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In this paper the existence of intersections for functions of several Brownian motions on the Carnot group is studied. A condition is presented for the existence of such intersections with Probability 1, which is in the form of the asymptotics of a measure on a specific family of small balls. The measure is arbitrary but can be chosen as a surface measure on the manifold related to intersections, and the balls are constructed using the distances related to the processes. Additionally, if the same condition holds in a weaker form, it is shown that there is a Hausdorff measure, such that the value of this Hausdorff measure on the set of intersection points is finite with Probability 1. [ABSTRACT FROM AUTHOR]

We study the hitting probabilities of the solution to a system of d stochastic heat equations with additive noise subject to Dirichlet boundary conditions. We show that for any bounded Borel set with positive (d - 6) -dimensional capacity, the solution visits this set almost surely. [ABSTRACT FROM AUTHOR]

We consider a multidimensional Markov chain X converging to a multidimensional Brownian motion. We construct a positive harmonic function for X killed on exiting the cone. We show that its asymptotic behaviour is similar to that of the harmonic function of Brownian motion. We use the harmonic function to study the asymptotic behaviour of the tail distribution of the exit time τ of X from a cone. [ABSTRACT FROM AUTHOR]

We study a quasi-ergodic theorem for moments of the mean-ratio of a discontinuous additive functional caused by the pure jump effects of an absorbing symmetric stable process on an open set D ⊂ R d . Our result depends only on a simple geometric condition on D and does not require that the total mass of D is finite. As a result, we prove that the result holds if D is a subset of a horn-shaped region with the reference function satisfying some condition. [ABSTRACT FROM AUTHOR]

The objective of this paper is to examine the restriction of a right process on a Radon topological space, excluding a negligible set, and investigate whether the restricted object can induce a Markov process with desirable properties. We address this question in three aspects: the induced process necessitates only right continuity; it is a right process, and the semi-Dirichlet form of the induced process is quasi-regular. The main findings characterize the negligible set that meets the requirements within a universally measurable framework. These characterizations can be employed to generate instances of Markov processes that are non-right or (semi-)Dirichlet forms that are non-quasi-regular. Specifically, we will construct an example of a non-tight, strong Feller, symmetric right process on a non-Lusin Radon topological space, whose Dirichlet form is not quasi-regular. [ABSTRACT FROM AUTHOR]

We consider an asymptotically stable multidimensional random walk S (n) = (S 1 (n) , ... , S d (n)) . For every vector x = (x 1 ... , x d) with x 1 ≥ 0 , let τ x : = min { n > 0 : x 1 + S 1 (n) ≤ 0 } be the first time the random walk x + S (n) leaves the upper half space. We obtain the asymptotics of p n (x , y) : = P (x + S (n) ∈ y + Δ , τ x > n) as n tends to infinity, where Δ is a fixed cube. From that, we obtain the local asymptotics for the Green function G (x , y) : = ∑ n p n (x , y) , as | y | and/or | x | tend to infinity. [ABSTRACT FROM AUTHOR]

Martingales with jumps on Riemannian manifolds and harmonic maps with respect to Markov processes are discussed in this paper. Discontinuous martingales on manifolds were introduced in Picard (Séminaire de Probabilités de Strasbourg 25:196–219, 1991). We obtain results about the convergence of martingales with finite quadratic variations on Riemannian submanifolds of higher-dimensional Euclidean space as t → ∞ and as t → 0 . Furthermore, we apply the result about martingales with jumps on submanifolds to harmonic maps with respect to Markov processes such as fractional harmonic maps. [ABSTRACT FROM AUTHOR]

We give a probabilistic representation of the solution to a semilinear elliptic Dirichlet problem with general (discontinuous) boundary data. The boundary behaviour of the solution is in the sense of the controlled convergence initiated by A. Cornea. Uniqueness results for the solution are also provided. [ABSTRACT FROM AUTHOR]

Using a biorthogonal technique (Appell system), the foremost aim of this study is to develop and highlight specific aspects of a new polynomial sequence known as Fractional Krawtchouk Appell polynomials associated with the fractional Pascal probability distribution. Additionally, the connection that exists between brand-new polynomials and the orthogonal Bell polynomials has been investigated. [ABSTRACT FROM AUTHOR]

