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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 > 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

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