Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the zero-noise limit for SDE's singular at the initial time

Jules Pitcho

TL;DR

The paper analyzes the zero-noise limit for SDEs with rough, divergence-free drifts singular at the initial time on the torus. It develops a framework where randomness is retained in the initial data and proves existence and uniqueness of a zero-noise flow η under BV regularity and Prodi–Serrin conditions on the drift. The main result shows η is concentrated on integral curves of b and incompressible for divergence-free fields, and for Depauw's DP field the disintegration is non-Dirac, demonstrating stochasticity in the limit. The work extends Ambrosio's regular Lagrangian flow theory to a broader class of vector fields and provides explicit construction validating sharpness through the DP example.

Abstract

We investigate the zero-noise limit for SDE's driven by Brownian motion with a divergence-free drift singular at the initial time and prove that a unique probability measure concentrated on the integral curves of the drift is selected. More precisely, we prove uniqueness of the zero-noise limit for divergence-free drifts in $L^1_{loc}((0,T];BV(\mathbb{T}^d;\mathbb{R}^d))\cap L^q((0,T);L^p(\mathbb{T}^d;\mathbb{R}^d))$ where $p$ and $q$ satisfy a Prodi-Serrin condition. The vector field constructed by Depauw [C. R. Acad. Sci. Paris, 2003] lies in this class and we show that for almost every intial datum, the zero-noise limit selects a probability measure concentrated on several distinct integral curves of this vector field.

On the zero-noise limit for SDE's singular at the initial time

TL;DR

The paper analyzes the zero-noise limit for SDEs with rough, divergence-free drifts singular at the initial time on the torus. It develops a framework where randomness is retained in the initial data and proves existence and uniqueness of a zero-noise flow η under BV regularity and Prodi–Serrin conditions on the drift. The main result shows η is concentrated on integral curves of b and incompressible for divergence-free fields, and for Depauw's DP field the disintegration is non-Dirac, demonstrating stochasticity in the limit. The work extends Ambrosio's regular Lagrangian flow theory to a broader class of vector fields and provides explicit construction validating sharpness through the DP example.

Abstract

We investigate the zero-noise limit for SDE's driven by Brownian motion with a divergence-free drift singular at the initial time and prove that a unique probability measure concentrated on the integral curves of the drift is selected. More precisely, we prove uniqueness of the zero-noise limit for divergence-free drifts in where and satisfy a Prodi-Serrin condition. The vector field constructed by Depauw [C. R. Acad. Sci. Paris, 2003] lies in this class and we show that for almost every intial datum, the zero-noise limit selects a probability measure concentrated on several distinct integral curves of this vector field.

Paper Structure

This paper contains 14 sections, 12 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Review of SDE theory
  4. Disintegration of a measure
  5. Statement of the result
  6. Our argument and organisation of this paper
  7. Existence and uniqueness of the zero-noise flow
  8. Existence of a zero-noise flow
  9. Uniqueness of the zero-noise flow
  10. Stochasticity of the zero-noise flow
  11. Disintegration of $\eta$ at the final time
  12. Disintegration of $\eta$ at the initial time
  13. Disintegration of $\eta$ for the Depauw vector field
  14. Construction of the Depauw vector field

Key Result

Proposition 1.2

Consider a Borel vector field $b:[0,T]\times\mathbb{T}^d\to \mathbb{R}^d$, and $p,q\in (1,+\infty]$ satisfying eqn_prodi_serrin. Assume that $b\in L^q_p(T)$. Then there exists a continuous stochastic process $X^\nu$ on $\mathscr{P}$ defined in eqn_proba_space such that Any two processes satisfying $(i)$ and $(ii)$ are indistinguishable. If we further assume that $b$ is divergence-free, then we ha

Figures (1)

  • Figure 1: Action of the flow of $u$ from $t=0$ to $t=T/2$. The shaded region denotes the set $\{\rho^B=1\}$. The figure is from DeLellis_Giri22.

Theorems & Definitions (26)

  • Definition 1.1
  • Proposition 1.2
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • Proposition 2.1
  • Remark 2.2
  • proof
  • Theorem 2.3
  • ...and 16 more