Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindrical α-stable Lévy processes

Abstract

In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical α-stable Lévy 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 Lévy 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 Lévy noise might lead to large error terms, and thus new estimates are needed to take this into account.

Amplitude equations for SPDEs driven by fractional additive noise with small Hurst parameter

“…Multiplicative Wiener noise was studied by [9], while for additive fractional noise with H > 1/2 certain results for Rayleigh-Bénard convection are available in [4], but these results rely heavily on the smoothness of the noise and fail even in the case H = 1/2. Additive α-stable Lévy noise was studied recently in [11], where the problem is not only the lack of smoothness, but also the lack of moments.…”