The Markov property of local times of Brownian motion indexed by the Brownian tree
Jean-François Le Gall
备注：
Revised version, taking account of all remarks made by referees. In particular, an error at the end of the proof of Lemma 8 has been corrected. A couple of arguments have also been simplified
We consider the model of Brownian motion indexed by the Brownian tree, which has appeared in a variety of different contexts in probability, statistical physics and combinatorics. For this model, the total occupation measure is known to have a continuously differentiable density. Although the density process indexed by nonnegative reals is not Markov, we prove that the pair consisting of the density and its derivative is a time-homogeneous Markov process. We also establish a similar result for the local times of one-dimensional super-Brownian motion. Our methods rely on the excursion theory for Brownian motion indexed by the Brownian tree.
We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Levy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no "really nice" set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, …
We derive several explicit distributions of functionals of Brownian motion indexed by the Brownian tree. In particular, we give a direct proof of a result of Bousquet-Mélou and Janson identifying the distribution of the density at 0 of the integrated super-Brownian excursion.
We prove that a set-indexed process is a set-indexed fractional Brownian motion if and only if its projections on all the increasing paths are one-parameter time changed fractional Brownian motions. As an application, we present an integral representation for such processes.
We consider the model of Brownian motion indexed by the Brownian tree. For every $r\geq 0$ and every connected component of the set of points where Brownian motion is greater than $r$, we define the boundary size of this component, and we then show that the collection of these boundary sizes evolves when $r$ varies like a well-identified growth-fragmentation process. We then prove that the same growth-fragmentation process appears when slicing a Brownian disk at height $r$ and considering the perimeters of the resulting connected components.
We give a necessary condition of geometric nature for the existence of the $H$-fractional Brownian field indexed by a Riemannian manifold. In the case of the Lévy Brownian field ($H=1/2$) indexed by manifolds with minimal closed geodesics it turns out to be very strong. In particular we show that compact manifolds admitting a Lévy Brownian field are simply connected. We also derive from our result the nondegenerescence of the Lévy Brownian field indexed by hyperbolic spaces. These results stress the need for alternative kernels on nonsimply connected manifolds to allow for Gaussian modelling o…
We prove that a square-integrable set-indexed stochastic process is a set-indexed Brownian motion if and only if its projection on all the strictly increasing continuous sequences are one-parameter $G$-time-changed Brownian motions. In addition, we study the "sequence-independent variation" property for group stationary-increment stochastic processes in general and for a set-indexed Brownian motion in particular. We present some applications.
Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] \times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Lévy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Hölder regularity prope…
We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical Itô theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connectedcomponent is itself a continuous tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with branching mechanism $ψ(u)=\sqrt{8/3}\,u^{3/2}$. Furthermo…
The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to statisfy a strenghtened definition of increment stationarity. This new definition for stationarity property allows to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dime…
In this article is introduced and studied a set-indexed Markov property named C-Markov. This new definition fulfils one important expectation for a Markov property: there exists a natural set-indexed generalization of the concept of transition operator which leads to characterization and construction theorems for C-Markov processes. Several other usual Markovian notions, including Feller and strong Markov properties, can also be developed in this framework. Actually, the C-Markov property turns out to be a natural extension of the two-parameter \ast-Markov property to the multiparameter and as…
We consider a type of Markov property for set-indexed processes which is satisfied by all processes with independent increments and which allows us to introduce a transition system theory leading to the construction of the process. A set-indexed generator is defined such that it completely characterizes the distribution of the process.
We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot's fractal percolation process.
