Planar Brownian motion and Gaussian multiplicative chaos
We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant the desired thickness level and show a simple relation between the two objects. Our results extend those of [BBK94] and in particular cover the entire -phase or subcritical regime. These results allow us to obtain a nondegenerate limit for the appropriately rescaled size of thick points, thereby considerably refining estimates of [DPRZ01].
- 1 Introduction
- 2 Preliminaries
- 3 First moment estimates
- 4 Uniform integrability
- 5 Convergence
- 6 Vague convergence, identification of the limits and properties of
- A Proof of Lemma 5.1
- B Proofs of lemmas on the zero-dimensional Bessel process
- C Continuity of the local times. Proof of Proposition 1.1
1.1 Main results
The Gaussian multiplicative chaos (GMC) introduced by Kahane [Kah85] consists in defining and studying the properties of random measures formally defined as the exponential of a log-correlated Gaussian field, such as the two-dimensional Gaussian free field (GFF). Since such a field is not defined pointwise but is rather a random generalised function, making sense of such a measure requires some nontrivial work. The theory has expanded significantly in recent years and by now it is relatively well understood, at least in the subcritical case [RV10, DS11, RV11, Sha16, Ber17] and even in the critical case [DRSV14a, DRSV14b, JS17, JSW18, Pow18]. Furthermore, Gaussian multiplicative chaos appears to be a universal feature of log-correlated fields going beyond the Gaussian theory discussed in these papers. Establishing universality for naturally arising models is a very active and important area of research. We mention the work of [SW16] on the Riemann function on the critical line and the work of [FK14, Web15, NSW18, LOS18, BWW18] on large random matrices.
The goal of this paper is to study the Gaussian multiplicative chaos for another natural non-Gaussian log-correlated field: (the square root of) the local times of two-dimensional Brownian motion.
Before stating our main results, we start by introducing a few notations. Let be the law under which is a planar Brownian motion starting from . Let be an open bounded simply connected domain, a starting point and be the first exit time of :
For all define the local time of at up to time (here stands for the euclidean norm):
One can use classical theory of one-dimensional semimartingales to get existence for a fixed of as a process. In this article, we need to make sense of jointly in and in . It is provided by Proposition 1.1 that we state at the end of this section. If the circle is not entirely included in , we will use the convention . For all we consider the sequence of random measures on defined by: for all Borel sets ,
The presence of the square root in the exponential may appear surprising at a first glance, but it is natural nevertheless in view of Dynkin-type isomorphisms (see [Ros14]).
To capture the fractal geometrical properties of a log-correlated field, another natural approach consists in encoding the so-called thick points (points where the field is unusually large) in flat measures supported on those thick points. At criticality, such measures are often called extremal processes. See for instance [BL16], [BL18] in the case of discrete two-dimensional GFF, see also [Abe18] in the case of simple random walk on trees. In our case, we can consider for all the sequence of random measures on defined by: for all Borel sets and ,
For all , the sequences of random measures and converge as in probability for the topology of vague convergence on and on respectively towards Borel measures and .
The measure can be decomposed as a product of a measure on and a measure on . Moreover, the component on agrees with and the component on is exponential:
For all , we have -a.s.,
Moreover, by denoting the conformal radius of from and the Green function of in , (see (1.8)), we have for all Borel set ,
The decomposition of and (1.3) justify that the square root of the local times is the right object to consider. These two properties are very similar to the case of the two-dimensional GFF (see [BL16] and [Ber16], Theorem 2.1 for instance).
Simulations of can be seen in Figure 1. They have been performed using simple random walk on the square lattice killed when it exits a square composed of vertices.
We decided to not include the case to ease the exposition, but notice that is also a sensible measure in this case. By modifying very few arguments in the proofs of Theorems 1.1 and 1.2, one can show that this sequence of random measures converges for the topology of vague convergence on towards a measure which can be decomposed as
for some random Borel measure on . With the help of (6.3) in Proposition 6.2 characterising the measure , it can be shown that is actually -a.s. absolutely continuous with respect to the occupation measure of Brownian motion, with a deterministic density. This last observation was already made in [AHS18], Section 7. See Section 1.2 for more details about the relation of our results to this paper.
Define the set of -thick points at level by
This is similar to the notion of thick points in [DPRZ01], except that they look at the occupation measure of small discs rather than small circles. In [Jeg18], the question to show the convergence of the rescaled number of thick points for the simple random walk on the two-dimensional square lattice was raised. As a direct corollary of Theorems 1.1 and 1.2, we answer the analogue of this question in the continuum:
For all , we have the following convergence in :
where denotes the Lebesgue measure of .
