Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime

Oleg Butkovsky, Samuel Gallay

TL;DR

The paper addresses weak existence for SDEs with additive noise driven by fractional Brownian motion or Lévy processes and drift in $L_q([0,T], L_p(\mathbb{R}^d))$, establishing existence under the sharp condition $(1/q) + (H d)/p < 1 - H$ that extends Krylov–Röckner theory to nonmartingale noise. It develops a Krylov-type moment bound via a refined stochastic sewing argument and the John–Nirenberg inequality, enabling tightness and stability analyses that yield weak existence; a parallel Lévy-noise theory is built under a Lévy analogue condition and a one-dimensional strong existence result is proved. A key counterexample demonstrates the optimality of the fractional Brownian condition, and the methods are shown to recover the Brownian Krylov regime when $H=1/2$. The work thus extends regularization by noise beyond the subcritical regime and provides a coherent framework for both Gaussian and Lévy driven SDEs with singular drifts.

Abstract

We study a multidimensional stochastic differential equation with additive noise: \[ d X_t=b(t, X_t) dt +d ξ_t, \] where the drift $b$ is integrable in space and time, and $ξ$ is either a fractional Brownian motion or a Lévy process. We show weak existence of solutions to this equation under the optimal condition on integrability indices of $b$, going beyond the subcritical Krylov--Röckner (Prodi--Serrin--Ladyzhenskaya) regime. This extends the recent results of Krylov (2020) to the fractional Brownian and Lévy cases. We also construct a counterexample to demonstrate the optimality of this condition. In the one-dimensional case, we show the existence of a strong solution under the same condition. Our methods are built upon a version of the stochastic sewing lemma of Lê and the John--Nirenberg inequality.

Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime

TL;DR

The paper addresses weak existence for SDEs with additive noise driven by fractional Brownian motion or Lévy processes and drift in , establishing existence under the sharp condition that extends Krylov–Röckner theory to nonmartingale noise. It develops a Krylov-type moment bound via a refined stochastic sewing argument and the John–Nirenberg inequality, enabling tightness and stability analyses that yield weak existence; a parallel Lévy-noise theory is built under a Lévy analogue condition and a one-dimensional strong existence result is proved. A key counterexample demonstrates the optimality of the fractional Brownian condition, and the methods are shown to recover the Brownian Krylov regime when . The work thus extends regularization by noise beyond the subcritical regime and provides a coherent framework for both Gaussian and Lévy driven SDEs with singular drifts.

Abstract

We study a multidimensional stochastic differential equation with additive noise: where the drift is integrable in space and time, and is either a fractional Brownian motion or a Lévy process. We show weak existence of solutions to this equation under the optimal condition on integrability indices of , going beyond the subcritical Krylov--Röckner (Prodi--Serrin--Ladyzhenskaya) regime. This extends the recent results of Krylov (2020) to the fractional Brownian and Lévy cases. We also construct a counterexample to demonstrate the optimality of this condition. In the one-dimensional case, we show the existence of a strong solution under the same condition. Our methods are built upon a version of the stochastic sewing lemma of Lê and the John--Nirenberg inequality.
Paper Structure (10 sections, 27 theorems, 194 equations)

This paper contains 10 sections, 27 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of the main results
  4. Proof of the Theorem \ref{['th:counter_example']}: no weak existence
  5. Recap of the sewing lemmas
  6. Main integral bound
  7. Verification of Assumption \ref{['main_assumption']} for Gaussian and Lévy processes
  8. Proofs of weak existence
  9. Proof of strong existence
  10. Appendix

Key Result

Theorem 2.1

Let $d\in\mathbb{N}$, $H \in (0, 1)$, $p \in [1, \infty )$, $q \in [1, \infty]$, $x \in \mathbb{R}^d$, $T>0$, $b\in L_q([0, T], L_p(\mathbb{R}^d))$. Suppose that the condition eq:main_cond is satisfied. Then the following holds.

Theorems & Definitions (53)

  • Theorem 2.1: Existence of weak solutions: fractional Brownian noise
  • Theorem 2.2
  • Theorem 2.3: Existence of strong solutions: fractional Brownian noise
  • Theorem 2.4: Existence of weak solutions: Lévy noise
  • Theorem 2.5: Existence of strong solutions: Lévy noise
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['th:counter_example']}
  • Definition 3.2
  • Definition 3.3
  • ...and 43 more