Local pathwise solutions and regularization by noises for the stochastic hyperbolic Keller-Segel equation

In this paper, we investigate the Cauchy problem associated with the stochastic hyperbolic Keller-Segel (SHKS) equation featuring multiplicative noises on the torus ${\mathbb{T}}^d$. First, we establish the local existence and uniqueness of pathwise solutions to the SHKS equation within Sobolev spaces $H^s({\mathbb{T}}^d)$ for $s>\frac{d}{2}+1$, under appropriate regularity conditions imposed on the nonlinear multiplicative noises. Subsequently, we explore two global results pertaining to noise-induced regularization: (1) The first result demonstrates that for polynomial-type nonlinear noises, when the noise intensity parameters meet specific threshold conditions, the SHKS equation possesses a unique pathwise solution for large initial data with probability one. This finding provides a partial answer to a question that has remained unresolved in the deterministic setting; (2) The second result reveals that, for small initial data, or equivalently when dealing with linear multiplicative noises with sufficiently large intensity (allowed to be negative), the SHKS equation admits a unique pathwise solution with high probability.