As mentioned in [Jeg18], this shows a fundamental difference in the structure of the thick points of GFF compare to those of planar Brownian motion. Indeed, for the GFF, the logarithmic power in the renormalisation is not the same (1/2 and not 1). This difference is initially surprising in view of the strong links between the two objects, see e.g. [Jeg18] for more about this.
The local time process , , , possesses a jointly continuous modification . In fact, this modification is -Hölder for all .
The proof of this proposition will be given in Appendix C. In the rest of the paper, when we write we actually always mean its continuous modification .
1.2 Relation with other works and further results
The construction of measures supported on the thick points of planar Brownian motion was initiated by the work of Bass, Burdzy and Khoshnevisan [BBK94]. The notion of thick points therein is defined through the number of excursions from which hit the circle , before the Brownian motion exits the domain : more precisely, for , they define the set
Note that our parametrisation is somewhat different; it is chosen to match the GMC theory. Informally, the relation between the two is given by . Next, we recall that the carrying dimension of a measure is the infimum of for which there exists a set such that and the Hausdorff dimension of is equal to . They showed:
Theorem A (Theorem 1.1 of [Bbk94]).
Assume that the domain is the unit disc of and that the starting point is the origin. For all , with probability one there exists a random measure , which is carried by and whose carrying dimension is equal to .
In [BBK94], the measure is constructed as the limit of measures as which are defined in a very similar manner as our measures using local times of circles (see the beginning of Section 3 of [BBK94]). We emphasise here the difference of renormalisation: the local times they consider are half of our local times. We also mention that the range for which they were able to show the convergence of is a strict subset of the so-called -phase of the GMC, which would correspond to or . This is the region where is bounded in , see Theorem 3.2 of [BBK94].
Bass, Burdzy and Khoshnevisan also gave an effective description of their measure in terms of a Poisson point process of excursions. More precisely, they define a probability distribution (written in [BBK94], defined just before Proposition 5.1 of [BBK94]) on continuous trajectories which can be understood heuristically as follows. The trajectory of a process under is composed of three independent parts. The first one is a Brownian motion starting from conditioned to visit before exiting and killed at the hitting time of . The third part is a Brownian motion starting from and killed when it exits for the first time . The second part is composed of an infinite number of excursions from generated by a Poisson point process with the intensity measure being the product of the Lebesgue measure on and an excursion law. In Proposition 5.1 of [BBK94], they roughly speaking show that for all , the law of the Brownian motion conditioned on the fact that is in the support of is . This characterises their measure (Theorem 5.2 of [BBK94]). Once Theorems 1.1 and 1.2 above are established, we can adapt their arguments for the proof of characterisation to conclude the same thing for our measure : see Proposition 6.2 for a precise statement. A consequence of Proposition 6.2 is the identification of our measure with their measure :
If the domain is the unit disc, the origin, and , we have -a.s. .
A consequence of Theorem A is a lower bound on the Hausdorff dimension of the set of thick points : for all , a.s. . The upper bound they obtained ([BBK94], Theorem 1.1 (ii)) is: for all , a.s. . They conjectured that the lower bound is sharp and holds for all . In 2001, Dembo, Peres, Rosen and Zeitouni [DPRZ01] answered positively the analogue of this question for thick points defined through the occupation measure of small discs:
In particular, their result went beyond the -phase to cover the entire -phase. This allowed them to solve a conjecture by Erdős and Taylor [ET60].
Very recently, Aïdékon, Hu and Shi [AHS18] made a link between the definitions of thick points of [BBK94] and [DPRZ01] (defined in (1.5) and (1.6) respectively) by constructing measures supported on these two sets of thick points. Their approach is superficially very different from ours but we will see that the measure we obtained is, perhaps surprisingly, related to theirs in a strong way (Corollary 1.3 below). Their measure is defined through a martingale approach for which the interpretation of the approximation is not immediately transparent (see [AHS18] (4.1), (4.2) and Corollary 3.6).
Let us describe this relation in more details. For technical reasons, in [AHS18], the boundary of is assumed to be a finite union of analytic curves. To compare our results with theirs, we will also make this extra assumption in the following and we will call such a domain a nice domain. Consider a boundary point such that the boundary of is analytic locally around ; we will call such a point a nice point. They denote by the law of a Brownian motion starting from and conditioned to exit through . They showed:
Theorem B (Theorem 1.1 of [Ahs18]).
For all , with -probability one there exists a random measure which is carried by and by and whose carrying dimension is equal to .
