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Stationary Points in Coalescing Stochastic Flows on $\Mathbb {R} $

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Stationary Points in Coalescing Stochastic Flows on $\Mathbb {R} $ Stationary points in coalescing stochastic flows on R A. A. Dorogovtsev, G. V. Riabov, B. Schmalfuß Abstract. This work is devoted to long-time properties of the Arratia flow with drift { a stochastic flow on R whose one-point motions are weak solutions to a stochastic differential equation dX(t) = a(X(t))dt + dw(t) that move independently before the meeting time and coalesce at the meeting time. We study special modification of such flow (constructed in [1]) that gives rise to a random dynamical system and thus allows to discuss stationary points. Existence of a unique stationary point is proved in the case of a strictly monotone Lipschitz drift by developing a variant of a pullback procedure. Connections between the existence of a stationary point and properties of a dual flow are discussed. 1 Introduction In the present paper we investigate a long-time behaviour of the Arratia flow [2, 3] and its generalizations { Arratia flows with drifts. These objects will be introduced in the framework of stochastic flows on R. Following [4], by a stochastic flow on R we understand a family f s;t : −∞ < s ≤ t < 1g of measurable random mappings of R that possess two properties. 1. Evolutionary property: for all r ≤ s ≤ t; x 2 R;! 2 Ω s;t(!; r;s(!; x)) = r;t(!; x) (1.1) and s;s(!; x) = x: 2. Independent and stationary increments: for t1 < : : : < tn mappings t1;t2 ;:::; tn−1;tn are independent and t1;t2 is equal in distribution to 0;t2−t1 : n For each n ≥ 1 an R -valued stochastic process t ! ( s;t(x1); : : : ; s;t(xn)); t ≥ s will be called an n−point motion of the stochastic flow f s;t : −∞ < s ≤ t < 1g. In [4, Th. 1.1] it is proved that distributions of all finite-point motions uniquely define the distribution of a stochastic flow, and, actually, one can construct a stochastic flow by specifying distributions of all its finite-point motions in a consistent way. Using this arXiv:1808.05969v1 [math.PR] 17 Aug 2018 approach we give a definition of the Arratia flow with drift. Throughout the paper a : R ! R is a Lipschitz function. The Borel σ-field on Rn will be denoted by B(Rn): Consider a SDE dX(t) = a(X(t))dt + dw(t); (1.2) where w is a Wiener process. For every x 2 R the equation (1.2) has a unique strong solution fXx(t): t ≥ 0g and defines a Feller semigroup of transition probabilities on R [5, Ch. V, Th. (24.1)] (1) Pt (x; A) = P(Xx(t) 2 A); 1 t ≥ 0; x 2 R;A 2 B(R): Further, n (n);ind: Y (1) P ((x1; : : : ; xn);A1 × ::: × An) = Pt (xi;Ai); i=1 n n where t ≥ 0; (x1; : : : ; xn) 2 R ;A1;:::;An 2 B(R ); defines a Feller transition probability on Rn that corresponds to an n-dimensional SDE dXi(t) = a(Xi(t))dt + dwi(t); 1 ≤ i ≤ n; (n);ind: where w1; : : : ; wn are independent Wiener processes. The sequence fP : n ≥ 1g is n k consistent [4], i.e. given t ≥ 0; x 2 R ; 1 ≤ i1 < i2 < : : : < ik ≤ n; Bk 2 B(R ) and n Cn = fy 2 R :(yi1 ; : : : ; yik ) 2 Bkg; one has (n);ind: (k);ind: Pt (x; Cn) = Pt ((xi1 ; : : : ; xik );Bk): Finite-point motions of the Arratia flow with drift a are specified via the result of [4] (see also [1, L. 4.1]). Lemma 1.1. [4, Th. 4.1] There exists a unique consistent sequence of Feller transition (n);c semigroups fPt : n ≥ 1g (the so-called coalescing transition semigroups) such that (n);c n 1. for every n ≥ 1 fPt : t ≥ 0g is a transition semigroup on R ; 2. for all x 2 R and t ≥ 0 (2);c Pt ((x; x); ∆) = 1; where ∆ = f(y; y): y 2 Rg is a diagonal; n n 3. Given x 2 R let X = (X1;:::;Xn) be an R -valued Feller process with the starting (n);c ~ ~ ~ n point x and transition probabilities fPt : t ≥ 0g and X = (X1;:::; Xn) be an R - (n);ind: valued Feller process with the starting point x and transition probabilities fPt : t ≥ 0g. Let τ = infft ≥ 0j9i; j : 1 ≤ i < j ≤ n; Xi(t) = Xj(t)g be the first meeting time for processes X1;:::;Xn; and ~ ~ τ~ = infft ≥ 0j9i; j : 1 ≤ i < j ≤ n; Xi(t) = Xj(t)g ~ ~ be the first meeting time for processes X1;:::; Xn: Then distributions of stopped ~ ~ processes f(X1(t^τ);:::;Xn(t^τ)) : t ≥ 0g and f(X1(t^τ~);:::; Xn(t^τ~)) : t ≥ 0g coincide. Definition 1.1. A stochastic flow f s;t : −∞ < s ≤ t < 1g is the Arratia flow with drift n a; if for all s 2 R; n ≥ 1 and x = (x1; : : : ; xn) 2 R the finite-point motion t ! ( s;s+t(x1); : : : ; s;s+t(xn)); t ≥ 0 (n);c is a Feller process with a starting point x and transition probabilities fPt : t ≥ 0g: 2 Remark 1.1. Because of the coalescence random mappings x ! s;t(x) are not bijections. 