A study of Landau damping with random initial inputs

TL;DR

This paper proves that for the Vlasov-Poisson equation with random initial data, the Landau damping solution maintains smooth dependence on randomness over time, extending deterministic results to stochastic settings.

Contribution

It generalizes the deterministic contraction argument to stochastic function spaces, showing persistence of regularity in random variables for long-time solutions.

Findings

Dependence of solutions on random inputs is smooth under certain conditions.

Random space regularity persists over long times in nonlinear kinetic equations.

Extension of deterministic Landau damping results to stochastic initial data.

Abstract

For the Vlasov-Poisson equation with random uncertain initial data, we prove that the Landau damping solution given by the deterministic counterpart (Caglioti and Maffei, {\it J. Stat. Phys.}, 92:301-323, 1998) depends smoothly on the random variable if the time asymptotic profile does, under the smoothness and smallness assumptions similar to the deterministic case. The main idea is to generalize the deterministic contraction argument to more complicated function spaces to estimate derivatives in space, velocity and random variables. This result suggests that the random space regularity can persist in long-time even in time-reversible nonlinear kinetic equations.