Stationary points in coalescing stochastic flows on R

On a Brownian motion conditioned to stay in an open set

“…In section 3 we prove Theorem 1.1 in full generality, by adapting the approach of [1]. Finally, in section 4 we apply the result to the distribution of boundaries of clusters in the Arratia flow, and obtain analogous results for an unbounded cluster in the Arratia flow with drift [5].…”

“…The following analogue of the lemma 4.1 follows from properties of the dual flow { ψt,s : −∞ < s ≤ t < ∞} obtained in [4]. In [5] it was proved that with probability 1,…”

“…For general properties of stochastis flows we refer to [2]. In our previous works [3,4,5] properties of clusters in certain coalescing stochastic flows were investigated. To illustrate the results and related questions, let us consider the Arratia flow on R. A stochastic flow {ψ s,t : −∞ < s ≤ t < ∞} is called the Arratia flow, if for all s ∈ R, n ≥ 1 and…”

“…The Arratia flow with drift a is a stochastic flow ψ such that each trajectory t → ψ s,t (x) is a weak solution of the stochastic differential equation dψ s,t (x) = a(ψ s,t (x))dt + dw s,x (t), every two trajectories move independently before they meet each other, at the meeting time trajectories coalesce (see section 4.2 for the precise definition). In [5] it was proved that if a ′ (x) ≤ −λ < 0 a.s., then there exists a unique stationary process {η t } t∈R such that for all s ≤ t, ψ s,t (η s ) = η t . At every moment t ≥ 0 there exists an interval of points that have coalesced into η 0 at time 0 :…”

“…[5] Let ψ be the Arratia flow with drift a. Assume that the drift a is Lipschitz and for some λ > 0 and all x, y ∈ R one has (a(x) − a(y))(x − y) ≤ −λ(x − y) 2 .…”