Resolvent approaches to elliptic regularity in stationary Fokker-Planck equations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914346192338944 |
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| author | Lee, Haesung |
| author_facet | Lee, Haesung |
| contents | This paper investigates the local regularity of solutions to stationary Fokker-Planck equations on an open set $U \subset \mathbb{R}^d$ with $d \geq 2$. A central objective is to relax the classical assumptions on the coefficients by focusing on the case where the drift vector field $\mathbf{G}$ is only assumed to be locally square-integrable, i.e. $\mathbf{G} \in L^2_{loc}(U, \mathbb{R}^d)$, the symmetric diffusion matrix $A = (a_{ij})_{1 \leq i,j \leq d}$ is assumed to be locally uniformly strictly elliptic and bounded, with coefficients satisfying $a_{ij} \in VMO_{loc}(U)$ for all $1 \leq i,j \leq d$ and ${\rm div}A \in L^2_{loc}(U, \mathbb{R}^d)$. Our main result shows that any locally bounded function $h \in L^\infty_{loc}(U)$ satisfying the stationary Fokker-Planck equation $L^*(h\, dx) = 0$ must in fact belong to the local Sobolev space $H^{1,2}_{loc}(U)$. The proof is based on the construction of a sub-Markovian resolvent associated with the principal elliptic operator $L^A := {\rm trace}(A \nabla^2)$, combined with delicate energy-type inequalities. In particular, we show that the density $h$ can be realized as the weak $H^{1,2}$-limit of images of resolvent operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13727 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Resolvent approaches to elliptic regularity in stationary Fokker-Planck equations Lee, Haesung Analysis of PDEs Primary: 35B65, 35Q84, Secondary: 60J35, 47D07 This paper investigates the local regularity of solutions to stationary Fokker-Planck equations on an open set $U \subset \mathbb{R}^d$ with $d \geq 2$. A central objective is to relax the classical assumptions on the coefficients by focusing on the case where the drift vector field $\mathbf{G}$ is only assumed to be locally square-integrable, i.e. $\mathbf{G} \in L^2_{loc}(U, \mathbb{R}^d)$, the symmetric diffusion matrix $A = (a_{ij})_{1 \leq i,j \leq d}$ is assumed to be locally uniformly strictly elliptic and bounded, with coefficients satisfying $a_{ij} \in VMO_{loc}(U)$ for all $1 \leq i,j \leq d$ and ${\rm div}A \in L^2_{loc}(U, \mathbb{R}^d)$. Our main result shows that any locally bounded function $h \in L^\infty_{loc}(U)$ satisfying the stationary Fokker-Planck equation $L^*(h\, dx) = 0$ must in fact belong to the local Sobolev space $H^{1,2}_{loc}(U)$. The proof is based on the construction of a sub-Markovian resolvent associated with the principal elliptic operator $L^A := {\rm trace}(A \nabla^2)$, combined with delicate energy-type inequalities. In particular, we show that the density $h$ can be realized as the weak $H^{1,2}$-limit of images of resolvent operators. |
| title | Resolvent approaches to elliptic regularity in stationary Fokker-Planck equations |
| topic | Analysis of PDEs Primary: 35B65, 35Q84, Secondary: 60J35, 47D07 |
| url | https://arxiv.org/abs/2602.13727 |