Type: Article

Well-Posedness and asymptotics for fractional CGL equations with multiplicative Lévy noise on the boundary and in the bulk

Juárez Campos, Beatriz., Kaikina EI.;

Abstract:

We investigate a class of stochastic complex Ginzburg--Landau (CGL) equations driven by both interior and boundary Lévy noise on multidimensional positive orthants. The deterministic model incorporates a nonlocal diffusion operator given by a Caputo-type fractional Laplacian of order ? ? ( 3 2 , 2 ) , alongside a cubic-type of nonlinearity. Stochastic forcing acts independently on each boundary hyperplane through non-Gaussian jump processes, modeled via Poisson random measures, and within the interior through multiplicative Lévy-type perturbations. We construct mild solutions in weighted Sobolev spaces under minimal regularity assumptions, employing Laplace transform techniques, stochastic convolutions, and trace estimates to handle both nonlocality and boundary singularities. A priori estimates and second moment bounds are established, and local and global well-posedness results are proved for L ? -integrable noise paths. Our analysis reveals how the interaction between Lévy jumps and anisotropic Dirichlet-type boundary conditions modifies the long-time asymptotic behavior of solutions. This work extends existing stochastic PDE theory to a new regime combining nonlocal diffusion, complex-valued nonlinearities, and discontinuous boundary noise, addressing significant analytical challenges associated with multidimensional geometry and jump-induced stochastic dynamics.