Kinetic SDEs with subcritical distributional drifts

Zikai Chen; Zimo Hao; Xicheng Zhang

doi:10.1515/ans-2023-0219

Kinetic SDEs with subcritical distributional drifts

Zikai Chen^*
, Zimo Hao
, Xicheng Zhang

^*Corresponding author for this work

Wuhan University
Kyoto University
Beijing Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in ℝ^2d(d ≥ 2) driven by Brownian motion: dX_t = V_tdt, dV_t = b(t,X_t,V_t)dt+ ^√ 2dW_t, where the subcritical distribution-valued drift b belongs to the weighted anisotropic Hölder space L^q_T^bC_a^αb(p_k) with parameters α_b ∈ (−1,0), q_b ∈ ⁽₁₊²_αb,∞ ^] , k ∈ ^[0,1 + α_b⁾ and div_υb is bounded. We establish the well-posedness of weak solutions to the associated integral equation: t t X_t = X₀ + _∫ V_sds, V_t = V₀ + _nlim _→∞∫ b_n(s,X_s,V_s)ds+ ^√ 2W_t, 0 0 where b_n :=b∗Γ_n denotes the mollification of b and the limit is taken in the L²-sense. As an application, we discuss examples of binvolving Gaussian random fields.

Original language	English
Journal	Advanced Nonlinear Studies
DOIs	https://doi.org/10.1515/ans-2023-0219
Publication status	Accepted/In press - 2026
Externally published	Yes

Keywords

anisotropic Hölder space
distributional drifts
Krylov’s estimate
subcritical kinetic SDEs

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@article{0c02834ea9cc4ed9991e401a3e9b24ea,

title = "Kinetic SDEs with subcritical distributional drifts",

abstract = "In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in ℝ2d(d ≥ 2) driven by Brownian motion: dXt = Vtdt, dVt = b(t,Xt,Vt)dt+ √ 2dWt, where the subcritical distribution-valued drift b belongs to the weighted anisotropic H{\"o}lder space LqTbCaαb(pk) with parameters αb ∈ (−1,0), qb ∈ (1+2αb,∞ ] , k ∈ [0,1 + αb) and divυb is bounded. We establish the well-posedness of weak solutions to the associated integral equation: t t Xt = X0 + ∫ Vsds, Vt = V0 + nlim →∞∫ bn(s,Xs,Vs)ds+ √ 2Wt, 0 0 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 binvolving Gaussian random fields.",

keywords = "anisotropic H{\"o}lder space, distributional drifts, Krylov{\textquoteright}s estimate, subcritical kinetic SDEs",

author = "Zikai Chen and Zimo Hao and Xicheng Zhang",

note = "Publisher Copyright: {\textcopyright} 2026 the author(s), published by De Gruyter, Berlin/Boston.",

year = "2026",

doi = "10.1515/ans-2023-0219",

language = "English",

journal = "Advanced Nonlinear Studies",

issn = "1536-1365",

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TY - JOUR

T1 - Kinetic SDEs with subcritical distributional drifts

AU - Chen, Zikai

AU - Hao, Zimo

AU - Zhang, Xicheng

PY - 2026

Y1 - 2026

N2 - In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in ℝ2d(d ≥ 2) driven by Brownian motion: dXt = Vtdt, dVt = b(t,Xt,Vt)dt+ √ 2dWt, where the subcritical distribution-valued drift b belongs to the weighted anisotropic Hölder space LqTbCaαb(pk) with parameters αb ∈ (−1,0), qb ∈ (1+2αb,∞ ] , k ∈ [0,1 + αb) and divυb is bounded. We establish the well-posedness of weak solutions to the associated integral equation: t t Xt = X0 + ∫ Vsds, Vt = V0 + nlim →∞∫ bn(s,Xs,Vs)ds+ √ 2Wt, 0 0 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 binvolving Gaussian random fields.

AB - In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in ℝ2d(d ≥ 2) driven by Brownian motion: dXt = Vtdt, dVt = b(t,Xt,Vt)dt+ √ 2dWt, where the subcritical distribution-valued drift b belongs to the weighted anisotropic Hölder space LqTbCaαb(pk) with parameters αb ∈ (−1,0), qb ∈ (1+2αb,∞ ] , k ∈ [0,1 + αb) and divυb is bounded. We establish the well-posedness of weak solutions to the associated integral equation: t t Xt = X0 + ∫ Vsds, Vt = V0 + nlim →∞∫ bn(s,Xs,Vs)ds+ √ 2Wt, 0 0 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 binvolving Gaussian random fields.

KW - anisotropic Hölder space

KW - distributional drifts

KW - Krylov’s estimate

KW - subcritical kinetic SDEs

UR - https://www.scopus.com/pages/publications/105041112998

U2 - 10.1515/ans-2023-0219

DO - 10.1515/ans-2023-0219

M3 - Article

AN - SCOPUS:105041112998

SN - 1536-1365

JO - Advanced Nonlinear Studies

JF - Advanced Nonlinear Studies

ER -

Kinetic SDEs with subcritical distributional drifts

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Keywords

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