SOLVING MCKEAN-VLASOV SDES VIA RELATIVE ENTROPY

In this paper we explore the merit of relative entropy in proving weak well-posedness of McKean-Vlasov SDEs and SPDEs, extending the technique introduced in Lacker (Probab. Math. Phys. 4 (2023) 377-432). In the SDE setting, we prove weak existence and uniqueness when the interaction is path dependent and only assumed to have linear growth. Meanwhile, we recover and extend the current results when the interaction has Krylov's L-t(q)-L-x(p) type singularity for (d)/(p)+(2)/(q)<1, where d is the dimension of space. We connect the aforementioned two cases which are traditionally disparate, and form a solution theory that is sufficiently robust to allow perturbations of sublinear growth at the presence of singularity, giving rise to the well-posedness of a new family of McKean-Vlasov SDEs. Our strategy naturally extends to the cases of a fractional Brownian driving noise BHBH for all H is an element of(0,1)H is an element of(0,1), obtaining new results in each separate case H is an element of(0,(1/)(2)) and H is an element of((1/)(2),1). In the SPDE setting, we construct McKean-Vlasov-type SPDEs with bounded measurable coefficients from the prototype of stochastic heat equation in spatial dimension one, and we do the same construction for the stochastic wave equation and a SPDE with white noise acting only on the boundary.