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.
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