We study the large time behavior of solutions to Cauchy problem for semilinear parabolic equations having quadratic nonlinearity in gradients. Equations this kind appear stochastic control theory. It turns out that as tends infinity solution converges a associated ergodic problem. Our approach relies on PDE and probabilistic arguments.

The paper is concerned with ergodic-type Bellman equations arising typically in linear (exponential) quadratic Gaussian control. We are interested giving recurrence-transience criteria for associated optimal feedback diffusions terms of qualitative properties solutions to the equation. To establish such criteria, we propose a new approach which based on Lyapunov method. It turns out that certain convexity equation plays key role our arguments.

We study the long time behavior of viscosity solutions Cauchy problem for Hamilton–Jacobi equations in ℝ n . prove that if Hamiltonian H(x, p) is coercive and strictly convex a mild sense p upper semi-periodic x, then any solution “converges” to an asymptotic lower semi-almost periodic initial function.

We discuss homogenization for stochastic partial differential equations (SPDEs) of Zakai type with periodic coefficients appearing typically in nonlinear filtering problems. prove such by two different approaches. One is rather analytic and the other comparatively probabilistic.

We investigate the large-time behavior of solutions Cauchy problem for Hamilton-Jacobi equations on real line R.We establish a result convergence to asymptotic as time t goes infinity.

We discuss the homogenization of stochastic partial differential equations whose coefficients are rapidly oscillating and perturbed by a diffusion process. Such class appear in nonlinear filtering problems with feedback. specify constant limit equation. The constants essentially different from case where do not contain factors.

In this paper we study the ergodic problem for viscous Hamilton--Jacobi equations with superlinear Hamiltonian and inward drift. We investigate (i) existence uniqueness of eigenfunctions associated generalized principal eigenvalue problem, (ii) relationships corresponding stochastic control both finite infinite time horizon, (iii) precise growth exponent respect to a perturbation potential function.

This paper is concerned with some long-run average cost problems for controlled Markov chains a denumerable state space. The criterion to be optimized contains both reward and penalty functions. As trade-off between penalty, we observe certain phase transition phenomena. Our results also provide stochastic optimal control interpretation transitions of discrete homopolymers finite attracting potentials.

This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations exponent m \textgreater{} 2. We prove that generalized principal eigenvalue of equation converges to a constant as $\rightarrow$ $\infty$, and limit coincides an gradient constraint. also investigate some qualitative properties respect perturbation potential function. It turns out different situations take place according = 2, 2 \textless{} limiting case $\infty$.