Solutions of semilinear parabolic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations. Parabolic equations are intended in a mild sense that reveals to be suitable also towards applications to optimal control.
"Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control." Ann. Probab. 30 (3) 1397 - 1465, July 2002. https://doi.org/10.1214/aop/1029867132