A single hydrodynamic vortex on a surface will in general move unless its Riemannian metric is a special “Steady Vortex Metric” (SVM). Metrics of constant curvature are SVM only in surfaces of genus zero and one. In this paper: I show that K. Okikiolu’s work on the regularization of the spectral zeta function leads to the conclusion that each conformal class of every compact surface with a genus of two or more possesses at least one steady vortex metric (SVM). I apply a probabilistic interpretation of the regularized zeta function for surfaces, as developed by P. G. Doyle and J. Steiner, to extend the concept of SVM to higher dimensions. The new special metric, which aligns with the Steady Vortex Metric (SVM) in two dimensions, has been termed the “Uniform Drainage Metric” for the following reason: For a compact Riemannian manifold M , the “narrow escape time” (NET) is defined as the expected time for a Brownian motion starting at a point p in M \ B ϵ (q) to remain within this region before escaping through the small ball B ϵ (q) , which is centered at q with radius ϵ and acts as the escape window. The manifold is said to possess a uniform drainage metric if, and only if, the spatial average of NET, calculated across a uniformly distributed set of initial points p , remains invariant regardless of the position of the escape window B ϵ (q) , as ϵ approaches 0 . [ABSTRACT FROM AUTHOR]

We prove the existence and establish the Lifschitz singularity of the integrated density of states for certain random Hamiltonians H ω = H 0 + V ω on fractal spaces of infinite diameter. The kinetic term H 0 is given by ϕ (- L) , where L is the Laplacian on the fractal and ϕ is a completely monotone function satisfying some mild regularity conditions. The random potential V ω is of alloy-type. [ABSTRACT FROM AUTHOR]

The Sierpinski gasket K has three line segments constituting a regular triangle as its border. This paper studies what will happen if one of them, which is called the bottom line and is denoted by I, is removed from K. At a glance, “the Sierpinski gasket minus the bottom line” K \ I has a structure of a tree of Sierpinski gaskets. This observation leads us to the results showing that the boundary of K \ I is not the line segment I but a Cantor set from viewpoints of geometry and analysis. As a by-product, we have an explicit expression of the jump kernel of the trace of the Brownian motion of K on the bottom line I. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to develop the concepts and ideas used in the theory of weighted networks on infinite graphs. Our basic ingredients are a standard Borel space (V , B) and symmetric σ -finite measure ρ on (V × V , B × B) supported by a symmetric Borel subset E ⊂ V × V . This setting is analogous to that of weighted networks and are used in various areas of mathematics such as Dirichlet forms, graphons, Borel equivalence relations, Guassian processes, determinantal point processes, joinings, etc. We define and study the properties of a measurable analogue of the graph Laplacian, finite energy Hilbert space, dissipation Hilbert space, and reversible Markov processes associated directly with the measurable framework. In the settings of networks on infinite graphs and standard Borel space, our present focus will be on the following two dichotomies: (i) Discrete vs continuous (or rather the measurable category); and (ii) deterministic vs stochastic. Both (i) and (ii) will be made precise inside our paper; in fact, this part constitutes a separate issue of independent interest. A main question we shall address in connection with both (i) and (ii) is that of limits. For example, what are the useful notions, and results, which yield measurable limits; more precisely, Borel structures arising as “limits” of systems of graph networks? We shall study it both in the abstract, and via concrete examples, especially models built on Bratteli diagrams. The question of limits is of basic interest; both as part of the study of dynamical systems on path- space; and also with regards to numerical considerations; and design of simulations. The particular contexts for the study of this question entails a careful design of appropriate Hilbert spaces, as well as operators acting between them. Our main results include: an explicit spectral theory for measure theoretic graph-Laplace operators; new properties of Green’s functions and corresponding reproducing kernel Hilbert spaces; structure theorems of the set of harmonic functions in L 2 -spaces and finite energy Hilbert spaces; the theory of reproducing kernel Hilbert spaces related to Laplace operators; a rigorous analysis of the Laplacian on Borel equivalence relations; a new decomposition theory; irreducibility criteria; dynamical systems governed by endomorphisms and measurable fields; and path-space measures and induced dissipation Hilbert spaces. We discuss also several applications of our results to other fields such as machine learning problems, ergodic theory, and reproducing kernel Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with a generalized N-urn Ehrenfest model, where balls perform independent random walks between N boxes uniformly laid on [0,1]. After a proper scaling of the transition rates function of the aforesaid random walk, we derive the hydrodynamic limit of the model, i.e., the law of large numbers which the empirical measure of the model follows, under an assumption where the initial number of balls in each box independently follows a Poisson distribution. We show that the empirical measure of the model converges weakly to a deterministic measure with density driven by an integral equation. Furthermore, we derive non-equilibrium fluctuation of the model, i.e, the central limit theorem from the above hydrodynamic limit. We show that the non-equilibrium fluctuation of the model is driven by a time-inhomogeneous generalized O-U process on the dual of C[0,1]. At last, we prove a large deviation principle from the hydrodynamic limit under an assumption where the transition rates function from [0,1] × [0,1] to [ 0 , + ∞) of the aforesaid random walk is a product of two marginal functions from [0,1] to [ 0 , + ∞) . [ABSTRACT FROM AUTHOR]

