Diffusion processes on branching Brownian motion
We construct a class of one-dimensional diffusion processes on the particles of branching Brownian motion that are symmetric with respect to the limits of random martingale measures. These measures are associated with the extended extremal process of branching Brownian motion and are supported on a Cantor-like set. The processes are obtained via a time-change of a standard one-dimensional reflected Brownian motion on in terms of the associated positive continuous additive functionals.
The processes introduced in this paper may be regarded as an analogue of the Liouville Brownian motion which has been recently constructed in the context of a Gaussian free field.
Key words and phrases:Branching Brownian motion, additive functional, extremal process, local time, random environment
2010 Mathematics Subject Classification:Primary: 60J55, 60J80, 60K37; Secondary: 60G55, 60G70, 60J60.
Over the last years diffusion processes in random environment, constructed by a random time-change of a standard Brownian motion in terms of singular measures, appeared in several situations. One prime example is the so-called FIN-diffusion (for Fontes, Isopi and Newman) introduced in  which appears for instance as the annealed scaling limit for one-dimensional trap models (see [22, 6, 7]) and for the one-dimensional random conductance model with heavy-tailed conductances (see [36, Appendix A]). Another example is the Liouville Brownian motion, recently constructed in [25, 8] as the natural diffusion process in the random geometry associated with two-dimensional Liouville quantum gravity.
In this paper we add one more class of examples to the collection. We consider a time change given by the right-continuous inverse of the positive continuous additive functional whose Revuz measure is the limit of certain random martingale measures that appear in the description of the extremal process of a branching Brownian motion (BBM for short). As a result we obtain a pure jump diffusion process on a Cantor-like set representing the positions of the BBM particles in the underlying Galton-Watson tree.
Branching Brownian motion has already been introduced in [30, 34] in the late 1950s and early 1960s. It is a continuous-time Markov branching process on a probability space which is constructed as follows. We start with a continuous-time Galton-Watson process (see e.g. ) with branching mechanism , normalised such that , and . At any time we may label the endpoints of the process , where is the number of branches at time . Observe that by our choice of normalisation we have that . BBM is then constructed by starting a Brownian motion at the origin at time zero, running it until the first time the GW process branches, and then starting independent Brownian motions for each branch of the GW process starting at the position of the original BM at the branching time. Each of these runs again until the next branching time of the GW occurs, and so on.
We denote the positions of the particles at time by . Note that, of course, the positions of these particles do not reflect the position of the particles ”in the tree”.
By a slight abuse of notation, we also denote by for the particle position of the ancestor of the particle at time .
for some constant , where is the -a.s. limit of the derivative martingale
For a truncated version of the derivative martingale
has been introduced in . Here we denote by an embedding of the particles into , which encodes the positions of the particles in the underlying Galton-Watson tree respecting the genealogical distance (see Section 2.1 below for the precise definition). The associated random measure on is given by
In  it has been shown that the vague limit
Furthermore, in [13, Theorem 3.1] an extended convergence result of the extremal process has been proven, namely
where are the atoms of a Cox process on with intensity measure and are the atoms of independent and identically distributed point processes with
Let denote a one-dimensional reflected standard Brownian motion on . Recall that is reversible w.r.t. the Lebesgue measure on . Then, the positive continuous additive functional (PCAF) of having Revuz measure (see Appendix A for definitions) is given by defined as
where denotes the family of local times of . Further, we define
-a.s., the following hold.
There exist a set with for all on which
for any . In particular, is continuous, increasing and satisfies and .
The functional is the (up to equivalence) unique PCAF of with Revuz measure .
We define the process as the time-changed Brownian motion
where denotes the right-continuous inverse of the PCAF in (1.9).
