Markov processes in the continuous model of stochastic synchronization

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

“…We need to show that φ is its stationary distribution. For that, it suffices to show that φGh = 0, ∀h ∈ H, which holds by ( 16) and (11). ✷…”

“…Subsequently to [5,6], there has been a line of work generally focused on modeling synchronization between elements (particles) of a large system, with mean-field interaction of the elements (particles), cf. [9][10][11][12][13][14].…”

“…Then, by Lemma 6, φ is Lipschitz. Then, it remains to observe that for any x, the LHS and RHS of (11) give the steady-state rates at which the particle of the corresponding process X(•) crosses point x from right to left, and from left to right, respectively. Another way to complete the proof of 'if' is to observe that, since φ is a stationary distribution of X(•), and using Lemma 6(ii), we have…”

“…Let us prove the 'only if' part. Suppose φ is Lipschitz and it satisfies (11). Consider the process X(•), defining Aφ.…”

Time scales in large systems of Brownian particles with stochastic synchronization

“…The idea of the proof is to show that x • (t) satisfies the Doeblin property. Similar arguments were used in [4,8]. So we omit here the proofs of Theorems 1 and 2.…”