1.3. EFFECTIVE DYNAMICS VIA DICRETE MARKOV CHAINS 15

consider a Markov chain Y

ε

= (Y

ε(k))k≥0

on the state space S = {−1, 1}. Let

PT (k) be the matrix of one-step transition probabilities at time k. If we denote

pT

−(k)

= P(Y

ε(k)

= −1), pT

+(k)

= P(Y

ε(k)

= 1), and write P

∗

for the transposed

matrix, we have

pT

−

(k + 1)

pT

+(k

+ 1)

= PT

∗(k)

pT

−

(k)

pT

+(k)

.

In order to model the periodic switching of the double-well potential in our

Markov chains, we define the transition matrix PT to be periodic in time with

period T . More precisely,

PT (k) =

Q1, 0 ≤ k mod T ≤

T

2

− 1,

Q2,

T

2

≤ k mod T ≤ T − 1,

with

(1.12)

Q1 =

1 − ϕ ϕ

ψ 1 − ψ

, Q2 =

1 − ψ ψ

ϕ 1 − ϕ

,

ϕ =

pe−V/ε,

ψ =

qe−v/ε,

where 0 ≤ p, q ≤ 1, 0 v V +∞, 0 ε ∞.

The entries of the transition matrices clearly are designed to mimic transition

rates between −1 and 1 or vice versa that correspond to the transition times of

the diffusion processes between the meta-stable equilibria, given according to the

preceding section by exp(

V

ε

) resp. exp(

v

ε

). The exponential factors in the one-step

transition probabilities are just chosen to be the inverses of those mean transition

times. This is exactly what elementary Markov chain theory requires in equilibrium.

The phenomenological prefactors p and q, chosen between 0 and 1, add asymmetry

to the picture.

It is well known that for a time-homogeneous Markov chain on {−1, 1} with

transition matrix PT one can talk about equilibrium, given by the stationary distri-

bution, to which the law of the chain converges exponentially fast. The stationary

distribution can be found by solving the matrix equation π = PT

∗π

with norming

condition

π−

+

π+

= 1.

For time non homogeneous Markov chains with time periodic transition matrix,

the situation is quite similar. Enlarging the state space S to ST = {−1, 1} ×

{0, 1,...,T − 1}, we recover a time homogeneous chain by setting

Zε(k)

= (Y

ε(k),k

mod T ), k ≥ 0,

to which the previous remarks apply. For convenience of notation, we assume ST

to be ordered in the following way:

ST = (−1, 0), (1, 0), (−1, 1), (1, 1),..., (−1,T − 1), (1,T − 1) .

Writing AT for the matrix of one-step transition probabilities of

Zε,

the station-

ary distribution R = (r(i,

j))∗

is obtained as a normalized solution of the matrix

equation (AT

∗

− E)R = 0, E being the identity matrix. We shall be dealing with

the following variant of stationary measure, which is not normalized in time. Let

πT (k) = (πT

−

(k),πT

+

(k))∗

=

(r(−1,k),r(1,k))∗,

0 ≤ k ≤ T − 1. We call the family

πT = (πT (k))0≤k≤T

−1

the stationary distribution of the Markov chain Y

ε.