Their starting point is the interpretation of the measure of [BBK94] described above in terms of Poisson point process of excursions. For , they define a measure on trajectories similar to mentioned above: the only difference is that the last part of the trajectory is a Brownian motion conditioned to exit the domain through . In a nutshell, they show the absolute continuity of with respect to and define a sequence of measures using the Radon-Nikodym derivative. Their convergence relies on martingales argument rather than on computations on moments. As in [BBK94], they obtain a characterisation of their measure in terms of ([AHS18], Proposition 5.1) matching with ours (Proposition 6.2). As a consequence, we are able to compare their measure with ours.
Before stating this comparison, let us notice that we can also make sense of our measure for the Brownian motion conditioned to exit through . Indeed, as noticed in [BBK94], Remark 5.1 (i), our measure is measurable with respect to the Brownian path and defined locally. is thus well defined for any process which is locally mutually absolutely continuous with respect to the two dimensional Brownian motion killed when it exits for the first time the domain . The Brownian motion conditioned to exit through being such a process, makes sense under as a measure on .
Let be a nice point and denote by the Poisson kernel of from at , that is the density of the harmonic measure with respect to the Lebesgue measure of at . For all , if , we have -a.s.,
For all , the following properties hold:
(i) Non-degeneracy: with -probability one, .
(ii) Thick points: with -probability one, is carried by and by .
(iii) Hausdorff dimension: with -probability one, the carrying dimension of is .
(iv) Conformal invariance: if is a conformal map between two nice domains, , and if we denote by and the measures built in Theorem 1 for the domains and respectively, we have
Let us mention that we present the previous properties (i)-(iii) as a consequence of Corollary 1.3 to avoid to repeat the arguments, but we could have obtained them without the help of [AHS18]: as in [BBK94], (i) and (ii) follow from the Poisson point process interpretation of the measure (Proposition 6.2) whereas (iii) follows from our second moment computations (Proposition 4.1). On the other hand, it is not clear that our approach yields the conformal invariance of the measure without the use of [AHS18].
Finally, while there are strong similarities between and the GMC measure associated to a GFF (indeed, our construction is motivated by this analogy), there are also essential differences. In fact, from the point of view of GMC theory, the measure is rather unusual in that it is carried by the random fractal set and does not need extra randomness to be constructed, unlike say Liouville Brownian motion or other instances of GMC on random fractals.
1.3 Organisation of the paper
We now explain the main ideas of our proofs of Theorems 1.1 and 1.2 and how the paper is organised. The overall strategy of the proof is inspired by [Ber17]. To prove the convergence of the measures and , it is enough to show that for any suitable and , the real valued random variables and converge in probability which is the content of Proposition 6.1 (we actually show that they converge in ). As in [Ber17], we will consider modified versions and of and by introducing good events (see (2.11) and (2.13)): at a given , the local times are required to be never too thick around at every scale. We will show that introducing these good events does not change the behaviour of the first moment (Propositions 3.1 and 3.2, Section 3) and it makes the sequences and bounded in (Propositions 4.1 and 4.2, Section 4). Furthermore, we will see that these two sequences are Cauchy sequences in (Proposition 5.1, Section 5) implying in particular that they converge in . Section 6 finishes the proof of Theorems 1.1 and 1.2 and demonstrates the links of our work with the ones of [BBK94] and [AHS18] (Corollaries 1.2, 1.3 and 1.4).
We now explain a few ideas underlying the proof. If the domain is a disc centred at , then it is easy to check (by rotational invariance of Brownian motion and second Ray-Knight isomorphism for local times of one-dimensional Brownian motion) that the local times have a Markovian structure. More precisely, for all and all , under and conditioned on ,
with being a zero-dimensional Bessel process starting from . This is an other clue that exponentiating the square root of the local times should yield an interesting object.
In the case of a general domain , such an exact description is of course not possible, yet for small enough radii, the behaviour of can be seen to be approximatively given by the one in (1.7). If we assume (1.7) then the construction of is similar to the GMC construction for GFF, with the Brownian motions describing circle averages replaced by Bessel processes of suitable dimension. It seems intuitive that the presence of the drift term in a Bessel process should not affect significantly the picture in [Ber17].
To implement our strategy and use (1.7), we need an argument. In the first moment computations (Propositions 3.1 and 3.2), we will need a rough upper bound on the local times; an obvious strategy consists in stopping the Brownian motion when it exits a large disc containing the domain. For the second moment (Proposition 5.1), we will need a much more precise estimate. Let us assume for instance that . We can decompose the local times according to the different macroscopic excursions from to before exiting the domain . To keep track of the overall number of excursions, we will condition on their initial and final points. Because of this conditioning, the local times of a specific excursion are no longer related to a zero-dimensional Bessel process. But if we now condition further on the fact that the excursion went deep inside , it will have forgotten its initial point and those local times will be again related to a zero-dimensional Bessel process: this is the content of Lemma 5.1 and Appendix A is dedicated to its proof. Let us mention that the spirit of Lemma 5.1 can be tracked back to Lemma 7.4 of [DPRZ01].