By the general theory of time changes of Markov processes, in particular cf. [24, Theorem 6.2.1], is a right-continuous strong Markov process on , which is -symmetric and induces a strongly continuous transition semigroup. Note that the empty set is the only polar set for the one-dimensional Brownian motion, so the measure does trivially not charge polar sets. Further, for any set
where denotes the right-continuous inverse of . Then, as and tend to infinity, the processes converge in law towards on the Skorohod space equipped with -topology (see Theorem 4.1 below). In a sense may be regarded as a random walk on the leaves of the underlying Galton-Watson tree. In addition, we also provide an approximation result for in terms of random walks on a lattice (see Theorem 4.5 below).
Similarly to the above procedure, for any , one obtains a measure from a truncation of the McKean martingale
Then one can define the process as with being the PCAF associated with . We refer to Section 5 for further details.
A diffusion process being similar to but different from is the FIN-diffusion introduced in . It is a one-dimensional singular diffusion in random environment given by a random speed measure , where is an inhomogeneous Poisson point process on with intensity measure for . Let be the PCAF
with denoting the family of local times of a one-dimensional Brownian motion . Then, the FIN-diffusion is the diffusion process defined as the time change of the Brownian motion . At first sight the measure and the process resemble strongly and , respectively. However, one significant difference is that is a discrete random measure with a set of atoms being dense in , so that has full support and has continous sample paths (see  or [7, Proposition 3.2]), while the measure is concentrated on a Cantor-like set and the sample paths of have jumps.
Another prominent example for a log-correlated process is the Gaussian Free Field (GFF) on a two-dimensional domain. In a sense the processes or introduced in this paper can be regarded as the BBM-analogue of the Liouville Brownian motion (LBM) recently constructed in  and in a weaker form in . More precisely, let denote a (massive) GFF on a domain , then in the subcritical case the analogue of the martingale measure can be constructed by using the theory of Gaussian multiplicative chaos established by Kahane in  (see also  for a review). On a formal level the resulting so-called Liouville measure on is given by
The associated PCAF , which can formally be written as
In the critical case the corresponding analogue of the derivative martingale measure can be interpreted as being given by
which has been introduced in [20, 21]. The corresponding PCAF and the critical Liouville Brownian motion have been constructed in . In the context of a discrete GFF such measures have been studied in [10, 9, 11], where in  an analogue of the extended convergence result in (1.6) has been established.
However, a major difference between the processes and is that for the LBM the functional and the planar Brownian motion are independent (cf. [25, Theorem 2.21]), while in the present paper the functional and the Brownian motion are dependent since is the local time of . A similar phenomenon can be observed in the context of trap models, where in dimension the underlying Brownian motion and the clock process of the FIN diffusion are dependent and in dimension the Brownian motion and the clock process of the scaling limit, known as the so-called fractional kinetics motion, are independent.
In  Croydon, Hambly and Kumagai consider time-changes of stochastic processes and their discrete approximations in a quite general framework for the case when the underlying process is point recurrent, meaning that it can be described in terms of its resistance form (examples include the one-dimensional standard Brownian motion or Brownian motion on tree-like spaces and certain low-dimensional fractals). The results cover the FIN-diffusion and a one-dimensional version of the LBM. However, the results of the present paper do not immediately follow from the approximation result in  since the required convergence of the measures towards in the Gromov-Hausdorff-vague topology on the non-compact space needs to be verified.
The rest of the paper is organised as follows. In Section 2 we provide the precise definition of the embedding and the (truncated) critical martingale measures. Then we prove Theorem 1.2 in Section 3 and we specify some properties of the process , in particular we describe its Dirichlet form. In Section 4 we show random walk approximations of . In Section 5 we sketch the construction of the process associated with the martingale measure obtained from the McKean martingale. Finally, in the appendix we recall the definitions of a PCAF and its Revuz measure and collect some properties of Brownian local times needed in the proofs.
2.1. Definition of the embedding
We start by recalling the definition of the embedding given in  which is a slight variant of the familiar Ulam-Neveu-Harris labelling (see e.g. ). We denote the set of (infinite) multi-indices by and let be the subset of multi-indices that contain only finitely many entries different from zero. Ignoring leading zeros, we see that
where is either the empty multi-index or the multi-index containing only zeros.
