The aim of this text is to establish some relations between Markov chains in Dirichlet Environments on directed graphs and certain hypergeometric integrals associated with a particular arrangement of hyperplanes. We deduce from these relations and the computation of the connexion obtained by moving one hyperplane of the arrangement some new relations on important functionals of the Markov chain.
The Lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a Dirichlet process. This approach leads to several results, which are illustrated here. Among these are a new and more direct procedure for determining the exact form of the distribution of the mean, a correspondence between the distribution of the mean and the parameter of a Dirichlet process, a characterization of the family of Cauchy distributions as the set of the fixed points of this correspondence, and an extension of the Markov-Krein identity. Moreov…
We consider one-dependent random walks on $\mathbb{Z}^d$ in random hypergeometric environment for $d\ge 3$. These are memory-one walks in a large class of environments parameterized by positive weights on directed edges and on pairs of directed edges which includes the class of Dirichlet environments as a special case. We show that the walk is a.s. transient for any choice of the parameters, and moreover that the return time has some finite positive moment. We then give a characterization for the existence of an invariant measure for the process from the point of view of the walker which is ab…
This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. We give the complete classification of such curves: conformal Markov property and symmetry single out a two-parameter family of random curves---Hypergeometric SLE---denoted by hSLE$_κ(ν)$ for $κ\in (0,4]$ and $ν<κ-6$. The proof relies crucially on Dubédat's commutation relation [Dub07] and a uniqueness result proved in [MS16b]. The classification indicates that hypergeometric SLE is the only possible scaling limit of the interfaces in critical lattice models (conjectured or…
We prove the equivalence of the local property for an irreducible regular Dirichlet form and the Markov property for the Gaussian field associated with the Dirichlet form. Moreover we introduce a strong Markov property for Gaussian fields and present some sufficient conditions for this to hold.
We give some relationships between the first Dirichlet eigenvalues and the exit time moments for the general symmetric Markov processes. As applications, we present some examples, including symmetric diffusions and $α$-stable processes, and provide the estimates of their first Dirichlet eigenvalues and the exit time moments.
Nakao's stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. Ito's formula in terms of the extended stochastic integrals is obtained.
We extend the Dirichlet principle to non-reversible Markov processes on countable state spaces. We present two variational formulas for the solution of the Poisson equation or, equivalently, for the capacity between two disjoint sets. As an application we prove a some recurrence theorems. In particular, we show the recurrence of two-dimensional cycle random walks under a second moment condition on the winding numbers.
This paper describes how one can use the well-known Bayesian prior to posterior analysis of the Dirichlet process, and less known results for the gamma process, to address the formidable problem of assessing the distribution of linear functionals of Dirichlet processes. In particular, in conjunction with a gamma identity, we show easily that a generalized Cauchy-Stieltjes transform of a linear functional of a Dirichlet process is equivalent to the Laplace functional of a class of, what we define as, beta-gamma processes. This represents a generalization of the Markov-Krein identity for mean fu…
The recently found hypergeometric multiple orthogonal polynomials on the step-line by Lima and Loureiro are shown to be random walk polynomials. It is proven that the corresponding Jacobi matrix and its transpose, which are nonnegative matrices and describe higher recurrence relations, can be normalized to two stochastic matrices, dual to each other. Using the Christoffel-Darboux formula on the step-line and the Poincaré theory for non-homogeneous recurrence relations it is proven that both stochastic matrices are related by transposition in the large $n$ limit. These random walks are beyond b…
We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal Lévy processes with weak scaling conditions. First, we establish sharp two-sided heat kernel estimates for these processes in $C^{1,1}$ open sets. As corollaries of our main results, we obtain sharp two-sided Green function estimates and a scale invariant boundary Harnack inequality with explicit decay rates in $C^{1,1}$ open sets.
