Abstract.
In this paper, we study the existence and uniqueness of mild solutions to a possibly degenerate elliptic partial differential equation \({\mathcal{L}}u(x) + \psi(x, u(x), \nabla u(x)G(x)) - \lambda u(x) = 0\) in Hilbert spaces. Our aim is, in the case in which ψ(·, 0, 0) is bounded, to drop the assumptions on the size of λ needed in [11]. The main tool will be existence, uniqueness and regular dependence on parameters of a bounded solution to a suitable backward stochastic differential equation with infinite horizon. Finally we apply the result to study an optimal control problem.
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Hu, Y., Tessitore, G. BSDE on an infinite horizon and elliptic PDEs in infinite dimension. Nonlinear differ. equ. appl. 14, 825–846 (2007). https://doi.org/10.1007/s00030-007-6029-5
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DOI: https://doi.org/10.1007/s00030-007-6029-5