PDEA Seminar: Singular limits for stochastic equations

Abstract: We study singular limits of nonlinear stochastic evolution equations in the interplay of disappearing strength of the noise and increasing roughness of the noise, so that the noise in the limit would be too rough to define a solution to the limiting equations. Simultaneously, the limit is singular in the sense that the leading order differential operator may vanish. Although the noise is disappearing in the limit, additional deterministic terms appear due to renormalization effects. This effect has first been observed by Hairer, Ryser and Weber, Electron. J. Probab. (2012). We give an abstract framework for the main error estimates, that first reduce to bounds on a residual and in a second step to bounds on the stochastic convolution. Moreover, we apply it to a singularly regularized Allen-Cahn equation and the Cahn-Hilliard/Allen-Cahn homotopy.

See article in Stochastics and Dynamics (2023+)

Joint work with Dirk Blömker, University of Augsburg, Germany.

This is a seminar in the PDEs and Applications seminar series