Well-posedness and Optimal Regularity of Stochastic Evolution Equations with Multiplicative Noises

In this paper, we establish the well-posedness and optimal trajectory regularity for the solution of stochastic evolution equations with generalized Lipschitz-type coefficients driven by general multiplicative noises. To ensure the well-posedness of the problem, the linear operator of the equations is only need to be a generator of a $\CC_0$-semigroup and the proposed noises are quite general, which include space-time white noise and rougher noises. When the linear operator generates an analytic $\CC_0$-semigroup, we derive the optimal trajectory regularity of the solution through a generalized criterion of factorization method.