Markov Chains and Mixing Times

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

“…Now, standard results [16] state that h y (w) := P w (X τ A = y) is the unique harmonic extension of the function 1 {y} from A to S. In other words, h y is the unique function so that P h y (w) = h y (w) for all w ∈ A c , and h y (w) = 1 {y} (w) on A. This can be rewritten as (P | A c − 1)h y (w) = −P 1 {y} (w) = −p(w, y) for all w ∈ A c .…”

“…When the Markov chain X itself is reversible, Proposition 2.1 is a direct consequence of the well established theory of electrical networks: for example, from Proposition 9.5 in [16] it follows that µ(x)P x (τ + y < τ + x ) = c 0 C(x ↔ y), where C(x ↔ y) is the effective conductance between x and y, and c 0 a global constant. Since C(x ↔ y) is symmetric in x and y, (2.1) follows in the reversible case.…”

“…The following statement could be taken as a definition of P 0 -essential classes and P 0 -transient states; the proof of equivalence to the traditional definition of essential classes (see e.g. [16]) is very easy, and omitted here. Here and below, we will say that a direct path γ is P 0 -relevant if P 0 (γ) = lim ε→0 P ε (γ) > 0, otherwise P 0 -irrelevant.…”

Fast convergence of quantized consensus using Metropolis chains

“…Remark 2: Given a nonempty subset B ⊂ Ω and a Markov chain with an irreducible transition matrix P , every function h(·) : Ω → R which is harmonic over Ω \ B and nonnegative on B must be nonnegative on Ω [20].…”

“…Proof: A short proof of this equality can be found in Lemma 10.10 of [20]. Note that this property is known as the transitivity of reversible Markov chains.…”