Core Concepts
This paper presents and analyzes convergent finite difference schemes for approximating weak L2 solutions of stochastic transport equations with low-regularity velocity fields. The schemes are shown to be stable and convergent under less restrictive conditions than those required for deterministic transport equations.
Abstract
The paper introduces a finite difference scheme for approximating weak L2 solutions of the stochastic transport equation:
du + V · ∇u dt + σ∇u ◦ dW = 0
where V is the velocity field and σ is the noise coefficient. The authors allow for low-regularity velocity fields V, only requiring V ∈ L2 ∩ L∞ with ∂x1V 1, ..., ∂xdV d ∈ Lp for some p > d, in contrast to the typical assumption of div V ∈ L∞.
The key aspects of the analysis are:
The discretization of the noise term σ∇u ◦ dW is designed to preserve the hyperbolic character of the equation, leading to a crucial structural property (1.3) that enables L2 stability.
A discrete duality argument is developed, building on a detailed analysis of a class of backward parabolic difference schemes. This allows the authors to obtain L2 stability estimates that are robust to the low regularity of the velocity field V.
Consistency and convergence of the scheme are established, showing that the numerical approximations converge to the unique weak L2 solution of the stochastic transport equation.
The paper represents one of the first attempts to leverage noise-based regularization to construct consistent and stable numerical schemes for stochastic transport equations with low-regularity coefficients.
Stats
The following sentences contain key metrics or figures:
The velocity field V lies in L2 ∩ L∞(Rd) with ∂x1V 1, ..., ∂xdV d ∈ Lp(Rd) for some p > d.
The noise coefficient σ lies in the Sobolev space W 3,∞(Rd) and satisfies σ(x) ≥ σ0 for all x ∈ Rd for some number σ0 > 0.
The initial data u0 lies in L2(Rd).