论文标题
可稳定的$γ$ - 有限状态马尔可夫连锁级的二级大偏差率功能
Metastable $Γ$-expansion of finite state Markov chains level two large deviations rate functions
论文作者
论文摘要
我们检查了马尔可夫链亚稳态行为的两个分析表征。第一个以其过渡概率表达，第二个以其较大偏差率的功能来表达。 考虑一系列连续的时间马尔可夫链$（x^{（n）} _ t：t \ ge 0）$在固定有限状态空间$ v $上演变。根据关于跳高率的假设，我们证明存在时间尺度$θ^{（（p）} _ n $，并且与脱节的概率度量支持$π^{（p）} _ j $，$ j $，$ j \ in s_p $，$ 1 \ le p \ le p \ le p \ le p \ le q $ q $ in（ $θ^{（k+1）} _ n/θ^{（（k）} _ n \ to \ infty $，（b）对于所有$ p $，$ x \ in v $，$ t> 0 $，从$ x $开始，从$ x $开始，$ x^{（n）} $ x^{（n）} _ {t uniftity n y Infty t unsiste概率的凸组合测量$π^{（p）} _ j $。凸组合的权重自然取决于$ x $和$ t $。 令$ i_n $为$ x^{（n）} _ t $的第二级大偏差率，为$ t \ to \ infty $。在相同的假设上，关于跳高速度和假设，此外，该过程是可逆的，我们证明$ i_n $可以写入$ i_n = i^{（0）} \，+\，\ sum_ {1 \ le le p \ le p \ le p \ le q}（1/θ^{（p） $ i^{（p）} $仅在凸的组合中以$π^{（p）} _ j $：$ i^{（p）}（μ）<\ infty $ if，仅在s_p} j $ in Motor s_p} op s__jj \，po）j $ j \ y y j $ j \ j $ j \ jjj \，po）的$π^{（p）} _ j $：$ i^{（p）<\ infty $ j $ j $ j $ j $ j $ j \ j \，jjjj \，po） $ω$ in $ s_p $。
We examine two analytical characterisation of the metastable behavior of a Markov chain. The first one expressed in terms of its transition probabilities, and the second one in terms of its large deviations rate functional. Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Under a hypothesis on the jump rates, we prove the existence of times-scales $θ^{(p)}_n$ and probability measures with disjoint supports $π^{(p)}_j$, $j\in S_p$, $1\le p \le q$, such that (a) $θ^{(1)}_n \to \infty$, $θ^{(k+1)}_n/θ^{(k)}_n \to \infty$, (b) for all $p$, $x\in V$, $t>0$, starting from $x$, the distribution of $X^{(n)}_{t θ^{(p)}_n}$ converges, as $n\to\infty$, to a convex combination of the probability measures $π^{(p)}_j$. The weights of the convex combination naturally depend on $x$ and $t$. Let $I_n$ be the level two large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under the same hypothesis on the jump rates and assuming, furthermore, that the process is reversible, we prove that $I_n$ can be written as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/θ^{(p)}_n) \, I^{(p)}$ for some rate functionals $I^{(p)}$ which take finite values only at convex combinations of the measures $π^{(p)}_j$: $I^{(p)}(μ) < \infty$ if, and only if, $μ= \sum_{j\in S_p} ω_j\, π^{(p)}_j$ for some probability measure $ω$ in $S_p$.
