Stochastic Aggregation Diffusion-Equation : Analysis via Dirichlet Forms
Abstract
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 Schr\"odinger operator associated with certain kernels is also provided.
Cite
@article{arxiv.2407.17239,
title = {Stochastic Aggregation Diffusion-Equation : Analysis via Dirichlet Forms},
author = {Jaouad Bourabiaa and Youssef Elmadani and Abdelouahab Hanine},
journal= {arXiv preprint arXiv:2407.17239},
year = {2024}
}