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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2604.27901 |
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| _version_ | 1866918475805491200 |
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| author | Colantoni, Fausto |
| author_facet | Colantoni, Fausto |
| contents | This paper investigates the heat equation on a bounded domain with a Robin boundary condition, where the reactivity parameter (or killing rate) is modeled as a continuous-time Markov chain. We analyze the system under two stochastic frameworks using a functional analytic approach.
First, we examine the annealed case, which accounts for the joint stochasticity of the diffusion and the switching mechanism. We describe the solution via a strongly continuous contraction semigroup on a product space. We identify its infinitesimal generator, which incorporates the state-dependent Robin conditions into its domain, and provide a corresponding Feynman-Kac formula.
Second, we study the quenched setting for fixed realizations of the switching paths. We characterize the solution through a non-autonomous evolution family (propagator) and derive a Feynman-Kac-type representation involving the boundary local time of a reflected Brownian motion. We prove an averaging principle in the fast-switching limit, showing that the system converges to a deterministic Robin problem. These results are applied to a biophysical model of stochastically gated receptors on cell membranes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27901 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Time-dependent Robin heat equation via Markovian switching Colantoni, Fausto Probability Analysis of PDEs This paper investigates the heat equation on a bounded domain with a Robin boundary condition, where the reactivity parameter (or killing rate) is modeled as a continuous-time Markov chain. We analyze the system under two stochastic frameworks using a functional analytic approach. First, we examine the annealed case, which accounts for the joint stochasticity of the diffusion and the switching mechanism. We describe the solution via a strongly continuous contraction semigroup on a product space. We identify its infinitesimal generator, which incorporates the state-dependent Robin conditions into its domain, and provide a corresponding Feynman-Kac formula. Second, we study the quenched setting for fixed realizations of the switching paths. We characterize the solution through a non-autonomous evolution family (propagator) and derive a Feynman-Kac-type representation involving the boundary local time of a reflected Brownian motion. We prove an averaging principle in the fast-switching limit, showing that the system converges to a deterministic Robin problem. These results are applied to a biophysical model of stochastically gated receptors on cell membranes. |
| title | Time-dependent Robin heat equation via Markovian switching |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2604.27901 |