We encode a continuous-time Galton-Watson process by the set of branching times, , where denotes the number of branching times up to time , and by a consistently assigned set of multi-indices for all times . To do so, (for a given tree) the sets of multi-indices, at time , are constructed as follows.
for all , for all , .
If then if , where
We use the convention that, if a given branch of the tree does not ”branch” at time , we add to the underlying Galton-Watson at this time an extra vertex where (see Figure 1). We call the resulting tree .
One relates the assignment of labels in the following backward consistent way. For , we define the function , through
Clearly, if and , then . This allows to define the boundary of the tree at infinity by . In this way we identify each leaf of the Galton-Watson tree at time , with , with some multi-label . We define the embedding by
For a given , the function describes a trajectory of a particle in , which converges to some point , as , -a.s. Hence also the sets converge, for any realisation of the tree, to some (random) set .
Recall that in BBM there is also the position of the Brownian motion of the -th particle at time . Thus to any ”particle” at time we can now associate the position , in . Hoping that there will not be too much confusion, we will identify with .
2.2. The critical martingale measure
A key object is the derivative martingale defined in . Recall the following result proven in .
The limit exists -a.s. and as -a.s.
For the truncated version
exists -a.s. Consider now the associated measures on given by
and denote by the Borel measure on defined via for all . Then, (2.6) implies that -a.s.
By [13, Proposition 3.2] is -a.s. non-atomic. Moreover, due to the recursive structure of the underlying GW-tree is supported on some Cantor-like set .
3. Approximation of the PCAF and properties of
3.1. Proof of Theorem 1.2
Let and let be the coordinate process on and set and , . Further, let be the family of probability measures on such that for each , under is a one-dimensional Brownian motion starting at . We denote by the minimum completed admissible filtration for and by the random field of local times of .
Now we set , , so that is a reflected Brownian motion on . Then, the family of local times of is given by
(cf. [31, Exercise VI.1.17]).
For -a.e. , there exists such that for all and the following hold.
The unique PCAF of with Revuz measure is given by
There exist a set with for all , on which is continuous, increasing and satisfies and .
We now turn to the proof of Theorem 1.2.
Proof of Theorem 1.2 (i).
Fix any environment such that Proposition 3.1 holds and converges vaguely to on . In particular,
for all continuous functions on with compact support.
By Lemma B.1 there exists a set with for all such that is jointly continuous for all . In particular, for any fixed we have that is continuous with compact support . Now, by choosing in (3.3) we obtain
and therefore pointwise convergence of towards on . Recall that by Proposition 3.1 the functionals are increasing for . Since pointwise convergence of continuous increasing functions towards a continuous function on a compact set implies uniform convergence, the claim follows. ∎
Alternatively, Theorem 1.2 (i) can also be derived from the result in [35, Theorem 1 (3)].
For the identification of as the unique PCAF with Revuz measure we need a preparatory lemma.
For -a.e. , there exists such that the following holds. For any , and any bounded Borel measurable function the family is uniformly -integrable.
Proof of Theorem 1.2 (ii).
Recall that only the empty set is polar for . In particular, the measure does trivially not charge polar sets, so by general theory (see e.g. [16, Theorem 4.1.1]) the PCAF with Revuz measure is (up to equivalence) unique. Thus, we need show that the limiting functional is -a.s. in Revuz correspondence with . In view of (A.3) it suffices to prove that -a.s.
for any non-negative Borel function . By a monotone class argument it is enough to consider continuous functions with compact support in . Note that for any and therefore
By Lemma B.2 we have and together with Lemma B.1 this implies that the mapping is bounded and continuous on . Furthermore, by (i) -a.s. the sequence converges weakly to on , -a.s. for any . We take limits in and on both sides of (3.9), where we use Lemma 3.3 for the left hand side and the vague convergence of towards for the right hand side, and obtain
3.2. First properties of
Recall that the process is defined as the time-changed Brownian motion
where is the PCAF in (1.9). First, we observe that the continuity of ensures that the process does not get stuck anywhere in the state space, and does not explode in finite time since, -a.s., . However, is not strictly increasing so that jumps occur.
