Integral Varadhan formula for non-linear heat flow
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915242178510848 |
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| author | Ohta, Shin-ichi Suzuki, Kohei |
| author_facet | Ohta, Shin-ichi Suzuki, Kohei |
| contents | We prove the integral Varadhan short-time formula for non-linear heat flow on measured Finsler manifolds. To the best of the authors' knowledge, this is the first result establishing a Varadhan-type formula for non-linear semigroups. We do not assume the reversibility of the metric, and the distance function can be asymmetric. In this generality, we reveal that the probabilistic interpretation is well-suited for our formula; the probability that a particle starting from a set $A$ can be found in another set $B$ describes the distance from $A$ to $B$. One side of the estimates (the upper bound of the probability) is also established in the nonsmooth setting of infinitesimally strictly convex metric measure spaces satisfying the local Sobolev-to-Lipschitz property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04102 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integral Varadhan formula for non-linear heat flow Ohta, Shin-ichi Suzuki, Kohei Probability Functional Analysis Metric Geometry We prove the integral Varadhan short-time formula for non-linear heat flow on measured Finsler manifolds. To the best of the authors' knowledge, this is the first result establishing a Varadhan-type formula for non-linear semigroups. We do not assume the reversibility of the metric, and the distance function can be asymmetric. In this generality, we reveal that the probabilistic interpretation is well-suited for our formula; the probability that a particle starting from a set $A$ can be found in another set $B$ describes the distance from $A$ to $B$. One side of the estimates (the upper bound of the probability) is also established in the nonsmooth setting of infinitesimally strictly convex metric measure spaces satisfying the local Sobolev-to-Lipschitz property. |
| title | Integral Varadhan formula for non-linear heat flow |
| topic | Probability Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2403.04102 |