In this work we prove the existence and uniqueness up to a stopping time for the stochastic counterpart of Tosio Kato's quasilinear evolutions in UMD Banach spaces. These class of evolutions are known to cover a large class of physically important nonlinear partial differential equations. Existence of a unique maximal solution as well as an estimate on the probability of positivity of stopping time is obtained. An example of stochastic Euler and Navier–Stokes equation is also given as an application of abstract theory to concrete models.
Mohan, M. T., & Sritharan, S. S. (2017). Stochastic quasilinear evolution equations in UMD Banach spaces. Mathematische Nachrichten, 290(13), 1971–1990. https://doi.org/10.1002/mana.201600015