Uniform contractivity of the Fisher infinitesimal model with strongly convex selection
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912505997033472 |
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| author | Calvez, Vincent Poyato, David Santambrogio, Filippo |
| author_facet | Calvez, Vincent Poyato, David Santambrogio, Filippo |
| contents | The Fisher infinitesimal model is a classical model of phenotypic trait inheritance in quantitative genetics. Here, we prove that it encompasses a remarkable convexity structure which is compatible with a selection function having a convex shape. It yields uniform contractivity along the flow, as measured by a $L^\infty$ version of the Fisher information. It induces in turn asynchronous exponential growth of solutions, associated with a well-defined, log-concave, equilibrium distribution. Although the equation is non-linear and non-conservative, our result shares some similarities with the Bakry-Emery approach to the exponential convergence of solutions to the Fokker-Planck equation with a convex potential. Indeed, the contraction takes place at the level of the Fisher information. Moreover, the key lemma for proving contraction involves the Wasserstein distance $W_\infty$ between two probability distributions of a (dual) backward-in-time process, and it is inspired by a maximum principle by Caffarelli for the Monge-Ampère equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12063 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniform contractivity of the Fisher infinitesimal model with strongly convex selection Calvez, Vincent Poyato, David Santambrogio, Filippo Probability Analysis of PDEs 35B40, 35P30, 35Q92, 47G20, 92D15 The Fisher infinitesimal model is a classical model of phenotypic trait inheritance in quantitative genetics. Here, we prove that it encompasses a remarkable convexity structure which is compatible with a selection function having a convex shape. It yields uniform contractivity along the flow, as measured by a $L^\infty$ version of the Fisher information. It induces in turn asynchronous exponential growth of solutions, associated with a well-defined, log-concave, equilibrium distribution. Although the equation is non-linear and non-conservative, our result shares some similarities with the Bakry-Emery approach to the exponential convergence of solutions to the Fokker-Planck equation with a convex potential. Indeed, the contraction takes place at the level of the Fisher information. Moreover, the key lemma for proving contraction involves the Wasserstein distance $W_\infty$ between two probability distributions of a (dual) backward-in-time process, and it is inspired by a maximum principle by Caffarelli for the Monge-Ampère equation. |
| title | Uniform contractivity of the Fisher infinitesimal model with strongly convex selection |
| topic | Probability Analysis of PDEs 35B40, 35P30, 35Q92, 47G20, 92D15 |
| url | https://arxiv.org/abs/2302.12063 |