We show in this paper that there exists no $H$-fractional Brownian field indexed by the cylinder $\mathbb{S}^1 \times ]0,\varepsilon[$ endowed with its product distance $d$ for any $\varepsilon>0$ and $H>0$. This is equivalent to say that $d^{2H}$ is not a negative definite kernel, which also leaves us without a proof that many classical stationary kernels, such that the Gaussian and exponential kernels, are positive definite kernels -- or covariances -- on the cylinder. We generalise this result from the cylinder to any Riemannian Cartesian product with a minimal closed geodesic. We also inve…
We present a satisfactory definition of the important class of Lévy processes indexed by a general collection of sets. We use a new definition for increment stationarity of set-indexed processes to obtain different characterizations of this class. As an example, the set-indexed compound Poisson process is introduced. The set-indexed Lévy process is characterized by infinitely divisible laws and a Lévy-Khintchine representation. Moreover, the following concepts are discussed: projections on flows, Markov properties, and pointwise continuity. Finally the study of sample paths leads to a Lévy-Itô…
We construct the analogue of the local time -- at a fixed point $x$ -- for Markov processes indexed by Levy trees. We start by proving that Markov processes indexed by Levy trees satisfy a special Markov property which can be thought as a spatial version of the classical Markov property. Then, we construct the analogue of the local time by an approximation procedure and we characterize the support of its Lebesgue-Stieltjes measure. We also give an equivalent construction in terms of a special family of exit local times. Finally, combining these results, we show that the points at which the Mar…
We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Holder exponent form a set with large intersection.
We investigate positive definiteness of the Brownian kernel K(x,y)=1/2(d(x,x_0) + d(y,x_0) - d(x,y)) on a compact group G and in particular for G=SO(n).
Two-sided infinite systems of Brownian particles with rank-dependent dynamics, indexed by all integers, exhibit different properties from their one-sided infinite counterparts, indexed by positive integers, and from finite systems. Consider the gap process, which is formed by spacings between adjacent particles. In stark contrast with finite and one-sided infinite systems, two-sided infinite systems can have one- or two-parameter family of stationary gap distributions, or the gap process weakly converging to zero as time goes to infinity.
The purpose of this article is a set-indexed extension of the well-known Ornstein-Uhlenbeck process. The first part is devoted to a stationary definition of the random field and ends up with the proof of a complete characterization by its $L^2$-continuity, stationarity and set-indexed Markov properties. This specific Markov transition system allows to define a general \emph{set-indexed Ornstein-Uhlenbeck (SIOU) process} with any initial probability measure. Finally, in the multiparameter case, the SIOU process is proved to admit a natural integral representation.
A family of Feller branching diffusions $Z^x$, $x \ge 0$, with nonlinear drift and initial value $x$ can, with a suitable coupling over the {\em ancestral masses} $x$, be viewed as a path-valued process indexed by $x$. For a coupling due to Dawson and Li, which in case of a linear drift describes the corresponding Feller branching diffusion, and in our case makes the path-valued process Markovian, we find an SDE solved by $Z$, which is driven by a random point measure on excursion space. In this way we are able to identify the infinitesimal generator of the path-valued process. We also establi…
We are interested in the increment stationarity property for $L^2$-indexed stochastic processes, which is a fairly general concern since many random fields can be interpreted as the restriction of a more generally defined $L^2$-indexed process. We first give a spectral representation theorem in the sense of \citet{Ito54}, and see potential applications on random fields, in particular on the $L^2$-indexed extension of the fractional Brownian motion. Then we prove that this latter process is characterized by its increment stationarity and self-similarity properties, as in the one-dimensional cas…
We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_σ)_{σ\in\mathcal T}$, indexed by the full binary tree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcal T_k$ is the $k$th generation of $\mathcal T$. In addition, let $(Σ_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $Σ_k \in \mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ with $\widetilde R_t^n := X_{Σ_{[tn]}}$, arising by observing the Markov chain $\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empir…
We consider a Markov-modulated Brownian motion $\{Y(t), ρ(t)\}$ with two boundaries at $0$ and $b > 0$, and allow for the controlling Markov chain $\{ρ(t)\}$ to instantaneously undergo a change of phase upon hitting either of the two boundaries at semi-regenerative epochs defined to be the first time the process reaches a boundary since it last hits the other boundary. We call this process a flexible Markov-modulated Brownian motion. Using the recently-established links between stochastic fluid models and Markov-modulated Brownian motions, we determine important characteristics of first exit t…
In this paper we study the stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs for short). We extend the notion of conditional $G$-expectation from deterministic time to the more general optional time situation. Then, via this conditional expectation, we develop the strong Markov property for $G$-SDEs. In particular, we obtain the strong Markov property for $G$-Brownian motion. Some applications including the reflection principle for $G$-Brownian motion are also provided.
