Stochastic flows in the Brownian web and net
It is known that certain one-dimensional nearest-neighbor random walks in i.i.d. random space-time environments have diffusive scaling limits. Here, in the continuum limit, the random environment is represented by a ‘stochastic flow of kernels’, which is a collection of random kernels that can be loosely interpreted as the transition probabilities of a Markov process in a random environment. The theory of stochastic flows of kernels was first developed by Le Jan and Raimond, who showed that each such flow is characterized by its -point motions. Our work focuses on a class of stochastic flows of kernels with Brownian -point motions which, after their inventors, will be called Howitt-Warren flows.
Our main result gives a graphical construction of general Howitt-Warren flows, where the underlying random environment takes on the form of a suitably marked Brownian web. This extends earlier work of Howitt and Warren who showed that a special case, the so-called ‘erosion flow’, can be constructed from two coupled ‘sticky Brownian webs’. Our construction for general Howitt-Warren flows is based on a Poisson marking procedure developed by Newman, Ravishankar and Schertzer for the Brownian web. Alternatively, we show that a special subclass of the Howitt-Warren flows can be constructed as random flows of mass in a Brownian net, introduced by Sun and Swart.
Using these constructions, we prove some new results for the Howitt-Warren flows. In particular, we show that the kernels spread with a finite speed and have a locally finite support at deterministic times if and only if the flow is embeddable in a Brownian net. We show that the kernels are always purely atomic at deterministic times, but, with the exception of the erosion flows, exhibit random times when the kernels are purely non-atomic. We moreover prove ergodic statements for a class of measure-valued processes induced by the Howitt-Warren flows.
Our work also yields some new results in the theory of the Brownian web and net. In particular, we prove several new results about coupled sticky Brownian webs and about a natural coupling of a Brownian web with a Brownian net. We also introduce a ‘finite graph representation’ which gives a precise description of how paths in the Brownian net move between deterministic times.
MSC 2010. Primary: 82C21 ; Secondary: 60K35, 60K37, 60D05.
Keywords. Brownian web, Brownian net, stochastic flow of kernels, measure-valued process, Howitt-Warren flow, linear system, random walk in random environment, finite graph representation.
Acknowledgement. R. Sun is supported by grants R-146-000-119-133 and R-146-000-148-112 from the National University of Singapore. J.M. Swart is sponsored by GAČR grants 201/07/0237 and 201/09/1931.
- 1 Introduction
- 2 Results for Howitt-Warren flows
- 3 Construction of Howitt-Warren flows in the Brownian web
- 4 Construction of Howitt-Warren flows in the Brownian net
- 5 Outline of the proofs
- 6 Coupling of the Brownian web and net
- 7 Construction and convergence of Howitt-Warren flows
- 8 Support properties
- 9 Atomic or non-atomic
- 10 Infinite starting mass and discrete approximation
- 11 Ergodic properties
- A The Howitt-Warren martingale problem
- B The Hausdorff topology
- C Some measurability issues
- D Thinning and Poissonization
- E A one-sided version of Kolmogorov’s moment criterion
In [LR04a], Le Jan and Raimond introduced the notion of a stochastic flow of kernels, which is a collection of random probability kernels that can be loosely viewed as the transition kernels of a Markov process in a random space-time environment, where restrictions of the environment to disjoint time intervals are independent and the environment is stationary in time. For suitable versions of such a stochastic flow of kernels (when they exist), this loose interpretation is exact, see Definition 2.1 below and the remark following it. Given the environment, one can sample independent copies of the Markov process and then average over the environment. This defines the -point motion for the flow, which satisfies a natural consistency condition: namely, the marginal distribution of any components of an -point motion is necessarily a -point motion. A fundamental result of Le Jan and Raimond [LR04a] shows that conversely, any family of Feller processes that is consistent in this way gives rise to an (essentially) unique stochastic flow of kernels.
