LDP and CLT for SPDEs with Transport Noise

TL;DR

This paper investigates the large deviations and Gaussian fluctuations of solutions to stochastic PDEs with transport noise, demonstrating convergence to deterministic PDEs with viscosity and establishing CLTs with explicit rates in specific models.

Contribution

It introduces new results on large deviations and CLTs for SPDEs with transport noise, using advanced analytical tools and covering both linear transport and 2D Euler equations.

Findings

Established CLTs with explicit convergence rates.

Proved large deviations principles for the models.

Demonstrated convergence to deterministic PDEs with viscosity.

Abstract

In this work we consider solutions to stochastic partial differential equations with transport noise, which are known to converge, in a suitable scaling limit, to solution of the corresponding deterministic PDE with an additional viscosity term. Large deviations and Gaussian fluctuations underlying such scaling limit are investigated in two cases of interest: stochastic linear transport equations in dimension $D\geq 2$ and $2$D Euler equations in vorticity form. In both cases, a central limit theorem with strong convergence and explicit rate is established. The proofs rely on nontrivial tools, like the solvability of transport equations with supercritical coefficients and $\Gamma$-convergence arguments.