Stochastic Aggregation Diffusion-Equation : Analysis via Dirichlet Forms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914885274697728 |
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| author | Bourabiaa, Jaouad Elmadani, Youssef Hanine, Abdelouahab |
| author_facet | Bourabiaa, Jaouad Elmadani, Youssef Hanine, Abdelouahab |
| contents | In this article, we study the stochastic aggregation-diffusion equation with a singular drift represented by a monotone radial kernel. We demonstrate the existence and uniqueness of a diffusion process that acts as a weak solution to our equation. This process can be described as a distorted Brownian motion originating from a delocalized point. Utilizing Dirichlet form theory, we prove the existence of a weak solution for a quasi-everywhere point in a state space. However uniqueness is not assured for solutions commencing from points outside polar sets, and explicitly characterizing these sets poses a significant challenge. To address this, we employ the H_2-condition introduced by Albeverio et al.(2003). This condition provides a more thorough understanding of the uniqueness issue within the framework of Dirichlet forms. Consequently the H_2-condition is pivotal in enhancing the analysis of weak solutions, ensuring a more detailed comprehension of the problem. An explicit expression for the generalized Schrödinger operator associated with certain kernels is also provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17239 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic Aggregation Diffusion-Equation : Analysis via Dirichlet Forms Bourabiaa, Jaouad Elmadani, Youssef Hanine, Abdelouahab Probability Analysis of PDEs 35R60, 60J60, 60J46, 31C25 In this article, we study the stochastic aggregation-diffusion equation with a singular drift represented by a monotone radial kernel. We demonstrate the existence and uniqueness of a diffusion process that acts as a weak solution to our equation. This process can be described as a distorted Brownian motion originating from a delocalized point. Utilizing Dirichlet form theory, we prove the existence of a weak solution for a quasi-everywhere point in a state space. However uniqueness is not assured for solutions commencing from points outside polar sets, and explicitly characterizing these sets poses a significant challenge. To address this, we employ the H_2-condition introduced by Albeverio et al.(2003). This condition provides a more thorough understanding of the uniqueness issue within the framework of Dirichlet forms. Consequently the H_2-condition is pivotal in enhancing the analysis of weak solutions, ensuring a more detailed comprehension of the problem. An explicit expression for the generalized Schrödinger operator associated with certain kernels is also provided. |
| title | Stochastic Aggregation Diffusion-Equation : Analysis via Dirichlet Forms |
| topic | Probability Analysis of PDEs 35R60, 60J60, 60J46, 31C25 |
| url | https://arxiv.org/abs/2407.17239 |