Kinetic SDEs with subcritical distributional drifts
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916904316174336 |
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| author | Chen, Zikai Hao, Zimo Zhang, Xicheng |
| author_facet | Chen, Zikai Hao, Zimo Zhang, Xicheng |
| contents | In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in $\mathbb R^{2d}(d\geq2)$ driven by Brownian motion: $$\mathord{\rm d} X_t=V_t\mathord{\rm d} t,\ \mathord{\rm d} V_t=b(t,X_t,V_t)\mathord{\rm d} t+\sqrt{2}\mathord{\rm d} W_t,$$ where the subcritical distribution-valued drift $b$ belongs to the weighted anisotropic Hölder space $\mathbb L_T^{q_b}\mathbf C_{\boldsymbol{a}}^{α_b}(ρ_κ)$ with parameters $α_b\in(-1,0)$, $q_b\in(\frac{2}{1+α_b},\infty]$, $κ\in[0,1+α_b)$ and $÷_v b$ is bounded. We establish the well-posedness of weak solutions to the associated integral equation: $$X_t=X_0+\int_0^t V_s\mathord{\rm d} s,\ V_t=V_0+\lim_{n\to\infty}\int_0^t b_n(s,X_s,V_s)\mathord{\rm d}+\sqrt{2}W_t,$$ where $b_n:=b*Γ_n$ denotes the mollification of $b$ and the limit is taken in the $L^2$-sense. As an application, we discuss examples of $b$ involving Gaussian random fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12234 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kinetic SDEs with subcritical distributional drifts Chen, Zikai Hao, Zimo Zhang, Xicheng Probability In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in $\mathbb R^{2d}(d\geq2)$ driven by Brownian motion: $$\mathord{\rm d} X_t=V_t\mathord{\rm d} t,\ \mathord{\rm d} V_t=b(t,X_t,V_t)\mathord{\rm d} t+\sqrt{2}\mathord{\rm d} W_t,$$ where the subcritical distribution-valued drift $b$ belongs to the weighted anisotropic Hölder space $\mathbb L_T^{q_b}\mathbf C_{\boldsymbol{a}}^{α_b}(ρ_κ)$ with parameters $α_b\in(-1,0)$, $q_b\in(\frac{2}{1+α_b},\infty]$, $κ\in[0,1+α_b)$ and $÷_v b$ is bounded. We establish the well-posedness of weak solutions to the associated integral equation: $$X_t=X_0+\int_0^t V_s\mathord{\rm d} s,\ V_t=V_0+\lim_{n\to\infty}\int_0^t b_n(s,X_s,V_s)\mathord{\rm d}+\sqrt{2}W_t,$$ where $b_n:=b*Γ_n$ denotes the mollification of $b$ and the limit is taken in the $L^2$-sense. As an application, we discuss examples of $b$ involving Gaussian random fields. |
| title | Kinetic SDEs with subcritical distributional drifts |
| topic | Probability |
| url | https://arxiv.org/abs/2508.12234 |