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September 24, 2025Open Access

Kinetic SDEs with subcritical distributional drifts

Key Points

Weak solutions to kinetic stochastic differential equations are shown to be well-posed under specific conditions.
Invariant parameters include the drift function belonging to a weighted anisotropic Hölder space.
The study utilizes mollification techniques to handle distribution-valued drifts within the kinetic framework.
Application examples featuring Gaussian random fields illustrate the practical relevance of these models.

Abstract

In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in R^2d (d2) driven by Brownian motion: d Xₜ=Vₜ d t, \ d Vₜ=b (t, Xₜ, Vₜ) d t+2 d Wₜ, where the subcritical distribution-valued drift b belongs to the weighted anisotropic Hölder space LT^qb C₀^αb (ρ_κ) with parameters αb (-1, 0), qb (21+αb, ], κ[0, 1+αb) and ÷ᵥ b is bounded. We establish the well-posedness of weak solutions to the associated integral equation: Xₜ=X₀+₀ᵗ Vₛ d s, \ Vₜ=V₀+₍₀ᵗ bₙ (s, Xₛ, Vₛ) d+2Wₜ, where bₙ: =b*Γₙ denotes the mollification of b and the limit is taken in the L²-sense. As an application, we discuss examples of b involving Gaussian random fields.

Kinetic SDEs with subcritical distributional drifts

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