More precisely, by the general theory of time changes of Markov processes we have the following properties of . First, in view of [24, Theorems A.2.12] is a right-continuous strong Markov process on and by [16, Proposition A.3.8] we have -a.s.
where denotes the support of the PCAF , i.e.
By general theory (cf. [24, Section 5.1]) we have (recall that only the empty set is polar) and has -measure zero.
3.3. The Dirichlet form
We can apply the general theory of Dirichlet forms to obtain a more precise description of the Dirichlet form associated with . For denote by the standard Sobolev space, that is
where the derivatives are in the distributional sense. On we define the form
Recall that can be regarded as a regular Dirichlet form on and the associated process is the reflected Brownian motion on . By we denote the extended Dirichlet space, that is the set of -equivalence classes of Borel measurable functions on such that -a.e. for some satisfying . By [16, Theorem 2.2.13] we have the following identification of :
Recall that denotes the support of the random measure . We define the hitting distribution
with for any non-negative Borel function on . Note that the function is uniquely determined by the restriction of to the set . Further, by [16, Theorem 3.4.8], we have and by [24, Lemma 6.2.1] whenever -a.e. for any . Therefore it makes sense to define the symmetric form on by
By [24, Theorem 6.2.1] is the regular Dirichlet form on associated with the process . Since has Lebesgue measure zero, it follows from the Beurling-Deny representation formula for (see [16, Theorem 5.5.9]) that has no diffusive part and is therefore a pure jump process.
4. Random walk approximations
4.1. Approximation by a random walk on the leaves
For any we define
where denotes the right-continuous inverse of . The process is taking values in and it may therefore be regarded as a random walk on the leaves of the underlying Galton-Watson tree represented by their values under the embedding .
Let (or , ) be the the space of -valued càdlàg paths on (or , ). We denote by and the metric w.r.t. Skorohod - and -topology, respectively. We refer to [37, Chapter 3] for the precise definitions. Further, let
where denotes the right-continuous inverse of . Finally, we set
equipped with the topology induced by supposing
Note that the -topology extends both the - and the -topology since it allows excursions in the approximating processes which are not present in the limit process provided they are of negligible -magnitude (cf. [18, Remark 1.3]).
-a.s., for every we have under ,
in distribution on , that is, -a.s., for every and for all bounded continuous functions on ,
Since the measures and do not have full support and and have discontinuities, the locally uniform convergence of the functionals only implies the -convergence of their inverses. In such a situation the composition mapping is only continuous in the -topology (see Lemma 4.3 below), which is why we obtain the approximation in Theorem 4.1 in the coarser -topology only. We refer to [17, Corollary 1.5 (b)] for a similar result and to [18, 23, 29] for examples of convergence results for trap models in the -topology (or slight modifications of it).
Before we prove Theorem 4.1 we recall some facts about the continuity of the inverse and the composition mapping on the space of càdlàg paths.
For any ,
Let be a sequence in such that in -topology for some . Then, in equipped with -topology, where and denote the right-continuous inverses of and , respectively.
Let and such that in -topology for some and in -topology for some . Then, in -topology.
Proof of Theorem 4.1.
Fix an environment such that Theorem 1.2 holds giving that for any , -a.s., locally uniformly as first and then . In particular, using Lemma 4.3 (i) we have that in -topology -a.s. In particular, for all bounded acting on which are continuous in -topology on a set with full -measure,
Now, observe that for any bounded continuous on ,
In the special case the convergence result in Theorem 4.1 can be extended to equipped with -topology. This is because the continuity of the inverse map stated in Lemma 4.3(ii) also holds in under the additional assumption that (cf. [37, Chapter 13.6]). Note that by construction the origin is contained in so that under . However, an arbitrary might not be contained in the support of the random measure , in which case does not hold.
4.2. Approximation by random walks on a lattice
Next we provide approximation results for in terms of random walks on the lattice , . For any let be the random measure
with the associated PCAF given by