For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up. [ABSTRACT FROM AUTHOR]

We consider on M(ℝd) (the set of all finite measures on ℝd) the evolution equation associated with the nonlinear operator F↦ΔF′+∑k⩾1bkFkF \mapsto \Delta F' + \sum\nolimits_{k \geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by means of the distribution of the d-dimensional Brownian motion and the non-local branching process on the finite configurations of M(ℝd), induced by the sequence (bk)k⩾1 of positive numbers such that ∑k⩾1bk⩽1\sum\nolimits_{k \geqslant 1} b_k \leqslant 1. It turns out that the representation also holds with the same branching process for the solution to the equation obtained replacing the Laplace operator by the generator of a Markov process on ℝd instead of the d-dimensional Brownian motion; more general, we can take the generator of a right Markov process on a Lusin topological space. We first investigate continuous flows driving branching processes. We show that if the branching mechanism of a superprocess is independent of the spatial variable, then the superprocess is obtained by introducing the branching in the time evolution of the right continuous flow on measures, canonically induced by a right continuous flow as spatial motion. A corresponding result holds for non-local branching processes on the set of all finite configurations of the state space of the spatial motion.

Consider finite state space irreducible and absorbing Markov processes. A general spectral criterion is provided for the absorbing time to be close to an exponential random variable, whatever the starting point. When exiting points are added to the state space, our criterion also insures that the exit time and position are almost independent. Since this is valid for any exiting extension of the state space, it corresponds to an instance of the metastability phenomenon. Simple examples at small temperature suggest that this new spectral criterion is quite sharp. But the main interest of the underlying quantitative approach, based on Poisson equations, is that it does not rely on a small parameter such as temperature, nor on reversibility. [ABSTRACT FROM AUTHOR]

We consider killed planar random walks on isoradial graphs. Contrary to the lattice case, isoradial graphs are not translation invariant, do not admit any group structure and are spatially non-homogeneous. Despite these crucial differences, we compute the asymptotics of the Martin kernel, deduce the Martin boundary and show that it is minimal. Similar results on the grid ℤ d are derived in a celebrated work of Ney and Spitzer. [ABSTRACT FROM AUTHOR]

In this paper, we establish the coincidence of two classes of L p -Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of L p -Kato class measures is defined by the pth power of positive order resolvent kernel, another is defined in terms of the pth power of Green kernel depending on some exponents related to the heat kernel estimates. We also prove that functions u such that sup x ∈ E ∫ B 1 (x) | u | q d m < ∞ are of L p -Kato class if q is greater than a constant related to p and the constants appeared in the upper and lower estimates of the heat kernel. These are complete extensions of some results by Aizenman–Simon and the recent results by the second named author in the framework of Brownian motions on Euclidean space. We further give necessary and sufficient conditions for a Radon measure with Ahlfors regularity to belong to L p -Kato class. Our results can be applicable to many examples, for instance, symmetric (relativistic) stable processes, jump processes on d-sets, Brownian motions on Riemannian manifolds, diffusions on fractals and so on. [ABSTRACT FROM AUTHOR]

We investigate level sets of the Gaussian free field on continuous transient metric graphs G ~ and study the capacity of its level set clusters. We prove, without any further assumption on the base graph G , that the capacity of sign clusters on G ~ is finite almost surely. This leads to a new and effective criterion to determine whether the sign clusters of the free field on G ~ are bounded or not. It also elucidates why the critical parameter for percolation of level sets on G ~ vanishes in most instances in the massless case and establishes the continuity of this phase transition in a wide range of cases, including all vertex-transitive graphs. When the sign clusters on G ~ do not percolate, we further determine by means of isomorphism theory the exact law of the capacity of compact clusters at any height. Specifically, we derive this law from an extension of Sznitman's refinement of Lupu's recent isomorphism theorem relating the free field and random interlacements, proved along the way, and which holds under the sole assumption that sign clusters on G ~ are bounded. Finally, we show that the law of the cluster capacity functionals obtained in this way actually characterizes the isomorphism theorem, i.e. the two are equivalent. [ABSTRACT FROM AUTHOR]

