Regularization by noise for SDEs and SPDEs beyond the Brownian case

The talk is based on joint works with Siva Athreya, Leonid Mytnik and Khoa Le. It is well-known that an SDE $$dX_t = b(X_t) dt +dW_t$$might have a unique solution even if the corresponding noiseless ODE$$dX_t =b(X_t)dt $$has no or infinitely many solutions. This is called regularization-by-noise. While this phenomenon is quite well understood in the case of Brownian forcing, much less is known if the forcing is non-Markovian (for example, fractional Brownian) or infinite-dimensional. This happens not because regularization-by-noise is specific to the Brownian case, but rather because there are (almost) no appropriate tools to study this problem in other setups. We will talk about new technique, stochastic sewing, and its latest modifications, which is surprisingly effective in tackling this problem in the non-Brownian setting.