Solutions of semilinear elliptic differential equations in infinite-dimensional spaces are obtained by means of forward and backward infinite-dimensional stochastic evolution equations. The backward equation is considered on an infinite time horizon and a suitable growth condition replaces the final condition. Elliptic equations are intended in a mild sense, suitable also for applications to optimal control. We finally notice that, due to the lack of smoothing properties, the elliptic partial differential equation considered here could not be treated by analytic methods.
"Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces." Ann. Probab. 32 (1B) 607 - 660, January 2004. https://doi.org/10.1214/aop/1079021459