As an example, in [LR04b], the authors used Dirichlet forms to construct a consistent family of reversible -point motions on the circle, which are -stable Lévy processes with some form of sticky interaction characterized by a real parameter . In particular, for , these are sticky Brownian motions. Subsequently, Howitt and Warren [HW09a] used a martingale problem approach to construct a much larger class of consistent Feller processes on , which are Brownian motions with some form of sticky interaction characterized by a finite measure on . In particular, if is a multiple of the Lebesgue measure, these are the sticky Brownian motions of Le Jan and Raimond. From now on, and throughout this paper, we specialize to the case of Browian underlying motions. By the general result of Le Jan and Raimond mentioned above, the sticky Brownian motions of Le Jan and Raimond, resp. Howitt and Warren, are the -point motions of an (essentially) unique stochastic flow of kernels on , which we call a Le Jan-Raimond flow, resp. Howitt-Warren flow (the former being a special case of the latter). It has been shown in [LL04, HW09a] that these objects can be obtained as diffusive scaling limits of one-dimensional random walks in i.i.d. random space-time environments.
The main goal of the present paper is to give a graphical construction of Howitt-Warren flows that follows as closely as possible the discrete construction of random walks in an i.i.d. random environment. In particular, we want to make explicit what represents the random environment in the continuum setting. The original construction of Howitt-Warren flows using -point motions does not tell us much about this. In [HW09b], it was shown that the Howitt-Warren flow with , known as the erosion flow, can be constructed using two coupled Brownian webs, where one Brownian web serves as the random space-time environment, while the conditional law of the second Brownian web determines the stochastic flow of kernels.
We will extend this construction to general Howitt-Warren flows, where in the general case, the random environment consists of a Brownian web together with a marked Poisson point process which is concentrated on the so-called points of type of the Brownian web. A central tool in this construction is a Poisson marking procedure invented by Newman, Ravishankar and Schertzer in [NRS10]. Of course, we also make extensive use of the theory of the Brownian web developed in [TW98, FINR04]. For a special subclass of the Howitt-Warren flows, we will show that alternatively the random space-time environment can be represented as a Brownian net, plus a countable collection of i.i.d. marks attached to its so-called separation points. Here, we use the theory of the Brownian net, which was developed in [SS08] and [SSS09].
Using our graphical construction, we prove a number of new properties for the Howitt-Warren flows. In particular, we give necessary and sufficient conditions in terms of the measure for the random kernels to spread with finite speed, for their support to consist of isolated points at deterministic times, and for the existence of random times when the kernels are non-atomic (Theorems 2.5, 2.7 and 2.8 below). We moreover use our construction to prove the existence of versions of Howitt-Warren flows with nice regularity properties (Proposition 3.8 below), in particular, versions which can be interpreted as bona fide transition kernels in a random space-time environment. Lastly, we study the invariant laws for measure-valued processes associated with the Howitt-Warren flows (Theorem 2.11).
Our graphical construction of the Howitt-Warren flows is to a large extent motivated by its discrete space-time counterpart, i.e., random walks in i.i.d. random space-time environments on . Many of our proofs will also be based on discrete approximation. Therefore, in the rest of the introduction, we will introduce a class of random walks in i.i.d. random space-time environments and some related objects of interest, and sketch heuristically how the Brownian web and the Brownian net will arise in the representation of the random space-time environment for the Howitt-Warren flows. An outline of the rest of the paper will be given at the end of the introduction.
Incidentally, we note that random walks in i.i.d. random space-time environments have been used in the physics literature to model the flow of stress in a granular medium, called the model, see e.g. [LMY01, JM11] and the references therein. The Howitt-Warren flows we consider are effectively scaling limits of so-called near-critical models.
1.2 Discrete Howitt-Warren flows
Let be the even sublattice of . We interpret the first coordinate as space and the second coordinate as time, which is plotted vertically in figures. Let be i.i.d. -valued random variables with common distribution . We view as a random space-time environment for a random walk, such that conditional on the environment , if the random walk is at time at the position , then in the next unit time step the walk jumps to with probability and to with the remaining probability (see Figure 1).
To formalize this, let denote the law of the environment and for each , let denote the conditional law, given the random environment , of the random walk in random environment we have just described, started at time at position . Since parts of the random environment belonging to different times are independent, it is not hard to see that under the averaged (or ‘annealed’) law , the process is still a Markov chain, which in each time step jumps to the right with probability and to the left with the remaining probability . Note that this is quite different from the usual random walk in random environment (RWRE) where the randomness is fixed for all time, and the averaged motion no longer has the Markov property.
