The well-posedness for SDEs with singularity in both space and distribution variables is derived, where the interacting drift term is bounded and Lipschitz continuous under total variation distance and the diffusion term is allowed to be Lipschitz continuous under $L^\eta$($\eta\in(0,1]$)-Wasserstein distance in the distribution variable. When the diffusion term is Lipschitz continuous under $L^k$-Wasserstein distance for some $k\geq 1$, the regularity estimate $$\|P_t^\ast\gamma^1-P_t^\ast\gamma^2\|_{ var}\leq ct^{ -\frac{ 1}{ 2}}\W_{ k}(\gamma^1,\gamma^2),\ \ t\in(0,T]$$ is established. This improves the results in \cite[Theorem 1.3]{ HRWJDE}.
Categories: math.PR