Stochastic Hamiltonian Systems: Exponential Convergence to the Invariant Measure, and Discretization by the Implicit Euler Scheme

#### D. Talay

2002, v.8, №2, 163-198

ABSTRACT

In this paper we carefully study the large time behaviour of $$u(t,x,y) := E_{x,y}f(X_t,Y_t)-\int f d\mu,$$ where $(X_t,Y_t)$ is the solution of a stochastic Hamiltonian dissipative system with non globally Lipschitz coefficients, $\mu$ its unique invariant law, and $f$ a smooth function with polynomial growth at infinity. Our aim is to prove the exponential decay to 0 of $u(t,x,y)$ and all its derivatives when $t$ goes to infinity, for all $(x,y)$ in $R^{2d}$. We apply our precise estimates on $u(t,x,y)$ to analyze the convergence rate of a probabilistic numerical method based upon the implicit Euler discretization scheme which approximates $\int f d\mu$.

Keywords: stochastic differential equations,stochastic Hamiltonian systems,parabolicpartial differential equation,invariant measure,Euler method,simulation