We will be interested in three objects associated with the random walks in the i.i.d. random space-time environment , namely: random transition kernels, -point motions, and a measure-valued process. The law of each of these objects is uniquely characterized by and, conversely, uniquely determines .
First of all, the random environment determines a family of random transition probability kernels,
For each , the random variables are independent.
and are equal in law for each .
We call the collection of random probability kernels the discrete Howitt-Warren flow with characteristic measure . Such a collection is a discrete time analogue of a stochastic flow of kernels as introduced by Le Jan and Raimond in [LR04a] (see Definition 2.1 below).
Next, given the environment , we can sample a collection of independent random walks
in the random environment , started at time zero from deterministic sites , respectively. It is easy to see that under the averaged law
the process is still a Markov chain, which we call the discrete -point motion. Its transition probabilities are given by
Note that these discrete -point motions are consistent in the sense that any coordinates of are distributed as a discrete -point motion. Each coordinate is distributed as a nearest-neighbor random walk thats makes jumps to the right with probability . Because of the spatial independence of the random environment, the coordinates evolve independently when they are at different positions. To see that there is some nontrivial interaction when they are at the same position, note that if coordinates are at position at time , then the probability that in the next time step the first coordinates jump to while the last coordinates jump to equals , which in general does not factor into . Note that the law of is uniquely determined by its moments, which are in turn determined by the transition probabilities of the discrete -point motions (for each ).
Finally, based on the family of kernels , we can define a measure-valued process
where is any locally finite initial measure on . Note that conditional on , the process evolves deterministically, with
Under the law , the process is a Markov chain, taking values alternatively in the spaces of finite measures on and . Note that (1.6) says that in the time step from to , an -fraction of the mass at is sent to and the rest is sent to . Obviously, this dynamics preserves the total mass. In particular, if is a probability measure, then is a probability measure for all . We call the discrete Howitt-Warren process.
We will be interested in the diffusive scaling limits of all these objects, which will be (continuum) Howitt-Warren flows and their associated -point motions and measure-valued processes, respectively. Note that the discrete Howitt-Warren flow determines the random environment a.s. uniquely. The law of is uniquely determined by either the law of its -point motions or the law of its associated measure-valued process.
1.3 Scaling limits of discrete Howitt-Warren flows
We now recall from [HW09a] the conditions under which the -point motions of a sequence of discrete Howitt-Warren flows converge to the -point motions of a (continuum) stochastic flow of kernels, which we call a Howitt-Warren flow. We will then use discrete approximation to sketch heuristically how such a Howitt-Warren flow can be constructed from a Brownian web or net.
Let be positive constants tending to zero, and let
be probability laws on satisfying
for some and finite measure on , where
denotes weak convergence. Howitt and Warren [HW09a]
Now let us use discrete approximation to explain heuristically how to construct a Howitt-Warren flow based on a Brownian web or net. The construction based on the Brownian net is conceptually easier, so we consider this case first.
Let and let be a finite measure on . Assuming, as we must in this case, that , we may define a sequence of probability measures on by
Then is a probability measure on for sufficiently large (such that ), and the satisfy (1.7). Thus, when space is rescaled by and time by , the discrete Howitt-Warren flow with characteristic measure approximates a Howitt-Warren flow with drift and characteristic measure .
Let be i.i.d. with common law , which serves as the random environment for a discrete Howitt-Warren flow with characteristic measure . We observe that for large , most of the are either zero or one. In view of this, it is convenient to alternatively encode as follows. For each , if , then we call a separation point, set , and we draw two arrows from , leading respectively to . When , resp. , we draw a single arrow from to , resp. . Note that the collection of arrows generates a branching-coalescing structure, called discrete net, on (see Figure 2) and conditional on , the at separation points of are independent with common law . Therefore the random environment can be represented by the pair , where a walk in such an environment must navigate along , and when it encounters a separation point , it jumps either left or right with probability , resp. .
