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 p−(k) T = P(Y ε (k) = −1), p+(k) T = P(Y ε (k) = 1), and write P for the transposed matrix, we have p−(k T + 1) p+(k T + 1) = PT (k) p−(k) T p+(k) T . 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 = P T 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) = (π−(k),π+(k))∗ T T = (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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