ON THE SMOLUCHOWSKI-KRAMERS APPROXIMATION OF STOCHASTIC DAMPED WAVE EQUATIONS

Abstract

In this thesis, we study three problems related to the small mass limit, also known as the Smoluchowski-Kramers diffusion approximation for stochastic damped wave equations. In the first part, we study the validity of a large deviation principle for a class of stochastic nonlinear damped wave equations, including equations of Klein-Gordon type, in the joint small mass and small noise limit. Additionally, we provide a proof of the Smoluchowski-Kramers approximation in the case of variable friction, non-Lipschitz nonlinear term, and unbounded diffusion. In the second part, we investigate the convergence, in the small mass limit, of the stationary solutions of a class of stochastic damped wave equations, where the friction coefficient depends on the state and the noisy perturbation is of multiplicative type. We demonstrate that the Smoluchowski-Kramers approximation, previously shown to hold for any fixed time interval, remains valid in the long-time regime. Specifically, we prove that the first marginals of any sequence of stationary solutions for the damped wave equation converge to the unique invariant measure of the limiting stochastic quasilinear parabolic equation.

The final result of this thesis concerns the Smoluchowski-Kramers approximation for a system of stochastic damped wave equations, whose solution is constrained to live on the unitary sphere in the space of square-integrable functions on any fixed interval. The stochastic perturbation is a nonlinear multiplicative Gaussian noise, with the stochastic differential interpreted in Stratonovich sense. Due to its particular structure, this noise not only conserves almost surely the constraint, but also preserves a suitable energy functional. In the small-mass limit, we derive a deterministic system, that remains confined to the unit sphere, but includes additional terms. These terms depend on the reproducing kernel of the noise and account for the interaction between the constraint and the conservative noise.