Stochastic Aggregation Diffusion-Equation : Analysis via Dirichlet Forms

TL;DR

This paper investigates the stochastic aggregation-diffusion equation with singular drift using Dirichlet form theory, establishing existence and partial uniqueness of weak solutions and providing explicit operator expressions.

Contribution

It introduces a novel analysis of the stochastic aggregation-diffusion equation via Dirichlet forms, addressing existence, partial uniqueness, and explicit operator characterization.

Findings

Existence of weak solutions for the stochastic aggregation-diffusion equation.

Partial uniqueness of solutions under the H_2-condition.

Explicit expression for the generalized Schrödinger operator.

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.