Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindrical {\alpha}-stable L\'evy processes

TL;DR

This paper develops a method to approximate nonlinear SPDEs driven by cylindrical lpha-stable Le9vy processes using modulation equations, analyzing the effects of jump noise on pattern dynamics near bifurcations.

Contribution

It introduces a novel approximation framework for nonlinear SPDEs with lpha-stable Le9vy noise, accounting for large jumps and their impact on pattern formation.

Findings

Amplitude equations effectively describe bifurcation dynamics.

Large jumps in Le9vy noise can cause significant deviations.

New estimates are developed to handle non-Gaussian noise effects.

Abstract

In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical {\alpha}-stable L\'evy processes via modulation or amplitude equations. We study SPDEs with a cubic nonlinearity, where the deterministic equation is close to a change of stability of the trivial solution. The natural separation of time-scales close to this bifurcation allows us to obtain an amplitude equation describing the essential dynamics of the bifurcating pattern, thus reducing the original infinite dimensional dynamics to a simpler finite-dimensional effective dynamics. In the presence of a multiplicative stable L\'evy noise that preserves the constant trivial solution we study the impact of noise on the approximation. In contrast to Gaussian noise, where non-dominant pattern are uniformly small in time due to averaging effects, large jumps in the L\'evy noise might lead to large error terms, and thus new estimates are needed to take this into account.