On 1-point densities for Arratia flows with drift

Abstract

We show that if drift coefficients of Arratia flows converge in L1(R) or L∞(R) then the 1-point densities associated with these flows converge to the density for the flow with the limit drift. The Arratia flow, a continual system of coalescing Wiener processes that are independent until they meet, was introduced independently as a limit of coalescing random walks in [1], as a system of reflecting Wiener processes in [2] and as a limit of stochastic homeomorphic flows in [3]. If interpreted as a collection of particles started at 0, it is a part of the Brownian web [4]. At the same time, one can construct the Arratia flow with drift using flows of kernels defined in [5] (see [6, §6] for a short explanation) or directly via martingale problems by adapting the method used in [7] to build coalescing stochastic flows with more general dependence between particles (see [8, Chapter 7] for this approach). Following [8, Chapter 7], we consider a modification of the Arratia flow that introduces drift affecting the motion of a particle within the flow. So by an Arratia flow X ≡ {Xa(u, t) | u ∈ R, t ∈ R+} with bounded measurable drift a we understand a collection of random variables such that (1) for every u the process X(u, ·) is an Itô process with diffusion coefficient 1 and drift a; (2) for all t ≥ 0 the mapping Xa(·, t) is monotonically increasing; (3) for any u1, u2 the joint quadratic covariation of the martingale parts of X (u1, ·) and X(u2, ·) equals (t − inf{s | X(u1, s) = X(u2, s)})+, with inf ∅ being equal ∞ by definition. Since the set {Xa(u, t) | u ∈ R} is known to be locally finite for all t > 0 [8, Chapter 7], one defines for any t > 0 the point process {|Xa(A, t)| | A ∈ B(R)}. Studying of such a point process can be performed using point densities (see: [9, 10] for a definition and a representation in terms of Pfaffians in the case of zero drift, respectively; [11, 12] for representations in the case of non-trivial drift; [13, 14] for applications to the study of the above-mentioned point process). In accordance with [9, Appendix B], the following definition is adopted in the present paper: the 1-point density at time t of the point 2020 Mathematics Subject Classification. Primary 60H10; Secondary 60K35, 60G55, 35C10.