Well-posedness of 2D stochastic parabolic-parabolic Keller-Segel system with logistic source

We consider a stochastic, fully parabolic Keller-Segel system with multiplicative noise in non-divergence form and a logistic source in a two-dimensional bounded domain, subject to random initial data. We construct a global-in-time, non-negative martingale solution by means of a suitable regularisation procedure, combined with Banach fixed-point arguments, a localisation procedure, and a stochastic compactness method. We further establish pathwise uniqueness and uniqueness in law for martingale solutions, and prove the existence of a unique global-in-time, non-negative weak solution without imposing any smallness condition on the initial data.