BSDE on an infinite horizon and elliptic PDEs in infinite dimension

The present paper studies the existence and the uniqueness of the mild solution of the partial differential equation (PDE) ${\scr L}u(x)+\psi(x, u(x),\nabla u(x)G(x))=0,$ $x\in H$, where $H$ is a Hilbert space, and the second-order differential operator ${\scr L}$ is of the form ${\scr L}=\frac12{\rm tr}(GG^*(x)D^2)+\langle Ax,\nabla\rangle + \langle F(x),\nabla\rangle$, $A$ is the generator of a $C_0$-semigroup of bounded linear operators in $H$, $F$ and $G$ are functions satisfying an appropriate Lipschitz assumption and $\lambda>0$. Unlike earlier works that studied the above PDE with space-independent $G$ with the help of a fixed point argument, M. Fuhrman and G. Tessitore [Ann. Probab. 32 (2004), no. 1B, 607--660; MR2039938 (2005b:60167)] followed the idea, which is widely spread in the finite-dimensional case, to interpret the PDE stochastically with the help of a Markovian forward-backward stochastic differential equation (SDE) with infinite time horizon, in which the usual final condition of the backward equation is replaced by a suitable growth condition. This allowed them to prove the existence and the uniqueness for $\lambda>0$ sufficiently large. In the present paper the authors show that, under suitable assumptions, one has the existence and uniqueness for all $\lambda>0$, even if one still allows $G$ to depend on $x$ or to be degenerate. For this the authors use recent results by P. Briand and Y. Hu [J. Funct. Anal. 155 (1998), no. 2, 455--494; MR1624569 (99e:35015)] and M. Royer [Stoch. Stoch. Rep. 76 (2004), no. 4, 281--307; MR2075474 (2005k:60202)] on the existence and the uniqueness of bounded solutions of backward SDEs with infinite or random time horizon. The main technical point consists in proving the differentiability of such bounded solutions with respect to the initial datum $x$ of the forward SDE. This is done by the authors by either assuming that $G$ is nondegenerate or by taking the hypothesis that $G$ is constant and $A+\nabla F$ is dissipative. In the second part of the paper the results are applied for the study of a stochastic control problem.