Extremes of branching Ornstein-Uhlenbeck processes
In this article, we focus on the asymptotic behaviour of extremal particles in a branching Ornstein-Uhlenbeck process: particles move according to an Ornstein-Uhlenbeck process, solution of , and branch at rate . We make depend on the time-horizon at which we observe the particles positions and we suppose that . We show that, properly centred and normalised, the extremal point process continuously interpolates between the extremal point process of the branching Brownian motion (case ) and the extremal point process of independent Gaussian random variables (case ). Along the way, we obtain several results on standard branching Brownian motion of intrinsic interest. In particular, we give a probabilistic representation of the main object of study in [DMS] which is the probability that the maximal position has an abnormally high velocity.
Spatial branching processes, and in particular, the behaviour of their extremal particles, have been at the centre of an enormous research activity over the past few years, both in the physics [BD09, BDMM06] and in the mathematical literature [Aid, ABBS, ABK, Madaule]. These models have a rich and complex structure that is of intrinsic interest, but they are also representatives of an intriguing “universality” class, the so-called log-correlated fields which includes the two-dimensional Gaussian free field [BDZ16, BL18], Gaussian multiplicative chaos [rhodes2013gaussian], random matrices [ABB2017] and others.
Perhaps the simplest model in this class is the branching Brownian motion, in which particles move in as Brownian motions, branch into two particles at rate one and are independent of each others. For the system started with a single particle at the origin, let be the set of particles alive at time and for let be its position. For we will also write for the position of the unique ancestor of at time so that is the path followed by the particle . Then, it was proved in [ABBS, ABK] that the point measure
converges in law, as toward a random intensity decorated Poisson point process (DPPP for short) .
In general, the law of a DPPP is characterized by a pair where is a random sigma-finite measure on and is the law of a random point process on . The point measure can be constructed, conditionally on , by first taking a realisation of a Poisson point process on with intensity , whose atoms are listed as , and an independent family of i.i.d. point processes with law . Then, each atom is replaced by the point process , shifted by (i.e. we decorate with a point process of law ). In other words, writing the atoms of the point process , we have
We refer to [SubZei] for an in-depth study of random intensity decorated Poisson point processes, and their occurrences as limit of extremal point measures.
With this notation, is the following DPPP
where is an implicit constant, is the a.s. positive limit of the so-called derivative martingale
and where the decoration law is the law of a point measure supported on , with an atom at , which belongs to the family , defined, for by the weak limits
where , see [BoH15] for a proof of the existence of . We set the law of the Dirac mass at . Moreover, it is well-known that converges in distribution toward , where is the position of the largest atom in a point process (see Lalley and Selke [LaS]).
The goal of this article is to study the same question — asymptotic behaviour of the extremal process — for a spatially inhomogeneous branching particle system: the branching Ornstein-Uhlenbeck process. As the name suggests, this is a continuous-time particle system in which particles move according to i.i.d. Ornstein-Uhlenbeck processes and split independently at rate .
An Ornstein-Uhlenbeck process with spring constant is the solution of the stochastic differential equation
where is a Brownian motion. It is well-known that Ornstein-Uhlenbeck processes may be represented, if , as a space-time scaled Brownian motion: given a standard Brownian motion, the process defined by
is an Ornstein-Uhlenbeck with spring constant and initial condition . Equation (1.7) shows that, if , the law of , conditionally on , is . In particular, is then strongly recurrent and its invariant measure is .
In a branching Ornstein-Uhlenbeck, since the genealogical structure of the process is independent of the motion of the particles, we continue to denote by the set of particles alive in a branching Ornstein-Uhlenbeck process with spring constant and we write for the positions of such particles. It will be convenient to work with a normalized version of that has variance so that things happen on the same scale as for the branching Brownian motion. This can be easily obtained by setting
With this notation, we define the extremal point process:
Note that here the logarithmic correction is instead of as in the branching Brownian motion case (, see (1.1)).
Throughout this paper, we will choose the spring constant as depending on the time-horizon at which we observe the positions of particles, in the sense that is kept fixed for the evolution of the branching process at all times . For reasons that will become clear later on in the paper, one should choose such that as which trivially covers the standard case where is fixed for all ’s.
Our main result is that converges in the appropriate sense to , a new random intensity decorated Poisson point process
the characteristic pair of which is defined as follows: Let be a branching Brownian motion and its maximal displacement at time . Then,
is the limit of the additive martingale:
As is a non-negative martingale, it converges a.s. to a limit . Moreover, it is well known that a.s. if, and only if, .