It turns out that the pair has a meaningful diffusive scaling limit. In particular, if space is scaled by and time by , then converges to a limiting branching-coalescing structure called the Brownian net, the theory of which was developed in [SS08, SSS09]. In particular, the separation points of have a continuum analogue, the so-called separation points of , where incoming trajectories can continue along two groups of outgoing trajectories. These separation points are dense in space and time, but countable. Conditional on , we can then assign i.i.d. random variables with common law to the separation points of . The pair provides a representation for the random space-time environment underlying the Howitt-Warren flow with drift and characteristic measure . A random motion in such a random environment must navigate along , and whenever it comes to a separation point , with probability resp. , it continues along the left resp. right of the two groups of outgoing trajectories in at . We will recall the formal definition of the Brownian net and give a rigorous construction of a random motion navigating in in Section 4.
We now consider Howitt-Warren flows whose characteristic measure is a general finite measure . Let satisfy (1.7) and let be an i.i.d. random space-time environment with common law . Contrary to the previous situation, it will now in general not be true that the most of the ’s are either zero or one. Nevertheless, it is still true that for large , most of the ’s are either close to zero or to one. To take advantage of this fact, conditional on , we sample independent -valued random variables such that with probability . For each , we draw an arrow from to . These arrows define a coalescing structure , called discrete web, on (see Figure 3). Think of these arrows as assigning to each point a preferred direction, which, in most cases, will be if is close to one and if is close to zero.
Now let us describe the joint law of differently. First of all, if we forget about , then the are just i.i.d. -valued random variables which take the value with probability . Second, conditional on , the random variables are independent with distribution
depending on whether resp. . Therefore, we can alternatively construct our random space-time environment in such a way, that first we construct an i.i.d. collection as above, and then conditional on , independently for each , we choose with law if and law if .
Let denote the coalescing structure on generated by the arrows associated with (see Figure 3). Then gives an alternative representation of the random environment . A random walk in such an environment navigates in such a way that whenever it comes to a point , the walk jumps to the right with probability and to the left with the remaining probability. The important thing to observe is that if is large, then is with large probability close to zero if and close to one if . In view of this, the random walk in the random environment will most of its time walk along paths in .
It turns out that has a
meaningful diffusive scaling limit. In particular, if space is scaled by
and time by , then the coalescing structure converges to a limit called the Brownian web (with
drift ), which loosely speaking is a collection of coalescing Brownian
motions starting from every point in space and time. These provide the default
paths a motion in the limiting random environment must follow. The
i.i.d. random variables turn out to converge to a
marked Poisson point process which is concentrated on so-called points of type
in , which are points where there is one incoming path and two
outgoing paths. These points are divided into points of type
and , depending on whether the incoming path continues on the
left or right. A random motion in such an environment follows paths in
by default, but whenever it comes to a marked point of type , it
continues along the left resp. right outgoing path with probability
1.4 Outline and discussion
The rest of the paper is organized as follows. Sections 2–4 provide an extended introduction where we rigorously state our results. In Section 2, we recall the notion of a stochastic flow of kernels, first introduced in [LR04a], and Howitt and Warren’s [HW09a] sticky Brownian motions, to give a rigorous definition of Howitt-Warren flows. We then state out main results for these Howitt-Warren flows, including properties for the kernels and results for the associated measure-valued processes. In Sections 3 and 4 we make the heuristics from Section 1.3 rigorous. In Section 3, in particular in Theorem 3.7, we present our construction of Howitt-Warren flows based on a ‘reference’ Brownian web with a Poisson marking, which is the main result of this paper. Along the way, we will recall the necessary background on the Brownian web. In Section 4, we show that a special subclass of the Howitt-Warren flows can be constructed alternatively as flows of mass in the Brownian net. Along the way, we will recall the necessary background on the Brownian net and establish some new results on a coupling between a Brownian web and a Brownian net. Sections 5–10 are devoted to proofs. In particular, we refer to Section 5 for an outline of the proofs. The paper concludes with a number of appendices and a list of notation.
Our work leaves several open problems. One question, for example, is how to characterize the measure-valued processes associated with a Howitt-Warren flow (see (2.1) below) by means of a well-posed martingale problem. Other questions (martingale problem formulation, path properties) refer to the duals (in the sense of linear systems duality) of these measure-valued processes, introduced in (11.1) below, which we have not investigated in much detail.