Let X be the constrained random walk on Z + 2 having increments (1, 0), (- 1 , 1) , (0 , - 1) with jump probabilities λ (M k) , μ 1 (M k) , and μ 2 (M k) where M is an irreducible aperiodic finite state Markov chain. The process X represents the lengths of two tandem queues with arrival rate λ (M k) , and service rates μ 1 (M k) , and μ 2 (M k) ; the process M represents the random environment within which the system operates. We assume that the average arrival rate with respect to the stationary measure of M is less than the average service rates, i.e., X is assumed stable. Let τ n be the first time when the sum of the components of X equals n for the first time. Let Y be the random walk on Z × Z + having increments (- 1 , 0) , (1, 1), (0 , - 1) with probabilities λ (M k) , μ 1 (M k) , and μ 2 (M k) . Supposing that the queues share a joint buffer of size n, p n = P (x n , m) (τ n < τ 0) is the probability that this buffer overflows during a busy cycle of the system. To the best of our knowledge, the only methods currently available for the approximation of p n are classical large deviations analysis giving the exponential decay rate of p n and rare event simulation. Let τ be the first time the components of Y are equal. For x ∈ R + 2 , x (1) + x (2) < 1 , x (1) > 0 , and x n = ⌊ n x ⌋ , we show that P (n - x n (1) , x n (2) , m) (τ < ∞) approximates P (x n , m) (τ n < τ 0) with exponentially vanishing relative error as n → ∞ . For the analysis we define a characteristic matrix in terms of the jump probabilities of (X, M). The 0-level set of the characteristic polynomial of this matrix defines the characteristic surface; conjugate points on this surface and the associated eigenvectors of the characteristic matrix are used to define (sub/super) harmonic functions which play a fundamental role both in our analysis and the computation/approximation of P (y , m) (τ < ∞) . [ABSTRACT FROM AUTHOR]

In this article we study stability properties of g O , the standard Green kernel for O an open regular set in ℝ d . In d ≥ 3 we show that g O β is again a Green kernel of a Markov Feller process, for any power β ∈ [1,d/(d − 2)). In dimension d = 2, we show the same result for g O β , for any β ≥ 1 and for kernels exp (α g O ) , exp (α g O ) − 1 , for α ∈ (0,2π), when O is an open Greenian regular set whose complement contains a ball. [ABSTRACT FROM AUTHOR]

The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field μ, there exist two natural infinite-volume Dirichlet forms (E u p r , D u p r ) and (E l w r , D l w r ) on L2(S,μ) describing interacting Brownian motions each with unlabeled equilibrium state μ. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the Markovian semi-group given by (E l w r , D l w r ) is associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide with each other by using the uniqueness of weak solutions of ISDE. We apply Theorem 3.1 to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle's class interaction potentials, and Theorem 3.2 to the sine2 interacting Brownian motion and interacting Brownian motions with Ruelle's class interaction potentials of C 0 3 -class. [ABSTRACT FROM AUTHOR]

Let { X t } t ≥ 0 be a transient α -stable process on R d and denote by H its generator. We consider the perturbation of the operator H by a positive measure μ from a certain class and denote by p μ (t , x , y) the corresponding fundamental solution. We obtain the precise asymptotics of p μ (t , x , y) as t → ∞ for critical μ , which is an extension of Pinchover (J Funct Anal 206:191–209, 2004). [ABSTRACT FROM AUTHOR]

We study semi-martingale obliquely reflected Brownian motion with drift in the first quadrant of the plane in the transient case. Our main result determines a general explicit integral expression for the moment generating function of Green's functions of this process. To that purpose we establish a new kernel functional equation connecting moment generating functions of Green's functions inside the quadrant and on its edges. This is reminiscent of the recurrent case where a functional equation derives from the basic adjoint relationship which characterizes the stationary distribution. This equation leads us to a non-homogeneous Carleman boundary value problem. Its resolution provides a formula for the moment generating function in terms of contour integrals and a conformal mapping. [ABSTRACT FROM AUTHOR]

We investigate stochastic fragmentation processes for particles with spatial position. The mathematical problem models the time evolution of a system of particles which move on an Euclidean surface driven by a given force (e.g., gravitational, fluid interaction, repulsion/attraction), and split in fragments with smaller masses and velocities. We establish a multiple-fragmentation process and we solve the corresponding stochastic integro-differential equation. Finally, we present several numerical simulations of such processes. [ABSTRACT FROM AUTHOR]