扩散过程代数式收敛定性的判别准则

Qualitative Criterion for Algebraic Convergence of Diffusion Processes

本文定性地讨论非紧空间中可逆扩散过程的代数式收敛的判定 .使用分裂空间的方法 .将全空间分裂成两个部分 :紧的子空间与非紧的余子空间 .在紧子空间中考虑边界反射的Neumann过程 ,它必然是代数式收敛的 .而在非紧子空间中考虑边界吸收的Dirichlet过程 ,如果这一Dirichlet过程以代数式的速度击中边界 ,那么就有原过程在全空间代数式收敛 ;反之 ,原过程代数式收敛 ,非紧子空间中的Dirichlet过程也是代数式收敛的 .因此过程在紧子空间的任意摄动不会影响在全空间的代数式收敛性 .

Algebraic L2 convergence is qualitatively studied for reversible diffusion processes on noncompact space.Using the skill of space splitting,the article obtains a qualitative criterion for algebraic L2 decay.That is,dividing the whole space into two parts,one is a compact domain,the other is noncompact.The conclusion is:the convergence rates of the process in the original space are consistent with the rates of hitting the boundary by Dirichlet process in the noncompact domian.All of them or non of them is algebraic L2 convergence.Thus,any perturbation of the operator in the compact domain can not affect the convergence rates.