Infinite horizon BSDEs with dissipative coefficients in Hilbert spaces and applications

In this paper we study a class of infinite horizon backward stochastic differential equations (BSDEs) of the form d Y ( t ) = λ Y ( t ) d t − f ( t , Y ( t ) , Z ( t ) ) d t + Z ( t ) d W ( t ) , 0 ⩽ t < ∞ , in a real separable Hilbert space, where λ is a given real parameter and the coefficient f is dissipative in y and Lipschitz in z. By Yosida approximation to dissipative mappings we show existence and uniqueness of the solutions for these equations. This result is applied to construct unique viscosity solutions to semilinear elliptic partial differential equations (PDEs).