This can be seen as a precise estimate on the large deviations for the maximal displacement of the branching Brownian motion, see [DMS, DerridaShi, GantertHof, Burac] for recent developments on this topic.
The family of laws is the family of point processes introduced in (1.5), the extremal point process in a branching Brownian motion, seen from , conditioned on , and is the image measure of by the application , dilating the positions of the atoms by a factor .
The constants and are given by
In the case, we set and , thus is an exponential random variable with mean . As is shown in Proposition 1.2, and a point measure drawn from is a.s. .
Assume that , then, with the above notations, we have that
where the convergence of the point process is in the sense of the topology of vague convergence.
Remark: Recall that a sequence of random point measures on converges to in law for the topology of vague convergence if and only if, for every compactly supported continuous function , the real valued random variables
converge in law to as . We prove in the forthcoming Lemma 4.1 that the convergence in law of a random point measure (for the topology of vague convergence) jointly with that of its maximum is equivalent to the convergence in law of to for all continuous functions with support bounded from the left. This notion of convergence forms a thinner topology on the space of point measures.
Remark: In the simplest case wherer is a constant, the Theorem with (1.8) and (1.9) implies the following behaviour for the unnormalised positions : the position of the rightmost particle is almost surely given by
and the next particles are at distance of order from the rightmost.
We shall call the case the uncorrelated case, because the extremal particles have the same distribution as the extremal particles of an i.i.d. sample of Gaussian random variables. Indeed, in this regime, the dilation factor diverges as , which prevents the existence of local correlations (decorations) in the limiting picture.
The cases interpolate between the uncorrelated case and the branching Brownian motion regime (). Notice, though, that the multiplicative factor of the logarithmic correction remains equal to (as in the uncorrelated case) and not (as in the branching Brownian motion). We believe that there is a second transition when where one gradually goes from the correction to . A similar phenomenon was observed by Bovier and Hartung [BoH18] for branching Brownian motion with piecewise constant variance.
Our model is notably different from the one studied by Kiestler and Schmidt [KS] which yields a different interpolation between the uncorrelated case and the branching Brownian motion.
The function defined in (1.12) is of intrinsic interest. Indeed, asymptotics of the probability were first studied in [ChRo88] (where the existence of the limit is implicit). It also plays a key role in [BoH15] where it is proven that and that . More recently, the same function is the focus of [DMS] where, in particular, the large and small asymptotics are given.
In the present work we will show that has a probabilistic representation. This will allow us to prove that is continuous and that the family of limiting point measure is vaguely continuous in . This is a key step in the proof of Theorem 1.1.
Let us now describe the representation of and in terms of a spine process. Let be a standard Brownian motion, be the ranked atoms of a Poisson point process with intensity on and for be i.i.d. branching Brownian motions. We shall assume that , the and the are independent of one another. Given and , we define the point process
In words, is the point process constructed using a Brownian motion with drift , that spawns branching Brownian motions at rate . A branching Brownian motion spawned at time then starts evolving backward in time until it hits time , the particles alive at that time are added to the point process.
. The function is continuous on . It also satisfies , for and .
. The family of point processes is continuous in the space of Radon point measures equipped with the topology of vague convergence.
The definition (1.5), in the case , is the one given in [ABK12] for the decoration of the extremal process of the branching Brownian motion. The above backbone description, in the case , is similar to the one given in [ABBS].
The convergences and as were already proved in [BoH15, proof of Lemma 3.3]. We believe the family of point measures to be vaguely continuous as , however this result is not as straightforward and is left open in the present paper.
1.1 Open questions and future work
One question that we have not explored in the present work is what happens when . We conjecture that in that case, the decoration measure is always which is the decoration of the branching Brownian motion. However, we believe that the precise constant in front of the logarithmic correction for the positions of the extremal particles will now depend on how fast decreases towards zero.
More precisely, it is predicted in [DMS] that as , with the same constant as in (1.3). Note that is also the constant such that , as . This constant is proved to exist for all branching random walks in [Aid, Proposition 4.1]. Note that in [DMS] the function defined by
where is the solution of the Fisher-KPP equation started from the Heavyside initial condition is the analogue of . The exact correspondence between the functions and is
Our factor is thus given by the constant denoted in [DMS] (see Equation (73) there).
On the other hand, we also know from [Madaule], that for the additive martingale
with the limit of the derivative martingale. Since and when , we see that
Since , the extremal point process is roughly , the centred extremal point process of the standard branching Brownian motion see (1.3), shifted to the left by (as ). This might suggest that the above-mentioned intermediate logarithmic corrections between and should appear for with , and the extremal point measure would be the same as for the branching Brownian motion as soon as . This would complement the recent work [BoH18].
