Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910574029307904 |
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| author | Chak, Martin |
| author_facet | Chak, Martin |
| contents | Given global Lipschitz continuity and differentiability of high enough order on the coefficients in Itô's equation, differentiability of associated semigroups, existence of twice differentiable solutions to Kolmogorov equations and weak convergence rates of numerical approximations are known results. In this work and against the counterexamples of Hairer et al.(2015), the drift and diffusion coefficients having Lipschitz constants that are $o(\log V)$ and $o(\sqrt{\log V})$ respectively for a function $V$ satisfying $(\partial_t + L)V\leq CV$ is shown to be a generalizing condition in place of global Lipschitz continuity for the above. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_05436 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions Chak, Martin Probability Analysis of PDEs 60H35 Given global Lipschitz continuity and differentiability of high enough order on the coefficients in Itô's equation, differentiability of associated semigroups, existence of twice differentiable solutions to Kolmogorov equations and weak convergence rates of numerical approximations are known results. In this work and against the counterexamples of Hairer et al.(2015), the drift and diffusion coefficients having Lipschitz constants that are $o(\log V)$ and $o(\sqrt{\log V})$ respectively for a function $V$ satisfying $(\partial_t + L)V\leq CV$ is shown to be a generalizing condition in place of global Lipschitz continuity for the above. |
| title | Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions |
| topic | Probability Analysis of PDEs 60H35 |
| url | https://arxiv.org/abs/2209.05436 |