As we have just explained, we will use (1.7) to transfer some computations from the local times to the zero-dimensional Bessel process. Throughout the paper, we will thus collect lemmas about this process (Lemmas 3.1, 3.2 and 5.2) that will be proven in Appendix B. Of course, we will not be able to transfer all the computations to the zero-dimensional Bessel process, for instance when we consider two circles which are not concentric. But we will be able to treat the local times as if they were the local times of a continuous time random walk: for a continuous time random walk starting at a given vertex and killed when it hits for the first time a given set , the time spent by the walk in is exactly an exponential variable which is independent of the hitting point of . We will show that it is also approximatively true for the local times of Brownian motion. This is the content of Section 2.
We end this introduction with some notations which will be used throughout the paper.
Notations: If , , , and , we will denote by:
the first hitting time of . In particular, ;
(resp. , ) the open disc (resp. closed disc, circle) with centre and radius ;
the Euclidean distance between and . If , we will simply write instead of ;
the conformal radius of seen from ;
the Green function in :
where is the transition probability of Brownian motion killed at . We recall its behaviour close to the diagonal (see Equation (1.2) of [Ber16] for instance):
where as ;
the law under which is a zero-dimensional Bessel process starting from ;
the set of integers .
Finally, we will write , etc, positive constants which may vary from one line to another. We will also write (resp. ) real-valued sequences which go to zero as (resp. which are bounded). If we want to emphasise that such a sequence may depend on a parameter , we will write (resp. ).
We start off with some preliminary results that will be used throughout the paper.
2.1 Green’s function
For all , so that and , we have:
We start by proving (2.1). By denoting the transition probability of Brownian motion killed at , we have:
But the Green function of the disc is equal to:
Because the last two integrals vanish, this gives (2.1). The proof of (2.2) is very similar. The only difference is that we consider the Green function of the general domain . Using the asymptotic (1.9), we conclude in the same way. ∎
2.2 Hitting probabilities
We now turn to the study of hitting probabilities. The following lemma gives estimates on the probability to hit a small circle before exiting the domain , whereas the next one gives estimates on the probability to hit a small circle before hitting another circle and before exiting the domain .
Let . For all small enough, for all such that and for all , we have:
A similar but weaker statement can be found in [BBK94] (Lemma 2.1) and our proof is really close to theirs. We will take smaller than to ensure that the circle stays far away from . If the domain were the unit disc and the origin, then the probability we are interested in is the probability to hit a small circle before hitting the unit circle. The two circles being concentric, we can use the fact that is a martingale to find that this probability is equal to:
In general, we come back to the previous situation by mapping onto the unit disc and to the origin with a conformal map . By conformal invariance of Brownian motion,
As is far away from the boundary of , the contour is included into a narrow annulus
for some depending on . In particular, using (2.4),
The lower bound is obtained is a similar manner which yields the stated claim (2.3) noticing that and that . ∎
For and define
For all , so that and are disjoint and included in , for all ,
By Markov property and by definition of , we have:
Combining those two inequalities yields
which is the first inequality stated in (2.6). The other inequality is similar. ∎
2.3 Approximation of local times by exponential variables
In this subsection, we explain how to approximate the local times by exponential variables. For and any event , define
where is the probability measure of Brownian motion starting at and killed when it hits for the first time the circle . For , we will denote the harmonic measure of from .
Let and . Assume that and that there exists such that for all and ,
Then for all and ,
The previous lemma states that we can approximate by an exponential variable which is independent of . This is similar to the case of random walks on discrete graphs. If we did not condition on , it would not have been necessary to add the multiplicative errors and . This statement without conditioning is also a consequence of Lemma 2.2 (i) of [BBK94].
Take small enough so that the annulus does not intersect . Consider the different excursions from to : denote and for all ,
The number of excursions before is related to by:
Hence, for any continuous bounded function, we have by dominated convergence theorem
and by a repeated application of Markov property, is at most
we have obtained
which is the required upper bound. The lower bound is obtained in a similar way. ∎
The next lemma explains how to compute the quantities appearing in the previous lemma. Again, particular cases of this can be found in [BBK94] (Lemmas 2.3, 2.5).
Let and such that and denote the distance between and . Assume . Let be either or and denote
We have for all , and
Moreover, denoting the first hitting time of after , we have for any ,
In this proof, we will consider such that .
Let us start by proving (2.7) for . Let . By Markov property applied to the first hitting time of , we have
But the measure is explicit and its density with respect to the Lebesgue measure on the circle is equal to
Hence, up to a multiplicative error , is independent of