2 Hence, the family f s;t : −∞ < s ≤ t < 1g cannot be extended to all pairs (s; t) 2 R in such a way that the evolutinary property (1.1) holds for all r; s; t 2 R. In the terminology of [6] the family f s;t : −∞ < s ≤ t < 1g should be called rather a semiflow. As we won't deal with bijective mappings in this paper, we will keep the term \stochastic flow” for the family . Informally, the Arratia flow with drift is a system of coalescing (weak) solutions of the stochastic differential equation (1.2) that start from every time-space point (t; x) 2 R2 and move independently up to the moment of meeting. The case a = 0 corresponds to the Arratia flow { a system of coalescing Brownian motions that are independent before meeting time. As it was mentioned, existence and uniqueness of the Arratia flow with drift follows from general results [4, Th. 1.1, 4.1]. The main objective of the present work is to study the long-time behaviour of random dynamical systems generated by the Arratia flow with drift. Appropriate modifications of coalescing stochastic flows on R that are random dynamical systems (in the sense of L. Arnold) was constructed in [1]. For Arratia flows with drift the existence result can be stated as follows. Theorem 1.1. [1, Th. 1.1] There exists a probability space (Ω; F; P) equipped with a measurable group (θt)t2R of measure-preserving transformations of Ω and a measurable mapping ' : R+ × Ω × R ! R such that 1. ' is a perfect cocycle over θ : 8s; t ≥ 0;! 2 Ω; x 2 R '(t + s; !; x) = '(t; θs!; '(s; !; x)); (1.3) 2. a stochastic flow f s;t : −∞ < s ≤ t < 1g defined by s;t(!; x) = '(t − s; θs!; x) is the Arratia flow with drift a in the sense of definition 1.1. Remark 1.2. In the terminology of random dynamical systems, evolutionary property (1.1) of can be rephrased as a perfect cocycle property of ': In this paper we deal only with stochastic flows that satisfes the evolutionary property with no exceptions. Such modifications of coalescing stochastic flows already appeared in [3] (for the Arratia flow) and in [4, 7] (for general stochastic flows). Main distinction of a theorem 1.1 modification is that it combines perfect cocycle property of ' with the measurability of the group of shifts θ: It must be noted that a number of various modifications of the Arratia flow that do not deal with the group of shifts of underlying probability space appeared in [8, 9, 10, 11, 12]. The long-time behaviour of the Arratia flow (in the driftless case a = 0) looks simple. Indeed, trajectories in the Arratia flow move like independent Wiener processes before meeting, hence with probability 1 each pair of trajectories meets in a finite time and 3 coalesces into a one trajectory. From this point of view it is natural to ask whether there is a path in the Arratia flow which is infinite in both directions, i.e. is there a (random) continuous function t ! ηt(!); t 2 R; such that for all t 2 R;! 2 Ω there exist (random) s ≤ t and x 2 R with ηt(!) = s;t(!; x); t ≥ s? If true this would indicate the existence of a stationary point for the corresponding random dynamical system in the sense of the following standard definition. Definition 1.2. [13, x1.4] A random variable η is a stationary point for the random dynamical system '; if there exists a forward-invariant set of full-measure Ω0 2 F (i.e. θt(Ω0) ⊂ Ω0 for all t ≥ 0), such that for all ! 2 Ω0 and t ≥ 0 '(t; !; η(!)) = η(θt!): Importance of stationary points for random dynamical systems stems from the fact that they define invariant measures for the skew-product flow Θt(!; x) = (θt!; φ(t; !; x)) by the relation µ(d!; dx) = δη(!)(dx)P(d!): Conversely, for a strictly monotone continuous random dynamical system on R every ergodic invariant measure for the skew-product flow Θ is generated by some stationary point [13, Th. 1.8.4]. The main question we address in this paper is the existence (and uniqueness) of a stationaty point for an Arratia flow with drift. Comparing to the well-studied case of continuous random dynamical systems [13, 14, 15] there are two new effects brought by coalescence.
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