Ramaswami showed recently that standard Brownian motion arises as the limit of a family of Markov-modulated linear fluid processes. We pursue this analysis with a fluid approximation for Markov-modulated Brownian motion. Furthermore, we prove that the stationary distribution of a Markov-modulated Brownian motion reflected at zero is the limit from the well-analyzed stationary distribution of approximating linear fluid processes. Key matrices in the limiting stationary distribution are shown to be solutions of a new quadratic equation, and we describe how this equation can be efficiently solved…
Fractional Brownian motion belongs to a class of long memory Gaussian processes that can be represented as linear functionals of an infinite dimensional Markov process. This representation leads naturally to: - An efficient algorithm to approximate the process. - An infinite dimensional ergodic theorem which applies to functionals of the type $integral_0^t phi(V_h(s)) ds $ where $V_h(s)=integral_0^t h(t-u) dB_u$ and $B$ is a standard Brownian motion.
We analyze the stationary distribution of regulated Markov modulated Brownian motions (MMBM) modified so that their evolution is slowed down when the process reaches level zero --- level zero is said to be {\em sticky}. To determine the stationary distribution, we extend to MMBMs a construction of Brownian motion with sticky boundary, and we follow a Markov-regenerative approach similar to the one developed in past years in the context of quasi-birth-and-death processes and fluid queues. We also rely on recent work showing that Markov-modulated Brownian motions may be analyzed as limits of a p…
In this paper we consider a class of stochastic differential equations driven by subordinate Brownian motion with Markovian switching. We use Malliavin calculus to study the smoothness of the density for the solution under uniform Hörmander's type condition.
In this article, we study the family of probability measures (indexed by a positive real number t), obtained by penalization of the Brownian motion by a given functional of its local times at time t. We prove that this family tends to a limit measure when t goes to infinity if the functional satisfies some conditions of domination, and we check these conditions in several particular cases.
We study continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. Burdzy and Pal in their paper proposed a continuous version of graphical models -- processes indexed by time-like graphs. We construct a more general family of continuous processes on a wider family of graphs, find new properties and solve the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. We present new results showing connections to the stochastic heat equation, Markov random…
In Latouche and Nguyen (2015), the authors constructed a sequence of stochastic fluid processes and showed that it converges weakly to a Markov-modulated Brownian motion (MMBM). Here, we construct a different sequence of stochastic fluid processes and show that it converges strongly to an MMBM. To the best of our knowledge, this is the first result on strong convergence to a Markov-modulated Brownian motion. We also prove that the rate of this almost sure convergence is $o(n^{-1/2} \log n)$. When reduced to the special case of standard Brownian motion, our convergence rate is an improvement ov…
In this paper we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain h…
We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton--Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time t. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. The latter has the same generator as the Markov process along the branches plus additional jumps, ass…
We introduce block Markov chains (BMCs) indexed by an infinite rooted tree. It turns out that BMCs define a new class of tree-indexed Markovian processes. We clarify the structure of BMCs in connection with Markov chains (MCs) and Markov random fields (MRFs). Mainly, show that probability measures which are BMCs for every root are indeed Markov chains (MCs) and yet they form a strict subclass of Markov random fields (MRFs) on the considered tree. Conversely, a class of MCs which are BMCs is characterized. Furthermore, we establish that in the one-dimensional case the class of BMCs coincides wi…
In this paper, we study the Hölder regularity of set-indexed stochastic processes defined in the framework of Ivanoff-Merzbach. The first key result is a Kolmogorov-like Hölder-continuity Theorem, whose novelty is illustrated on an example which could not have been treated with anterior tools. Increments for set-indexed processes are usually not simply written as $X_U-X_V$, hence we considered different notions of Hölder-continuity. Then, the localization of these properties leads to various definitions of Hölder exponents, which we compare to one another. In the case of Gaussian processes, al…
We consider a Markov-modulated Brownian motion reflected to stay in a strip [0,B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explaining its simplicity. Moreover, this argument allows for generalizations including the distribution of the reflected process at an independent exponentially distributed epoch. Our second contribution concerns transient behavior of the reflected system. We identify the joint law of the processes t,X(t),J(t) at inverse local times.