We consider a general class of discrete, two-dimensional Markov chains modeling the dynamics of a population with two types, without mutation or immigration, and neutral in the sense that type has no influence on each individual's birth or death parameters. We prove that all the eigenvectors of the corresponding transition matrix or infinitesimal generator Π can be expressed as the product of "universal" polynomials of two variables, depending on each type's size but not on the specific transitions of the dynamics, and functions depending only on the total population size. These eigenvectors a…
Laplace transforms for integrals of stochastic processes have been known in analytically closed form for just a handful of Markov processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In this paper, we construct broad extensions of these process classes. We show how the known models fit into a classification scheme for diffusion processes for which Laplace transforms for integrals of the diffusion processes and transitiona…
Let $(\mathcal{E},D(\mathcal{E}))$ be a quasi-regular semi-Dirichlet form and $(X_t)_{t\geq0}$ be the associated Markov process. For $u\in D(\mathcal{E})_{loc}$, denote $A_t^{[u]}:=\tilde{u}(X_{t})-\tilde{u}(X_{0})$ and $F^{[u]}_t:=\sum_{0<s\leq t}(\tilde u(X_{s})-\tilde u(X_{s-}))1_{\{|\tilde u(X_{s})-\tilde u(X_{s-})|>1\}}$, where $\tilde{u}$ is a quasi-continuous version of $u$. We show that there exist a unique locally square integrable martingale additive functional $Y^{[u]}$ and a unique continuous local additive functional $Z^{[u]}$ of zero quadratic variation such that $$A_t^{[u]}=Y_t^…
We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $\mathbb{R}^{n^2}$. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean $(1/n,...,1/n)$. We show that if $\bM$ is such a random matrix, then the empirical distribution built from the singular values of$\sqrt{n} \bM$ tends as $n\to\infty$ to a Wigner quarter--circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis…
It is well known that a regular diffusion on an interval $I$ without killing inside is uniquely determined by a canonical scale function $s$ and a canonical speed measure $m$. Note that $s$ is a strictly increasing and continuous function and $m$ is a fully supported Radon measure on $I$. In this paper we will associate a general triple $(I,s,m)$, where $s$ is only assumed to be increasing and $m$ is not necessarily fully supported, to certain Markov processes by way of Dirichlet forms. A straightforward generalization of Dirichlet form associated to regular diffusion will be first put forward…
Let $(S,ρ)$ be an ultrametric space with certain conditions and $S^k$ be the quotient space of $S$ with respect to the partition by balls with a fixed radius $ϕ(k)$. We prove that, for a Hunt process $X$ on $S$ associated with a Dirichlet form $(\mathcal E, \mathcal F)$, a Hunt process $X^k$ on $S^k$ associated with the averaged Dirichlet form $(\mathcal E^k, \mathcal F^k)$ is Mosco convergent to $X$, and under certain additional conditions, $X^k$ converges weakly to $X$. Moreover, we give a sufficient condition for the Markov property of $X$ to be preserved under the canonical projection $π^k…
We consider a Markov chain on the space of (countable) partitions of the interval [0,1], obtained first by size biased sampling twice (allowing repetitions) and then merging the parts (if the sampled parts are distinct) or splitting the part uniformly (if the same part was sampled twice). We prove a conjecture of Vershik stating that the Poisson-Dirichlet law with parameter theta=1 is the unique invariant distribution for this Markov chain. Our proof uses a combination of probabilistic, combinatoric, and representation-theoretic arguments.
In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains.
In this paper we consider a large class of symmetric Markov processes $X=(X_t)_{t\ge0}$ on $\R^d$ generated by non-local Dirichlet forms, which include jump processes with small jumps of $α$-stable-like type and with large jumps of super-exponential decay. Let $D\subset \R^d$ be an open (not necessarily bounded and connected) set, and $X^D=(X_t^D)_{t\ge0}$ be the killed process of $X$ on exiting $D$. We obtain explicit criterion for the compactness and the intrinsic ultracontractivity of the Dirichlet Markov semigroup $(P^{D}_t)_{t\ge0}$ of $X^D$. When $D$ is a horn-shaped region, we further o…
We combine the notion of free Stein kernel and the free Malliavin calculus to provide quantitative bounds under the free (quadratic) Wasserstein distance in the multivariate semicircular approximations for self-adjoint vector-valued multiple Wigner integrals. On the way, we deduce an HSI inequality for a modified non-microstates free entropy with respect to the potential associated with these semicircular families in the case of non-degeneracy of the covariance matrix. The strategy of the proofs is based on functional inequalities involving the free Stein discrepancy. We obtain a bound which d…
Strong and Markov uniqueness problems in $L^2$ for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on a--priori estimates is used. The extension of the problem to the $L^p$-setting is discussed. As a direct application essential self--adjointness and strong uniqueness in $L^p$ is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume $Λ\subset \R^2$.
The networks of this -- primarily (but not exclusively) expository -- compendium are strongly connected, finite directed graphs $X$, where each oriented edge $(x,y)$ is equipped with a positive weight (conductance) $a(x,y)$. We are not assuming symmetry of this function, and in general we do not require that along with $(x,y)$, also $(y,x)$ is an edge. The weights give rise to a difference operator, the normalised version of which we consider as our Laplace operator. It is associated with a Markov chain with state space $X$. A non-empty subset of $X$ is designated as the boundary. We provide a…
We proposed in \cite{bl2} a new approach to prove the metastable behavior of reversible dynamics based on potential theory and local ergodicity. In this article we extend this theory to nonreversible dynamics based on the Dirichlet principle proved in \cite{gl2}.
The Markov chain approximation of a one-dimensional symmetric diffusion is investigated in this paper. Given an irreducible reflecting diffusion on a closed interval with scale function $s$ and speed measure $m$, the approximating Markov chains are constructed explicitly through the trace of the Dirichlet form corresponding to the diffusion. One feature of our approach is that it does not require uniform ellipticity on diffusion coefficient of the limit object or uniform regularity on conductances of the approximative Markov chains, as imposed usually in the previous related works.
We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis. A key technical ingredient in this work is the use of a novel Gromov--Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion.
