Generalized Feynman-Kac Formula and Associated Heat Kernel
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866918255643328512 |
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| author | Ndumu, Martin N. |
| author_facet | Ndumu, Martin N. |
| contents | Let M be a smooth closed (compact without boundary) Riemannian manifold of dimension n and P a q-dimensional smooth submanifold of M. U will denote the tubular neighborhood of P in M. Let E be a smooth vector bundle over M. Here we will obtain a vector bundle Generalized Feynman-Kac formula associated to U and the vector bundle differential operator L consisteing of half the generalized Laplacian, a vector field X on M and a potential term V on M. From this formula, we shall deduce the usual Feynman-Kac formula as well as a stochastic representation of the Generalized Elworthy-Truman Heat Kernel formula, and ultimately the heat kernel formula. The Feynman-Kac expression can be expanded and from this expansion we shall deduce both the generalized heat trace and heat content expansions. The Generalized Feynman-Kac Formula is thus at the center of several other previously known results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_04740 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalized Feynman-Kac Formula and Associated Heat Kernel Ndumu, Martin N. General Mathematics 2020: 60D05 Let M be a smooth closed (compact without boundary) Riemannian manifold of dimension n and P a q-dimensional smooth submanifold of M. U will denote the tubular neighborhood of P in M. Let E be a smooth vector bundle over M. Here we will obtain a vector bundle Generalized Feynman-Kac formula associated to U and the vector bundle differential operator L consisteing of half the generalized Laplacian, a vector field X on M and a potential term V on M. From this formula, we shall deduce the usual Feynman-Kac formula as well as a stochastic representation of the Generalized Elworthy-Truman Heat Kernel formula, and ultimately the heat kernel formula. The Feynman-Kac expression can be expanded and from this expansion we shall deduce both the generalized heat trace and heat content expansions. The Generalized Feynman-Kac Formula is thus at the center of several other previously known results. |
| title | Generalized Feynman-Kac Formula and Associated Heat Kernel |
| topic | General Mathematics 2020: 60D05 |
| url | https://arxiv.org/abs/2306.04740 |