Stochastic Hamiltonian Systems : Exponential Convergence to the Invariant Measure , and Discretization by the Implicit Euler Scheme
Stochastic Hamiltonian Systems : Exponential Convergence to the Invariant Measure , and Discretization by the Implicit Euler Scheme
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发表时间：
2002
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通讯作者：
D. Talay
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作者：
D. Talay
本文研究了u（t, x,y）的大时间性质：= Ex,y f(Xt, Yt)−∫f dμ，其中（Xt, Yt）是具有非全局Lipschitz系数的随机Hamiltonian耗散系统的解，μ是其唯一不变律，f是在无穷远处多项式增长的光滑函数。我们的目的是证明当t趋于无穷时，对于r中的所有（x, y）， u（t, x, y）及其所有导数的指数衰减到0。我们应用我们对u（t, x, y）的精确估计来分析基于近似∫f dμ的隐式欧拉离散格式的概率数值方法的收敛速度。
In this paper we carefully study the large time behaviour of u(t, x, y) := Ex,y f(Xt, Yt)− ∫ f dμ, where (Xt, Yt) is the solution of a stochastic Hamiltonian dissipative system with non gbally Lipschitz coefficients, μ 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. 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 ∫ f dμ.