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Advanced seminar AG Mathematical Modeling

Monday, 16.12.2024 at 17.00 in B7 - 210

 

Talk by Katharina Klioba (Hamburg) on the topic "Optimal pathwise uniform convergence rates in time for hyperbolic SPDEs"

Abstract:

In this talk, I will present optimal bounds for the pathwise uniform strong error arising from temporal discretization of semi-linear hyperbolic stochastic evolution equations. Up to a square-root-logarithmic factor, we recover the convergence rates for the whole path from the semigroup corresponding to the semi-linear SPDE with globally Lipschitz nonlinearity and noise. This extends and improves previous results from exponential Euler to general contractive time discretization schemes and from the group to the semigroup case. Furthermore, the square-root-logarithmic factor is shown to be optimal.

 

We illustrate how novel maximal inequalities for stochastic convolutions and path regularity results were used to obtain these results, which are applicable to a large class of hyperbolic equations. As an example, we discuss the convergence rates of implicit and exponential Euler for the nonlinear Schrödinger equation with multiplicative noise.

 

This talk is based on joint work with Mark Veraar (TU Delft).

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