Moving away from the Brownian case, we note that it is an open problem whether our methods can be generalized to other stochastic flows of kernels than those introduced by Howitt and Warren. In particular, this applies to the stochastic flows of kernels with -stable Lévy -point motions introduced in [LR04b] for . A first step on this road would be the construction of an -stable Lévy web which should generalize the presently known Brownian web. Some first steps in this direction have recently been taken in [EMS13].
2 Results for Howitt-Warren flows
In this section, we recall the notion of a stochastic flow of kernels, define the Howitt-Warren flows, and state our results on these Howitt-Warren flows, which include almost sure path properties and ergodic theorems for the associated measure-valued processes. The proofs of these results are based on our graphical construction of the Howitt-Warren flows, which we postpone to Sections 3–4 due to the extensive background we need to recall.
2.1 Stochastic flows of kernels
In [LR04a], Le Jan and Raimond developed a theory of stochastic flows of kernels, which may admit versions that can be interpreted as the random transition probability kernels of a Markov process in a stationary random space-time environment. The notion of a stochastic flow of kernels generalizes the usual notion of a stochastic flow, which is a family of random mappings from a space to itself. In the special case that all kernels are delta-measures, a stochastic flow of kernels reduces to a stochastic flow in the usual sense of the word.
Since stochastic flows of kernels play a central role in our work, we take some time to recall their defintion. For any Polish space , we let denote the Borel -field on and write and for the spaces of finite measures and probability measures on , respectively, equipped with the topology of weak convergence and the associated Borel -field. By definition, a probability kernel on is a function such that the map from to is measurable. By a random probability kernel, defined on some probability space , we will mean a function such that the map from to is measurable. We say that two random probability kernels are equal in finite dimensional distributions if for each , the -tuple of random probability measures is equally distributed with . We say that two or more random probability kernels are independent if their finite-dimensional distributions are independent.
(Stochastic flow of kernels)
A stochastic flow of kernels on is a collection of random probability kernels on such that
For all and , a.s. and for all .
For each , the random probability kernels are independent.
and are equal in finite-dimensional distributions for each real and .
The finite-dimensional distributions of a stochastic flow of kernels are the laws of -tuples of random probability measures of the form , where and , .
Remark. If the random set of probability on which Definition 2.1 (i) holds can be chosen uniformly for all and , then we can interpret as bona fide transition kernels of a random motion in random environment. For the stochastic flows of kernels we are interested in, we will prove the existence of a version of which satisfies this property (see Proposition 2.3 below). To the best of our knowledge, it is not known whether such a version always exists for general stochastic flows of kernels, even if we restrict ourselves to those defined by a consistent family of Feller processes.
If is a stochastic flow of kernels and is a finite measure on , then setting
defines an -valued Markov process . Moreover, setting
defines a Markov transition function on . We call the Markov process with
these transition probabilities the -point motion associated with the
stochastic flow of kernels . We observe that the
-point motions of a stochastic flow of kernels satisfy a natural
consistency condition: namely, the marginal distribution of any components
of an -point motion is necessarily a -point motion for the flow. A
fundamental result of Le Jan and Raimond [LR04a, Thm 2.1] states that
conversely, any consistent family of Feller processes on a locally compact
space gives rise to a stochastic flow of kernels on which is unique in
2.2 Howitt-Warren flows
As will be proved in Proposition A.5 below, under the condition (1.7), if space and time are rescaled respectively by and , then the -point motions associated with the discrete Howitt-Warren flow introduced in Section 1.2 with characteristic measure converge to a collection of Brownian motions with drift and some form of sticky interaction characterized by the measure . These Brownian motions solve a well-posed martingale problem, which we formulate now.
Let , a finite measure on , and define constants by
We note that in terms of these constants, (1.7) is equivalent to
For , we define
where denotes the cardinality of a set.
The martingale problem we are about to formulate was invented by Howitt and Warren [HW09a]. We have reformulated their definition in terms of the functions in (2.5), which form a basis of the vector space of test functions used in [HW09a, Def 2.1] (see Appendix A for a proof). This greatly simplifies the statement of the martingale problem and also facilitates our proof of the convergence of the -point motions of discrete Howitt-Warren flows.
(Howitt-Warren martingale problem) We say that an -valued process solves the Howitt-Warren martingale problem with drift and characteristic measure if is a continuous, square-integrable semimartingale, the covariance process between and is given by
and, for each nonempty ,
is a martingale with respect to the filtration generated by .