We extend to Markov-modulated Brownian motion (MMBM) the renewal approach which has been successfully applied to the analysis of Markov-modulated fluid models. It has recently been shown that MMBM may be expressed as the limit of a parameterized family of Markov-modulated fluid models. We prove that the weak convergence also holds for systems with two reflecting boundaries, one at zero and one at $b >0$, and that the stationary distributions of the approximating fluid models converge to the stationary distribution of the two-sided reflected MMBM. Thus, we obtain a new representation for the st…
We study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.
Markovian maximal couplings of Markov processes are characterized by an equality of total variation and a distance of Wasserstein type. If a Markovian maximal coupling is a Feller process, the generator can be calculated, e.g. for reflection coupled Brownian motion. Apart from processes with continuous paths also jump processes are treated for the first time. For subordinated Brownian motion a Markovian maximal coupling is constructed by subordinating reflection coupled Brownian motion. This coupling is the unique Markovian maximal coupling and its generator is determined by state-space depend…
We consider a rough differential equation indexed by a small parameter $\varepsilon>0$. When the rough differential equation is driven by fractional Brownian motion with Hurst parameter $H$ ($1/4<H<1/2$), we prove the Laplace-type asymptotics for the solution as the parameter $\varepsilon$ tends to zero.
We discuss subordination of random compact R-trees. We focus on the case of the Brownian tree, where the subordination function is given by the past maximum process of Brownian motion indexed by the tree. In that particular case, the subordinate tree is identified as a stable Levy tree with index 3/2. As a more precise alternative formulation, we show that the maximum process of the Brownian snake is a time change of the height process coding the Levy tree. We then apply our results to properties of the Brownian map. In particular, we recover, in a more precise form, a recent result of Miller …
Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam and van Moerbeke that the process of eigenvalues of two consecutive minors of an Hermitian Brownian motion is a Markov process, whereas if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.
The purpose of this paper is to highlight some hidden Markovian structure of the concave majorant of the Brownian motion. Several distributional identities are implied by the joint law of a standard one-dimensional Brownian motion $B$ and its almost surely unique concave majorant $K$ on $[0,\infty)$. In particular, the one-dimensional distribution of $2 K_t - B_t$ is that of $R_5(t)$, where $R_5$ is a $5-$dimensional Bessel process with $R_5(0) = 0$. The process $2K-B$ shares a number of other properties with $R_5$, and we conjecture that it may have the distribution of $R_5$. We also describe…
We first consider the additive Brownian motion process $(X(s_1,s_2),\ (s_1,s_2) \in \mathbb{R}^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random set $\{(s_1,s_2)\in \mathbb{R}^2: X(s_1,s_2) >0\}$ is equal to $$ \frac{1}{4}\left(1 + \sqrt{13 + 4 \sqrt{5}}\right) \simeq 1.421\, . $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the nonnegative quadrant.
In this paper we show that an $H$-Brownian Gibbsian line ensemble is completely characterized by the finite-dimensional marginals of its lowest indexed curve for a large class of interaction Hamiltonians $H$. A particular consequence of our result is that the KPZ line ensemble is the unique line ensemble that satisfies the $H_1$-Brownian Gibbs property with $H_1(x) = e^x$ and whose lowest indexed curve is equal to the Hopf-Cole solution to the narrow wedge KPZ equation.
We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class ${\mathcal{F}}$ is different from the class of functions $f$ for which we have good properties of the observables $(f(X_i))_{i\geq0}$. We introduce a new bracketing number to measure the size of the index…
Let $X$ be a super-Brownian motion (SBM) defined on a domain $E\subset R^n$ and $(X_D)$ be its exit measures indexed by sub-domains of $E$. The relationship between the equation $1/2 Δu=2 u^2$ and Super-Brownian motion (SBM) is analogous to the relationship between Brownian motion and the Laplace's equation, and substantial progress has been made on the study of the solutions of this semi-linear p.d.e. exploring this analogy. An area that remains to be explored is Martin boundary theory. Martin boundary in the semi-linear case is defined as the convex set of extreme $X$-harmonic functions whic…
