Modulation and Amplitude Equations on Bounded Domains for Nonlinear SPDEs Driven by Cylindrical α-stable Levy Processes

In the present work, we establish the approximation via modulation or amplitude equations of the nonlinear stochastic partial differential equation (SPDE) driven by cylindrical alpha-stable Levy processes. 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 simpler finite dimensional effective dynamics. In the presence of a multiplicative stable Levy noise that preserves the constant trivial solution we study the impact of noise on the approximation. In contrast to Gaussian noise, where due to averaging effects nondominant patterns are uniformly small in time, large jumps in the Levy noise might lead to large error terms, and thus new estimates are needed to take this into account.