Oleg ButkovskyWeierstrass Institute for Applied Analysis and Stochastics, Berlin
Le 6 févr. 2025
Titre :
New developments in regularization by noise for stochastic differential equations
Résumé :
It is well-known that a differential equation $dX_t =b(X_t)dt$ may have no solutions or infinitely many solutions if the driving vector field $b$ is non-regular, e.g., $b(x)=|x|^\alpha$ with $\alpha\in(0,1)$. On the other hand, if an extra source of random noise is injected into the system, the corresponding stochastic differential equation $dX_t = b(X_t) dt +d\xi_t$, where $\xi$ is a random noise, may have a unique solution. This is called ``regularization by noise''.
While this phenomenon is well understood in the case of Brownian forcing, much less is known when the forcing is non-Markovian (for example, fractional Brownian motion) or infinite-dimensional (for example, space-time white noise). This is not because regularization by noise is inherently specific to the Brownian case, but rather because there are few tools available to study this problem in other settings, and PDE techniques become unavailable.
We will discuss new methods, including stochastic sewing and its latest modifications, which effectively tackle this problem in the non-Brownian setting.