The case is also interesting and is not covered in the present work. Notice that in the case we rely heavily on the results from Bovier and Hartung [BoH15]. However we think that the case corresponds to that of decreasing variances for the variable speed branching Brownian motion for which results concerning the position of the maximum are known (see e.g. Maillard and Zeitouni [MaZ]), but not concerning the full extremal point process.
Our interest in the extremal point measure of branching Ornstein-Uhlenbeck processes was sparked by a conjecture in [CoM18], that the genealogy of a branching Ornstein-Uhlenbeck with selection is given by a Beta-coalescent whose parameters can be tuned by the spring constant. The study of the extremal point process is a first step towards a better understanding of the relevant objects.
Organisation of the paper.
The rest of the article is organised as follows. In the next section, we observe that in the particular case for some , Theorem 1.1 is a direct consequence of the result of Bovier and Hartung [BoH15] on the convergence of the extremal process of time-inhomogeneous branching Brownian motions.
In Section 3, we prove Theorem 1.2, i.e. the probabilistic representation of the law of the decoration. Then, in Section LABEL:sec:gaussian, we introduce some technical tools: Gaussian tail estimates in Section LABEL:sec:tailGaussian and comparison theorems in Section LABEL:subsec:normal.
The proof of Theorem 1.1 when is in Section 4. Section 5 is devoted to a simple and self-contained proof of the convergence of the extremal point process when the spring force is constant. We conclude in Section 1.1 with some open questions, conjectures and natural ways to extend the present work.
For further reference we gather here the meaning of some of our notations:
The point process is the normalised positions in the branching Ornstein-Uhlenbeck at time , seen from .
The point process , of law , is the limit point process of positions in a branching Brownian motion conditioned to have a particle right of , seen from the rightmost particle.
The point process is a decorated Poisson point process with random exponential intensity, where the decoration is given by for some
The point process is the one obtained by the spine construction : branching Brownian motions started at rate 2 on a drifted Brownian spine.
2 The case and variable speed branching Brownian motion
In this section, we prove Theorem 1.1 assuming that , where is a fixed constant. Our approach relies on earlier work on the extremal process of variable speed branching Brownian motions. These processes were introduced in [FaZ] and further studied in [BoH14, BoH15], where the convergence of the extremal point measure for a wide class of variance profiles is established. In particular, if then the convergence stated in Theorem 1.1 above follows readily from [BoH15]. We start with a brief introduction to the variable speed branching Brownian motion, and state the convergence result [BoH15, Theorem 1.2]. We then explain the connection between a branching Ornstein-Uhlenbeck process with spring constant and variable speed branching Brownian motion, before using this result to prove Theorem 1.1 in that particular case.
Let be a twice differentiable increasing function with and . Then, the variable speed branching Brownian motion with variance profile and time horizon is defined in the same way as a branching Brownian motion, except that particles move as Brownian motions with time-dependent variance where is the time of the process. In particular, the position of a particle at time is a Gaussian random variable with variance .
The following result on the convergence of the extremal point measure of branching Brownian motion with variance profile is proved in [BoH15].
Theorem A (Bovier and Hartung [BoH15] Theorem 1.2).
Assume that the twice differentiable increasing function satisfies
, and for all ;
Let denote the variable speed branching Brownian motion with variance profile and
be its extremal point measure at time . Then
the extremal process converges in law for the topology of the vague convergence to , which is a
the maximal displacement of the process converges in law, and for all ,
Remark: It is well-known (and we give a fairly general proof of that fact in Lemma 4.1) that if and its maximum converge toward and its maximum in law, then there is joint convergence of the two quantities, just as in our Theorem 1.1.
Proof of Theorem 1.1 in the case.
Recall from (1.7) that, an Ornstein-Uhlenbeck at time with spring-constant started from 0 can be written as
Recall also that given and , we denote by a branching Ornstein-Uhlenbeck process on with spring constant .
For any , we define by
Clearly, has variance . It is easily checked that the whole process is then a variable speed branching Brownian motion, with variance profile , which is a function satisfying the assumptions of Theorem A with
Note that when but we do not assume that , it is not possible to apply directly the same approach since in that case the variance profile of will be a function of and and not just of . Instead, we need to rely on comparison and continuity results.
3 Spine representation of and weak continuity of the cluster distribution
In this section, we prove Theorem 1.2, that is the weak continuity in of the cluster point process as well as the continuity of the function and their spine representation.
