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arXiv · 2106.15186

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

Shenglan Yuan·Dirk Blömker
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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.

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Shenglan Yuan, Dirk Blömker. 2021-06-29. Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindrical {\alpha}-stable L\'evy processes. https://arxiv.org/abs/2106.15186

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