On the $f$-Norm Ergodicity of Markov Processes in Continuous Time
On the $f$-Norm Ergodicity of Markov Processes in Continuous Time
复制标题
连续时间内马尔可夫过程的 $f$-范数遍历性
DOI：
--
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
S. Meyn
中科院分区：
文献类型：
--
作者：
Ioannis Kontoyiannis;S. Meyn
考虑一个马尔可夫过程${Phi(t): tgeq 0}$在波兰空间${sf X}$上的演化。得到$f$-范数遍历定理的一个版本：假设过程是$psi$-不可约且非周期的。对于给定的函数$fcolon{sf X}: 0 [1,infty)$，在适当的条件下，下列过程是等价的：egin{enumerate} item[(i)]有一个唯一的不变概率度量$pi$满足$int f,dpi 0$，它是‘ ’ self $f$-regular。存在一个函数$Vcolon{sf X} o (0, inty]$，它在${sf X}$中至少有一个点是有限的，满足以下Lyapunov漂移条件，[{cal D} Vleq - f+bfield{I}_C,, eqno{hbox{(V3)}}]，其中$C$是一个闭小集，${cal D}$是过程的扩展生成器。对于离散时间链，结果是众所周知的。此外，还得到了$fPhi$在合适范数下的遍历性：对于满足$V(X)< inty $的初始条件$xin{sf X}$，对于任意函数$gcolon{sf X} ore $，其中$|g|$以$f$为界，[lim_{t oinfty} {sf E}_x[g(Phi(t))] = int g,dpi。在过程${Phi(t)}$或函数$g$的适当假设下，探索在连续时间内建立相应结果的适当版本的可能方法。
Consider a Markov process ${Phi(t) : tgeq 0}$ evolving on a Polish space ${sf X}$. A version of the $f$-Norm Ergodic Theorem is obtained: Suppose that the process is $psi$-irreducible and aperiodic. For a given function $fcolon{sf X}: o[1,infty)$, under suitable conditions on the process the following are equivalent: egin{enumerate} item[(i)] There is a unique invariant probability measure $pi$ satisfying $int f,dpi 0$ that is ``self $f$-regular.' item There is a function $Vcolon{sf X} o (0,infty]$ that is finite on at least one point in ${sf X}$, for which the following Lyapunov drift condition is satisfied, [ {cal D} Vleq - f+bfield{I}_C, , eqno{hbox{(V3)}} ] where $C$ is a closed small set and ${cal D}$ is the extended generator of the process. end{enumerate} For discrete-time chains the result is well-known. Moreover, in that case, the ergodicity of $fPhi$ under a suitable norm is also obtained: For each initial condition $xin{sf X}$ satisfying $V(x)<infty$, and any function $gcolon{sf X} oRe$ for which $|g|$ is bounded by $f$, [ lim_{t oinfty} {sf E}_x[g(Phi(t))] = int g,dpi. ] Possible approaches are explored for establishing appropriate versions of corresponding results in continuous time, under appropriate assumptions on the process ${Phi(t)}$ or on the function $g$.