Path-Dependent Hamilton--Jacobi Equations with Super-Quadratic Growth in the Gradient and the Vanishing Viscosity Method

The nonexponential Schilder-type theorem in Backhoff-Veraguas, Lacker, and Tangpi [Ann.\Appl. Probab., 30 (2020), pp. 1321--1367] is expressed as a convergence result for path-dependent partial differential equations with appropriate notions of generalized solutions. This entails non-Markovian counterpart to the vanishing viscosity method. We show uniqueness maximal subsolutions viscous Hamilton--Jacobi related convex super-quadratic backward stochastic equations. establish well-posedness Hamilton--Jacobi--Bellman equation associated Bolza problem calculus variations terminal cost. In particular, among lower semicontinuous solutions holds, state constraints are admitted.