  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.
