Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910692182851584 |
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| author | Hambly, Ben Kreher, Dörte Starovoitovs, Konstantins |
| author_facet | Hambly, Ben Kreher, Dörte Starovoitovs, Konstantins |
| contents | We construct non-negative martingale solutions to the stochastic porous medium equation in one dimension with homogeneous Dirichlet boundary conditions which exhibit a type of sticky behavior at zero. The construction uses the stochastic Faedo--Galerkin method via spatial semidiscretization, so that the pre-limiting system is given by a finite-dimensional diffusion with Wentzell boundary condition. We derive uniform moment estimates for the discrete systems by an Aubin-Lions-type interpolation argument, which enables us to implement a general weak convergence approach for the construction of martingale solutions of an SPDE using a Skorokhod representation-type result for non-metrizable spaces. We rely on a stochastic argument based on the occupation time formula for continuous semimartingales for the identification of the diffusion coefficient in the presence of an indicator function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05924 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior Hambly, Ben Kreher, Dörte Starovoitovs, Konstantins Probability 60H15, 35R60, 76S05 We construct non-negative martingale solutions to the stochastic porous medium equation in one dimension with homogeneous Dirichlet boundary conditions which exhibit a type of sticky behavior at zero. The construction uses the stochastic Faedo--Galerkin method via spatial semidiscretization, so that the pre-limiting system is given by a finite-dimensional diffusion with Wentzell boundary condition. We derive uniform moment estimates for the discrete systems by an Aubin-Lions-type interpolation argument, which enables us to implement a general weak convergence approach for the construction of martingale solutions of an SPDE using a Skorokhod representation-type result for non-metrizable spaces. We rely on a stochastic argument based on the occupation time formula for continuous semimartingales for the identification of the diffusion coefficient in the presence of an indicator function. |
| title | Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior |
| topic | Probability 60H15, 35R60, 76S05 |
| url | https://arxiv.org/abs/2411.05924 |