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Average dissipation for stochastic transport equations with Lévy noise

Franco Flandoli, Andrea Papini, Marco Rehmeier

Abstract

We show that, in one spatial and arbitrary jump dimension, the averaged solution of a Marcustype SPDE with pure jump Lévy transport noise satisfies a dissipative deterministic equation involving a fractional Laplace-type operator. To this end, we identify the correct associated Lévy measure for the driving noise. We consider this a first step in the direction of a non-local version of enhanced dissipation, a phenomenon recently proven to occur for Brownian transport noise and the associated local parabolic PDE by the first author. Moreover, we present numerical simulations, supporting the fact that dissipation occurs for the averaged solution, with a behavior akin to the diffusion due to a fractional Laplacian, but not in a pathwise sense.

Average dissipation for stochastic transport equations with Lévy noise

Abstract

We show that, in one spatial and arbitrary jump dimension, the averaged solution of a Marcustype SPDE with pure jump Lévy transport noise satisfies a dissipative deterministic equation involving a fractional Laplace-type operator. To this end, we identify the correct associated Lévy measure for the driving noise. We consider this a first step in the direction of a non-local version of enhanced dissipation, a phenomenon recently proven to occur for Brownian transport noise and the associated local parabolic PDE by the first author. Moreover, we present numerical simulations, supporting the fact that dissipation occurs for the averaged solution, with a behavior akin to the diffusion due to a fractional Laplacian, but not in a pathwise sense.
Paper Structure (7 sections, 3 theorems, 24 equations, 3 figures)

This paper contains 7 sections, 3 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Stochastic transport equations of Marcus-type
  3. Special cases
  4. Averaged enhanced dissipation
  5. Numerical results
  6. Open questions
  7. Acknowledgements.

Key Result

Proposition 2.2

If $u_0 \in C^2_b(\mathbb{R}^d)$ and $\sigma\in C^4_b(\mathbb{R}^d,\mathbb{R}^{d\times m})$, then there is a unique solution to M-SPDE, and it is given by where $(t,x)\mapsto \varphi_{t,0}(x)$ denotes the inverse of the stochastic flow of the Marcus-SDE

Figures (3)

  • Figure 1: Solution trajectory at several times
  • Figure 2: Averaged solution to equation \ref{['M-SPDE']}
  • Figure 3: Nonlinear regression on \ref{['eq:det-eq']}

Theorems & Definitions (6)

  • Definition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Remark 3.3