Local elliptic regularity for solutions to stationary Fokker-Planck equations via Dirichlet forms and resolvents
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915242535026688 |
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| author | Lee, Haesung |
| author_facet | Lee, Haesung |
| contents | In this paper, we show that, for a solution to the stationary Fokker-Planck equation with general coefficients, defined as a measure with an $L^2$-density, this density not only exhibits $H^{1,2}$-regularity but also Hölder continuity. To achieve this, we first construct a reference measure $μ=ρdx$ by utilizing existence and elliptic regularity results, ensuring that the given divergence-type operator corresponds to a sectorial Dirichlet form. By employing elliptic regularity results for homogeneous boundary value problems in both divergence and non-divergence type equations, we demonstrate that the image of the resolvent operator associated with the sectorial Dirichlet form has $H^{2,2}$-regularity. Furthermore, through calculations based on the Dirichlet form and the $H^{2,2}$-regularity of the resolvent operator, we prove that the density of the solution measure for the stationary Fokker-Planck equation is, indeed, the weak limit of $H^{1,2}$-functions defined via the resolvent operator. Our results highlight the central role of Dirichlet form theory and resolvent approximations in establishing the regularity of solutions to stationary Fokker-Planck equations with general coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local elliptic regularity for solutions to stationary Fokker-Planck equations via Dirichlet forms and resolvents Lee, Haesung Analysis of PDEs Probability Primary: 35B65, 35Q84, 31C25, Secondary: 60J60, 60J35, 35D30 In this paper, we show that, for a solution to the stationary Fokker-Planck equation with general coefficients, defined as a measure with an $L^2$-density, this density not only exhibits $H^{1,2}$-regularity but also Hölder continuity. To achieve this, we first construct a reference measure $μ=ρdx$ by utilizing existence and elliptic regularity results, ensuring that the given divergence-type operator corresponds to a sectorial Dirichlet form. By employing elliptic regularity results for homogeneous boundary value problems in both divergence and non-divergence type equations, we demonstrate that the image of the resolvent operator associated with the sectorial Dirichlet form has $H^{2,2}$-regularity. Furthermore, through calculations based on the Dirichlet form and the $H^{2,2}$-regularity of the resolvent operator, we prove that the density of the solution measure for the stationary Fokker-Planck equation is, indeed, the weak limit of $H^{1,2}$-functions defined via the resolvent operator. Our results highlight the central role of Dirichlet form theory and resolvent approximations in establishing the regularity of solutions to stationary Fokker-Planck equations with general coefficients. |
| title | Local elliptic regularity for solutions to stationary Fokker-Planck equations via Dirichlet forms and resolvents |
| topic | Analysis of PDEs Probability Primary: 35B65, 35Q84, 31C25, Secondary: 60J60, 60J35, 35D30 |
| url | https://arxiv.org/abs/2412.14636 |