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On well-posedness of stable-driven McKean-Vlasov stochastic differential equations with Besov interaction kernel of non-positive regularity

Anna Bahrii

TL;DR

This work develops a rigorous framework for well-posedness of time-inhomogeneous stable-driven McKean–Vlasov SDEs with a convolution drift whose interaction kernel lies in Besov spaces of negative regularity, pushing beyond the previous threshold of $-1$ by employing a sharp Besov product rule and a distributional-drift approach. The authors construct solutions via mollified problems, prove a priori bounds for mollified densities in suitable weighted Besov spaces, and pass to the limit using a stochastic nonlinear Young integral to handle the integrated drift, yielding martingale and weak optimality results and, in 1D, strong well-posedness. They establish short-time results and extend to global time under smallness assumptions on the initial data, with detailed regularity results for the time-evolving law, including density estimates in $B^{-{\beta}-\theta}_{p',1}$ with explicit time weights. The analysis hinges on an extensive Besov-space toolkit, including product rules for negative regularity, heat-kernel Besov estimates, embeddings, and mollification of kernels, bridging linear SDE theory with nonlinear McKean–Vlasov dynamics in rough function spaces. The findings advance the understanding of distributional interactions in MV SDEs driven by $\alpha$-stable noise and provide a pathway to modeling long-range, singular interactions in probabilistic PDEs and related applications.

Abstract

We prove well-posedness results for time-inhomogeneous stable-driven McKean-Vlasov stochastic differential equations with a convolution drift where the interaction kernel belongs to some Lebesgue-Besov space. The novelty of this work is that we manage to go below -1 in space regularity for such a kernel. This is achieved under additional smoothness conditions on the initial data and divergence conditions on the kernel. The proof heavily relies on a suitable product rule in Besov space. We prove smoothing properties of the law, which allow us to have the drift in a Lebesgue-Besov space of non-positive regularity.

On well-posedness of stable-driven McKean-Vlasov stochastic differential equations with Besov interaction kernel of non-positive regularity

TL;DR

This work develops a rigorous framework for well-posedness of time-inhomogeneous stable-driven McKean–Vlasov SDEs with a convolution drift whose interaction kernel lies in Besov spaces of negative regularity, pushing beyond the previous threshold of by employing a sharp Besov product rule and a distributional-drift approach. The authors construct solutions via mollified problems, prove a priori bounds for mollified densities in suitable weighted Besov spaces, and pass to the limit using a stochastic nonlinear Young integral to handle the integrated drift, yielding martingale and weak optimality results and, in 1D, strong well-posedness. They establish short-time results and extend to global time under smallness assumptions on the initial data, with detailed regularity results for the time-evolving law, including density estimates in with explicit time weights. The analysis hinges on an extensive Besov-space toolkit, including product rules for negative regularity, heat-kernel Besov estimates, embeddings, and mollification of kernels, bridging linear SDE theory with nonlinear McKean–Vlasov dynamics in rough function spaces. The findings advance the understanding of distributional interactions in MV SDEs driven by -stable noise and provide a pathway to modeling long-range, singular interactions in probabilistic PDEs and related applications.

Abstract

We prove well-posedness results for time-inhomogeneous stable-driven McKean-Vlasov stochastic differential equations with a convolution drift where the interaction kernel belongs to some Lebesgue-Besov space. The novelty of this work is that we manage to go below -1 in space regularity for such a kernel. This is achieved under additional smoothness conditions on the initial data and divergence conditions on the kernel. The proof heavily relies on a suitable product rule in Besov space. We prove smoothing properties of the law, which allow us to have the drift in a Lebesgue-Besov space of non-positive regularity.
Paper Structure (16 sections, 32 theorems, 298 equations)

This paper contains 16 sections, 32 theorems, 298 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and literature overview
  3. Organization of the paper.
  4. Main results and examples
  5. Strategy of the proof
  6. Auxiliary tools
  7. Well-posedness and regularity of the Fokker-Planck equation in short time
  8. Well-posedness of McKean-Vlasov SDE in short time
  9. Martingale problem well-posedness
  10. Weak well-posedness
  11. Strong well-posedness: dimension one
  12. Global well-posedness
  13. Auxiliary tools and lemmas
  14. Fokker-Planck equation well-posedness and regularity in global time
  15. Global SDE well-posedness
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let where are such that $\zeta_0>0$, where $\zeta_0$ is given by def-zeta0, and $\bar{\theta}>0$ is such that for some $\theta\in[0,\frac{1}{2})$. Let $\alpha\in(1,2]$, $\beta\in(-2,-1)$, $p,r\in(1,+\infty]$, $q\in[1,+\infty]$. Finally, let where where $\bar{\beta}_0=\beta_0(1\wedge\frac{p_0'}{p})$.

Theorems & Definitions (71)

  • Definition 1.1: Martingale problem
  • Definition 1.2: Stochastic non-linear Young integral
  • Definition 1.3: Weak solution
  • Theorem 1.1: Short time well-posedness
  • Theorem 1.2: Long time well-posedness
  • Remark 1.2: About a choice of $\theta$
  • Theorem 1.3: Classical well-posedness in short time
  • Example 1.1: Thresholds in short time
  • Remark 1.3: Comparison of \ref{['C3']} and \ref{['C4']}
  • Example 1.2: Thresholds in long time
  • ...and 61 more