Limiting behavior of inertial manifolds for stochastic differential equations driven by non-Gaussian Levy noise
Longyu Wu, Ji Shu
TL;DR
The paper addresses the limiting behavior of stochastic differential equations in a Hilbert space when driven by non-Gaussian $\alpha$-stable Lévy noise as $\alpha$ approaches 2, i.e., the Gaussian Brownian limit. It develops a random dynamical systems framework and constructs $C^{1}$ inertial manifolds for both the $\alpha$-stable and Brownian systems, proving that the manifolds converge in probability to the Gaussian case as $\alpha\to2$. The key contributions are: (i) a rigorous demonstration that solutions of the $\alpha$-stable system converge to Brownian-driven solutions, (ii) the existence of $C^{1}$ inertial manifolds for both systems under a spectral-gap condition, and (iii) the convergence in probability of these manifolds and their derivatives to the Brownian inertial manifold. This work provides a robust finite-dimensional reduction and a principled Gaussian limit for long-time dynamics in infinite-dimensional stochastic systems with non-Gaussian noise, with implications for applications in physics, biology, and engineering.
Abstract
In this paper, we study the limiting behavior for stochastic differential equations driven by non-Gaussian alpha-stable Levy noise as alpha approaches 2. We first prove the convergence of solutions for system driven by alpha-stable Levy noise to those of the system driven by Brownian motion. Then we construct the C^1 inertial manifolds for both systems and show that these inertial manifolds converge in probability as alpha rightarrow2.