{rmd}t+\text{rmd}W,$$ again for $b$ non-smooth.