Background Vlasov equations and Young measures for passive scalar and vector advection equations under special stochastic scaling limits

TL;DR

This paper introduces stochastic Vlasov equations to analyze fluctuations in passive scalar and vector advection equations under stochastic scaling, revealing new statistical insights beyond deterministic limits.

Contribution

It develops stochastic Vlasov equations that provide detailed statistical information on fluctuations and oscillations in passive advection models, extending previous convergence results.

Findings

Convergence of stochastic models to Young measures satisfying PDEs with diffusion.

Introduction of stochastic Vlasov equations for detailed fluctuation analysis.

New statistical insights for passive vector fields in stochastic advection.

Abstract

In the last few years it was proved that scalar passive quantities subject to suitable stochastic transport noise, and more recently that also vector passive quantities subject to suitable stochastic transport and stretching noise, weakly converge to the solutions of deterministic equations with a diffusion term. In the background of these stochastic models, we introduce stochastic Vlasov equations which gives additional information on the fluctuations and oscillations of solutions: we prove convergence to non-trivial Young measures satisfying limit PDEs with suitable diffusion terms. In the case of a passive vector field the background Vlasov equation adds completely new statistical information to the stochastic advection equation.