It is not hard to prove that both martingale problems are equivalent.
Remark. When , condition (2.7) is equivalent to the condition that
are martingales. In [HW09a], such are called -coupled Brownian motions, with . In this case, is a Brownian motion with stickiness at the origin. Such a process can be constructed by time-changing a standard Brownian motion in such a way that it spends positive Lebesgue time at the origin. More generally, for solutions to the Howitt-Warren martingale problem started in , the set of times such that is a nowhere dense set with positive Lebesgue measure. The measure then determines a two-parameter family of constants (see formula (A.4) in the Appendix), which can be interpreted as the rate, in a certain excursion theoretic sense, at which split into two groups, and , with .
Howitt and Warren [HW09a, Prop. 8.1] proved that their martingale problem is well-posed and its solutions form a consistent family of Feller processes. Therefore, by the already mentioned result of Le Jan and Raimond [LR04a, Thm 2.1], there exists a stochastic flow of kernels on , unique in finite-dimensional distributions, such that the -point motions of (in the sense of (2.2)) are given by the unique solutions of the Howitt-Warren martingale problem. We call this stochastic flow of kernels the Howitt-Warren flow with drift and characteristic measure . It can be shown that Howitt-Warren flows are the diffusive scaling limits, in the sense of weak convergence of finite dimensional distributions, of the discrete Howitt-Warren flows with characteristic measures satisfying (1.7). (Indeed, this is a direct consequence of Proposition A.5 below on the convergence of -point motions.)
We will show that it is possible to construct versions of Howitt-Warren flows which are bona fide transition probability kernels of a random motion in a random space-time environment, and the kernels have ‘regular’ parameter dependence.
(Regular parameter dependence) For each and finite measure on , there exists a version of the Howitt-Warren flow with drift and characteristic measure such that in addition to the properties (i)–(iii) from Definition 2.1:
A.s., for all , and .
A.s., the map from to is continuous for each .
When the characteristic measure , solutions to the Howitt-Warren martingale problem are coalescing Brownian motions. In this case, the associated stochastic flow of kernels is a stochastic flow (in the usual sense), which is known as the Arratia flow. In the special case that and is Lebesgue measure, the Howitt-Warren flow and its -point motions are reversible. This stochastic flow of kernels has been constructed before (on the unit circle instead of ) by Le Jan and Raimond in [LR04b] using Dirichlet forms. We will call any stochastic flow of kernels with for some a Le Jan-Raimond flow. In [HW09b], Howitt and Warren constructed a stochastic flow of kernels with and , which they called the erosion flow. In this paper, we will call this flow the symmetric erosion flow and more generally, we will say that a Howitt-Warren flow is an erosion flow if with . The paper [HW09b] gives an explicit construction of the symmetric erosion flow based on coupled Brownian webs. Their construction can actually be extended to any erosion flow and can be seen as a precursor and special case of our construction of general Howitt-Warren flows in this paper.
2.3 Path properties
In this subsection, we state a number of results on the almost sure path properties of the measure-valued Markov process defined in terms of a Howitt-Warren flow by (2.1). Throughout this subsection, we will assume that is a finite measure, and is defined using a version of the Howitt-Warren flow , which satisfies property (iv) in Proposition 2.3, but not necessarily property (i)’. Then it is not hard to see that for any , the Markov process defined in (2.1) has continuous sample paths in . We call this process the Howitt-Warren process with drift and characteristic measure .
See Figures 4 and 5 for some simulations of Howitt-Warren processes for various choices of the characteristic measure . There are a number of parameters that are important for the behavior of these processes. First of all, following [HW09a], we define
In a certain excursion theoretic sense, describes the rate at which a group of coordinates of the -point motion that are at the same position splits into two groups consisting of and specified coordinates, respectively. In particular, following again notation in [HW09a], we set
and we call the stickiness parameter of the Howitt-Warren flow. Note that when is increased, particles separate with a higher rate, hence the flow is less sticky. The next proposition shows that with the exception of the Arratia flow, by a simple transformation of space-time, we can always scale our flow such that and . Below, for any and we write and .
