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Title: Limit Theorems - for Metastable Markov Chains
Author: Samarth Tiwari, Zsolt Pajor-Gyulai

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Limit Theorems
for Metastable Markov Chains
Samarth Tiwari, Zsolt Pajor-Gyulai
Courant Institute of Mathematical Sciences

27 October, 2017

Courant Institute of Mathematical Sciences

Formulation of the Problem

What is a discrete-time Markov Chain?
What is a continuous-time Markov Process?
What is a continuous-time semi-Markov Process?
How do we analyze families of stochastic processes?
How is their behavior dependent on time-scales?

Courant Institute of Mathematical Sciences

We only consider stochastic processes on a Finite State
Space S.
For discrete-time, transition probabilities of the Markov chain
Z are given as pi,j := probability of transitioning to state j if
process is currently at state i.
Process is Markov: transition probabilities do not depend on
the history. In other words, if at time t the process is at state
i, the path it took to get to this state is irrelevant for future
transitions.

Courant Institute of Mathematical Sciences

Well known result: for every state i ∈ S, ∃µi : S → [0, 1] such
that

lim P(Z (n) = j Z (0) = i) = µi (j)
n→∞

This µi is the Ergodic measure of Z starting at i.

Courant Institute of Mathematical Sciences

First generalization: to continuous time.
Now the transitions of the Markov process do not occur at
every integer time. Transitions are modeled through N0 − 1
exponential random variables

Courant Institute of Mathematical Sciences

Suppose Z (0) = i is the initial point of the Markov Process.
Let zj be pairwise independent exponential random variables
with parameters pi,j Their PDFs are:
fzk (t) = pi,j exp(−pi,j t)
These can be considered ”exponential alarm clocks”. The
moment an alarm clock goes off, the process transitions to the
respective state, and a new set of alarm clocks are considered.

Courant Institute of Mathematical Sciences

Property of exponential random variables: the minimum of
finitely many independent exponential r.v.s is itself an
exponential r.v.
Thus, the amount of time the process spends at a state is an
exponential r.v. There is a similar ergodic theorem for
continuous time Markov chains.

Courant Institute of Mathematical Sciences

Second generalization: Asymptotics
We consider a family of continuous-time Markov processes Z ε ,
for ε &gt; 0. Each of these processes have transition probabilities
ε
pi,j
, and we analyze the limiting behavior, as ε → 0.

Courant Institute of Mathematical Sciences

Notion of time-scales: time-scales are suitably defined
functions f : (0, ∞) → (0, ∞) that satisfy
lim f (ε) = ∞

ε→∞

Courant Institute of Mathematical Sciences