Recall the construction (1.14) of the point process
The measure is the increasing limit of . The first thing to show is that is a sigma-finite point measure, meaning that for every , we have a.s. as . We start with the case .
For all , is a well-defined point measure. Moreover, we have
Let , the point measure can be rewritten as
We observe that is a random walk with negative drift . Moreover, for all the position of the largest atom in the point measure is, for large values of , typically around position . Thus, heuristically, if , the random walk drifts to at positive speed such that only a finite number of branching Brownian motions put particles in any given compact set. On the other hand, when , the random walk has drift zero and we show that it implies that an infinite number of particles are to be found in any finite neighbourhood of .
To make the above argument rigorous, we write for the maximal displacement at time in a branching Brownian motion. Setting , It is well-known [ABBS, ABK] that is tight and has uniform exponential tails. More precisely, it is proved in [fang2012] (in a much more general settings) there exists and such that
Given , we denote by the maximal displacement of at time . Using the bounds from (3.3), we observe immediately, using the Borel-Cantelli Lemma and the fact that that, with probability one,
In view of (3.4) and the law of large numbers, we deduce that
In particular, it implies that given one can find a random such that
in which case for all and . This proves that is locally finite a.s. and that as , as claimed. ∎
Next we show the weak continuity of the family .
The family of point processes is a.s. continuous in . Moreover, for all , and
To prove the a.s. continuity of , it is enough to show that for all continuous function with compact support, the function is continuous a.s. This is a direct consequence of the fact that there are only finitely atoms in any compact interval, and that the position of these atoms in are decreasing and continuous with , by (1.14). Hence, for any , there is only a finite number of atoms to follow as increases to compute . Hence this function is continuous, which completes the proof of the first statement. For the second statement, it suffices to observe that for there is positive probability that and that
We now focus on the case . By law of iterated logarithms for the random walk, we have that
which together with (3.4) yields
This shows that the event has probability for every . In particular it implies that a.s. as .
To conclude the proof, we observe that for all , there exists such that . At the same time it follows from (3.1) that is continuous in , hence for all small enough, we have
which shows that , completing the proof. ∎
We now prove that the cluster law associated by Bovier and Hartung [BoH14] to the extremal point process of a variable speed branching Brownian motion is related to the point process defined in (1.14). This connection is based on the well-known spinal decomposition for branching Brownian motions and so-called probability tilting techniques based on the additive martingale . Those ideas were pioneered by Lyons, Peamantle and Peres in [LPP95], then generalized to branching random walks by Lyons [Lyo97] and to general branching processes in [BiK04].
Let be the filtration associated to the branching Brownian motion, defined by
For and , we introduce the size-biased law as
and call under the size biased process. The spinal decomposition links the size biased process with the so-called branching Brownian motion with spine. It describes the evolution of a branching particle system with a distinguished particle , which behaves differently from the others. The system starts with the spine particle at position . This particle moves according to a Brownian motion with drift and produces children at rate . Each of its children starts an independent (standard) branching Brownian motion from its birth place. We shall use the same notation for the set of particles alive at time in this process (it is not a Yule process anymore), and write for the label of the spine particle. The law of this branching Brownian motion with spine is denoted by . The spinal decomposition can be stated as follows.
Theorem B (Spinal decomposition [Lpp95, Lyo97]).
With the above notation, we have for all . Moreover, for all ,
In words: the law of the marked tree has same law under probability and . Moreover, conditionally on this process, one can choose to distinguish at random an individual with probability proportional to to construct the law of the branching Brownian motion with spine.
With the spinal decomposition in hands we link the extremal point measure law of the branching Brownian motion with the point measure .
Let , the point measure defined in (3.1) and
the extremal process of the branching Brownian motion seen from the rightmost individual. For all non-negative measurable function , we have
For , denote by the label of the largest particle alive at time (which is a.s. unique). We observe that we can write
where is the extremal point measure seen from particle . Thanks to the spinal decomposition and using (3.6), the above reads
Next, we use the definition of the branching Brownian motion with spine to rewrite the above expression. For let where and for all , is the th instant at which the spine gives birth to a new particle when running time backward from (i.e. is the last time before at which the spine branches).Then, under , is a standard Brownian motion and are the atoms of a Poisson point process on with intensity measure . For each branching event , the spine gives birth to a standard branching Brownian motion that we call . With these notation we then get
All that is left to do is thus to realise that under , the pair of variables jointly have the same law as from (3.1). Thus substituting by and