(Scaling and removal of the drift) Let be a Howitt-Warren flow with drift and characteristic measure . Then:
For each , the stochastic flow of kernels defined by is a Howitt-Warren flow with drift and characteristic measure .
For each , the stochastic flow of kernels defined by is a Howitt-Warren flow with drift and characteristic measure .
There are two more parameters that are important for the behavior of a Howitt-Warren flow. We define
Note that , where are the constants defined in (2.3). We call and the left speed and right speed of a Howitt-Warren flow, respectively. The next theorem shows that these names are justified. Below, denotes the support of a measure , i.e., the smallest closed set that contains all mass.
(Left and right speeds) Let be a Howitt-Warren process with drift and characteristic measure , and let be defined as in (2.12). Set . Then:
If and , then is a Brownian motion with drift . If and , then for all .
If , then for all .
Analogue statements hold for , with replaced by .
It turns out that the support of a Howitt-Warren process is itself a Markov process. Let be the space of closed subsets of . We equip with a topology such that if and only if , where denotes the closure of a set in and means convergence of compact subsets of in the Hausdorff topology. The branching-coalescing point set is a -valued Markov process that has been introduced in [SS08, Thm 1.11]. Its definition involves the Brownian net; see formula (4.6) below. The following proposition, which we cite from [SS08, Thm 1.11 and Prop. 1.15] and [SSS09, Prop. 3.14], lists some of its elementary properties.
(Properties of the branching-coalescing point set) Let be the branching-coalescing point set defined in (4.6), started in any initial state . Then:
The process is a -valued Markov process with continuous sample paths.
If , then is a Brownian motion with drift . Likewise, if , then is a Brownian motion with drift .
The law of a Poisson point set with intensity is a reversible invariant law for and the limit law of as for any initial state .
For each deterministic time , a.s. is a locally finite subset of .
Almost surely, there exists a dense set such that for each , the set contains no isolated points.
Our next result shows how Howitt-Warren processes and the branching-coalescing point set are related. Note that this result covers all possible values of , except the case which corresponds to the Arratia flow. In (2.13) below, we continue to use the notation .
(Support process) Let be a Howitt-Warren process with drift and characteristic measure and let be defined as in (2.12). Then:
If , then a.s. for all ,
where is a branching-coalescing point set.
If and , then a.s. for all , where . An analogue statement holds when and .
If and , then a.s. for all .
Proposition 2.6 (d) and Theorem 2.7 (a) imply that if the left and right speeds of a Howitt-Warren process are finite, then at deterministic times the process is purely atomic. The next theorem generalizes this statement to any Howitt-Warren process, but shows that if the characteristic measure puts mass on the open interval , then there are random times when the statement fails to hold.
(Atomicness) Let be a Howitt-Warren process with drift and characteristic measure . Then:
For each , the measure is a.s. purely atomic.
If , then a.s. there exists a dense set of random times when is purely non-atomic.
If , then a.s. is purely atomic at all .
In the special case that is (a multiple of) Lebesgue measure, a weaker version of part (a) has been proved in [LR04b, Prop. 9 (c)]. Part (b) is similar to Proposition 2.6 (e) and in fact, by Theorem 2.7 (a), implies the latter. Note that parts (b) and (c) of the theorem reveal an interesting dichotomy between erosion flows (where is nonzero and concentrated on ) and all other Howitt-Warren flows (except the Arratia flow, for which atomicness is trivial). The reason is that atoms in erosion flows lose mass continuously (see the footnote in Section 1.3 and the construction in Section 3.4 below), while in all other flows atoms can be split into smaller atoms. This latter mechanism turns out to be more effective at destroying atoms. For erosion flows, we have an exact description of the set of space-time points where has an atom in terms of an underlying Brownian web, see Theorem 9.6 below.
2.4 Infinite starting measures and discrete approximation
The ergodic behavior of the branching-coalescing point set is well-understood (see Proposition 2.6 (c)). As a consequence, by Theorem 2.7 (a), it is known that if we start a Howitt-Warren process with left and right speeds , in any nonzero initial state, then its support will converge in law to a Poisson point process with intensity . This does not mean, however, that the Howitt-Warren process itself converges in law. Indeed, since its 1-point motion is Brownian motion, it is easy to see that any Howitt-Warren process started in a finite initial measure satisfies