Large time behavior of solutions to semi-linear equations with quadratic growth in the gradient

This paper studies the large time behavior of solutions to semi-linear Cauchy problems with quadratic nonlinearity in gradients. The Cauchy problem considered has a general state space and may degenerate on the boundary of the state space. Two types of large time behavior are obtained: i) pointwise convergence of the solution and its gradient; ii) convergence of solutions to associated backward stochastic di�erential equations. When the state space is Rd or the space of positive de�nite matrices, both types of convergence are obtained under growth conditions on coe�cients. These large time convergence results have direct applications in risk sensitive control and long term portfolio choice problems.