The quantitative long time behavior of absorbing, finite, irreducible Markov processes is considered. Via Doob transforms, it is shown that only the knowledge of the ratio of the values of the underlying first Dirichlet eigenvector is necessary to come back to the well-investigated situation of the convergence to equilibrium of ergodic finite Markov processes. This leads to explicit estimates on the convergence to quasi-stationarity, in particular via functional inequalities. When the process is reversible, the optimal exponential rate consisting of the spectral gap between the two first Diric…
We consider Markov chains on the space of (countable) partitions of the interval $[0,1]$, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $β_m$ (if the sampled parts are distinct) or splitting the part with probability $β_s$ according to a law $σ$ (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if $σ$ is the uniform measure then the Poisson-Dirichlet law is an invariant probability measure, and it is unique within a suitably defined class of ``analytic'' invarian…
The aim of this article is to prove that diffusion processes in $\mathbb{R}^d$ with a drift can be approximated by suitable Markov chains on $n^{-1}\mathbb{Z}^d$. Moreover, we investigate sufficient conditions on the conductances which guarantee convergence of the associated Markov chains to such Markov processes. Analogous questions are answered for a large class of nonsymmetric jump processes. The proofs of our results rely on regularity estimates for weak solutions to the corresponding nonsymmetric parabolic equations and Dirichlet form techniques.
Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an Itô formula for Dirichlet processes is obtained.
We consider the problem of learning two families of time-evolving random measures from indirect observations. In the first model, the signal is a Fleming--Viot diffusion, which is reversible with respect to the law of a Dirichlet process, and the data is a sequence of random samples from the state at discrete times. In the second model, the signal is a Dawson--Watanabe diffusion, which is reversible with respect to the law of a gamma random measure, and the data is a sequence of Poisson point configurations whose intensity is given by the state at discrete times. A common methodology is develo…
In this paper, we first develop a new family of conjugate prior distributions for the cell parameters of discrete graphical models Markov with respect to a set P of moral directed acyclic graphs with skeleton a given decomposable graph G. Such families arise when the set of conditional independences between discrete variables is given and can be represented by a decomposable graph and additionally, the direction of certain edges is imposed by the practitioner. This family, which we call the P-Dirichlet, is a generalization of the hyper Dirichlet given in Dawid and Lauritzen (1993): it keeps th…
For any discrete target distribution, we exploit the connection between Markov chains and Stein's method via the generator approach and express the solution of Stein's equation in terms of expected hitting time. This yields new upper bounds of Stein's factors in terms of the parameters of the Markov chain, such as mixing time and the gradient of expected hitting time. We compare the performance of these bounds with those in the literature, and in particular we consider Stein's method for discrete uniform, binomial, geometric and hypergeometric distribution. As another application, the same met…
Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. Perron-Frobenius theory ensures the existence of a corresponding positive eigenvector $φ$. The goal of the paper is to give bounds on the amplitude $\max φ/\minφ$. Two approaches are proposed: one using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative st…
Chen, Fitzsimmons, Kuwae and Zhang (Ann. Probab. 36 (2008) 931-970) have established an Ito formula consisting in the development of F(u(X)) for a symmetric Markov process X, a function u in the Dirichlet space of X and any $\mathcal{C}^2$-function F. We give here an extension of this formula for u locally in the Dirichlet space of X and F admitting a locally bounded Radon-Nikodym derivative. This formula has some analogies with various extended Ito formulas for semi-martingales using the local time stochastic calculus. But here the part of the local time is played by a process $(Γ^a_t,a\in \m…
For a Markov process associated with a diffusion type Dirichlet form an upper bound is shown for the law of the finite dimensional distributions of the process. Under some more assumptions on the underlaying space this is also shown for the law of the Markov process itself. In the last section we want to give an application to the Wasserstein diffusion.
We provide a sharp nonasymptotic analysis of the rates of convergence for some standard multivariate Markov chains using spectral techniques. All chains under consideration have multivariate orthogonal polynomial as eigenfunctions. Our examples include the Moran model in population genetics and its variants in community ecology, the Dirichlet-multinomial Gibbs sampler, a class of generalized Bernoulli--Laplace processes, a generalized Ehrenfest urn model and the multivariate normal autoregressive process.
Let X be a spectrally negative self-similar Markov process with 0 as an absorbing state. In this paper, we show that the distribution of the absorption time is absolutely continuous with an infinitely continuously differentiable density. We provide a power series and a contour integral representation of this density. Then, by means of probabilistic arguments, we deduce some interesting analytical properties satisfied by these functions, which include, for instance, several types of hypergeometric functions. We also give several characterizations of the Kesten's constant appearing in the study …
Motivated by Lévy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be ``continuous'' in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator…
Stochastic models for collections of interacting populations have crucial roles in scientific fields such as epidemiology and ecology, yet the standard approach to extending an ordinary differential equation model to a Markov chain does not have sufficient flexibility in the mean-variance relationship to match data. To handle that, over-dispersed Markov chains have previously been constructed using gamma white noise on the rates. We develop new approaches using Dirichlet noise to construct collections of independent or dependent noise processes. This permits the modeling of high-frequency vari…
