SDEs with supercritical distributional drifts
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866912517563875328 |
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| author | Hao, Zimo Zhang, Xicheng |
| author_facet | Hao, Zimo Zhang, Xicheng |
| contents | Let $d\geq 2$. In this paper, we investigate the following stochastic differential equation (SDE) in ${\mathbb R}^d$ driven by Brownian motion $$ {\rm d} X_t=b(t,X_t){\rm d} t+\sqrt{2}{\rm d} W_t, $$ where $b$ belongs to the space ${\mathbb L}_T^q \mathbf{H}_p^α$ with $α\in [-1, 0]$ and $p,q\in[2, \infty]$, which is a distribution-valued and divergence-free vector field. In the subcritical case $\frac dp+\frac 2q<1+α$, we establish the existence and uniqueness of a weak solution to the integral equation: $$ X_t=X_0+\lim_{n\to\infty}\int^t_0b_n(s,X_s){\rm d} s+\sqrt{2} W_t. $$ Here, $b_n:=b*ϕ_n$ represents the mollifying approximation, and the limit is taken in the $L^2$-sense. In the critical and supercritical case $1+α\leq\frac dp+\frac 2q<2+α$, assuming the initial distribution has an $L^2$-density, we show the existence of weak solutions and associated Markov processes. Moreover, under the additional assumption that $b=b_1+b_2+{\rm div} a$, where $b_1\in {\mathbb L}^\infty_T{\mathbf B}^{-1}_{\infty,2}$, $b_2\in {\mathbb L}^2_TL^2$, and $a$ is a bounded antisymmetric matrix-valued function, we establish the convergence of mollifying approximation solutions without the need to subtract a subsequence. To illustrate our results, we provide examples of Gaussian random fields and singular interacting particle systems, including the two-dimensional vortex models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11145 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | SDEs with supercritical distributional drifts Hao, Zimo Zhang, Xicheng Probability Analysis of PDEs 60H10 Let $d\geq 2$. In this paper, we investigate the following stochastic differential equation (SDE) in ${\mathbb R}^d$ driven by Brownian motion $$ {\rm d} X_t=b(t,X_t){\rm d} t+\sqrt{2}{\rm d} W_t, $$ where $b$ belongs to the space ${\mathbb L}_T^q \mathbf{H}_p^α$ with $α\in [-1, 0]$ and $p,q\in[2, \infty]$, which is a distribution-valued and divergence-free vector field. In the subcritical case $\frac dp+\frac 2q<1+α$, we establish the existence and uniqueness of a weak solution to the integral equation: $$ X_t=X_0+\lim_{n\to\infty}\int^t_0b_n(s,X_s){\rm d} s+\sqrt{2} W_t. $$ Here, $b_n:=b*ϕ_n$ represents the mollifying approximation, and the limit is taken in the $L^2$-sense. In the critical and supercritical case $1+α\leq\frac dp+\frac 2q<2+α$, assuming the initial distribution has an $L^2$-density, we show the existence of weak solutions and associated Markov processes. Moreover, under the additional assumption that $b=b_1+b_2+{\rm div} a$, where $b_1\in {\mathbb L}^\infty_T{\mathbf B}^{-1}_{\infty,2}$, $b_2\in {\mathbb L}^2_TL^2$, and $a$ is a bounded antisymmetric matrix-valued function, we establish the convergence of mollifying approximation solutions without the need to subtract a subsequence. To illustrate our results, we provide examples of Gaussian random fields and singular interacting particle systems, including the two-dimensional vortex models. |
| title | SDEs with supercritical distributional drifts |
| topic | Probability Analysis of PDEs 60H10 |
| url | https://arxiv.org/abs/2312.11145 |