Weak solution for distribution dependent SDEs driven by Lévy noise

Abstract: In this paper, we establish the existence of weak solutions for distribution-dependent stochastic differential equations (DDSDEs) driven by a broad class of Lévy noises, where the drift coefficients satisfy specific integrability conditions. This is achieved through the Krylov-type estimate and tightness argument.

• The paper establishes the existence of weak solutions for distribution-dependent SDEs driven by Lévy noise under weak integrability conditions on the drift.
• It extends Krylov-type L^q(L^p) estimates and tightness methods to the Lévy framework, accommodating nonlocal and singular jump behaviors.
• The approach unifies results for Brownian and stable noise, offering improved parameter ranges and new insights for mean-field and nonlocal models.

Weak Solutions for Distribution Dependent SDEs Driven by Lévy Noise

Introduction and Context

This paper addresses the existence problem for weak solutions of distribution-dependent stochastic differential equations (DDSDEs) driven by Lévy noise, with drift coefficients satisfying weak integrability conditions. This context is motivated by substantial developments in the theory of SDEs with singular drift, initially for Brownian noise under classical Krylov-Röckner and Zvonkin-type regularization approaches, and more recently extended to McKean-Vlasov and mean-field models. The novelty here lies in treating the nonlocal and potentially degenerate noise structures generated by general (possibly pure-jump) Lévy processes.

A decisive analytic obstacle in the Lévy framework is the weaker smoothing and more singular behavior of the associated transition semigroups, which impedes direct analogues of classical results. The paper systematically extends Krylov-type $L^q(L^p)$ estimates, tightness, and compactness arguments in this more singular setting and establishes weak existence under relaxed drift assumptions.

Mathematical Framework and Main Assumptions

Let $L_t$ be a $d$-dimensional càdlàg Lévy process on a filtered probability space, generating the transition semigroup $P_t f(x) = \mathbb{E}[f(x + L_t)]$ with infinitesimal generator $\mathcal{L}$.

The paper works with DDSDEs of the form

$dX_t = b(t, X_t, \mathscr{L}_{X_t})\,dt + dL_t,$

where $b$ depends measurably on time, space, and the law $\mathscr{L}_{X_t}$.

Key assumptions on the noise include:

• Gradient bounds: For all $p > 1$, there exists $M$ such that $L_t$0 for $L_t$1.
• Moment controls: $L_t$2, ensuring sufficient moment regularity.
• Smooth densities: $L_t$3 admits a smooth density for $L_t$4, established via Lévy symbol analysis.
• Non-degeneracy: The Lévy measure is not supported on any proper subspace, and the generator may be a mixture of jump and diffusive components.

These noise hypotheses are verified for a broad class including symmetric and non-degenerate $L_t$5-stable processes (with or without truncation or tempering), their linear combinations, and Brownian motion.

On the drift, the main regularity is a two-component assumption:

• Local boundedness or strong integrability: $L_t$6, with precise exponents depending on $L_t$7 and $L_t$8.
• Distributional continuity: $L_t$9 must be continuous in the law variable, formulated in the Wasserstein topology.

Krylov-Type Estimate and Analytic Tools

The core of the proof strategy is a Krylov-type stochastic estimate for the semimartingale $d$0 governed by Lévy noise and measurable drift. The framework determines conditions on $d$1 such that for all measurable $d$2,

$d$3

This critical estimate holds for $d$4 and $d$5, and is proved via precise semigroup interpolation inequalities, Sobolev embeddings, and maximal function estimates. The proof requires careful analysis of the smoothing properties of the Lévy semigroup and the interplay between spatial regularity and temporal integrability.

Weak Existence via Approximation and Tightness

To construct weak solutions, the drift is approximated by a sequence of Lipschitz, continuous functions $d$6, each yielding a unique strong solution due to classical results in the Lévy-SDE framework. Tightness of the approximating solutions is deduced via Aldous' criterion, drawing on the uniform moment estimates afforded by the Krylov bound and the integrability of $d$7.

Convergence in law (in the Skor okhod space, for pathwise càdlàg processes), plus a refined passage to the limit in the drift terms, yields a weak solution to the original DDSDE. The limiting procedure handles the measure-dependence via convergence in Wasserstein distance and employs a generalized Yamada-Watanabe strategy to link weak and strong solutions under additional uniqueness hypotheses.

Main Results and Explicit Parameter Conditions

The core existence theorem asserts:

Let $d$8, $d$9, $P_t f(x) = \mathbb{E}[f(x + L_t)]$0, $P_t f(x) = \mathbb{E}[f(x + L_t)]$1, and $P_t f(x) = \mathbb{E}[f(x + L_t)]$2. Then, for any initial condition, there exists a weak solution to the DDSDE with Lévy noise.

For DDSDEs, the law-dependence is accommodated by strengthening the integrability and continuity of $P_t f(x) = \mathbb{E}[f(x + L_t)]$3 with respect to the measure variable (Wasserstein). The required parameter ranges are optimal up to the limitations of existing nonlocal regularization estimates. The drift regularity threshold is explicit and, for linear combinations of Lévy processes with different stability indices, the allowed $P_t f(x) = \mathbb{E}[f(x + L_t)]$4 pairs improve, relaxing the assumptions.

The approach generalizes and unifies previous results for Brownian-driven DDSDEs and classic McKean-Vlasov equations, as well as extending methods known for pure-jump noise.

Theoretical and Practical Implications

The methods developed here broaden the class of mean-field and interacting particle systems for which weak solutions can be constructed robustly under low regularity assumptions. In particular:

• The systematic use of Krylov-type estimates enables treatment of kinetic equations, nonlinear Fokker-Planck flows, and mean-field models with jumps, as in mathematical physics and quantitative finance.
• The explicit parameter dependencies guide extensions to high-dimensional or strongly nonlocal systems, revealing the limits of integrability and non-degeneracy essential for probabilistic well-posedness.
• The results bridge to nonlocal PDE theory for porous medium and nonlinear fractional Fokker-Planck equations.

Potential future directions include extending the approach to uniqueness (via regularization techniques or smoothing transformations for Lévy noise), further relaxing the drift regularity (e.g., to Kato-type classes), and treating multiplicative noise with fully non-symmetric jump generators.

Conclusion

This work establishes the existence of weak solutions for distribution-dependent SDEs driven by a broad class of Lévy processes, under integrability and non-degeneracy conditions, via a generalized Krylov estimate and compactness arguments. The framework unifies and extends earlier results for Brownian and stable noise, providing a foundation for further study of nonlocal McKean-Vlasov models with singular interactions and jump-driven stochastic dynamics.

Reference: "Weak solution for distribution dependent SDEs driven by Lévy noise" (2604.12317).