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Abstract: Abstract We prove a quantitative Russo–Seymour–Welsh (RSW)-type result for random walks on two natural examples of random planar graphs: the supercritical percolation cluster in \(\mathbb {Z}^2\) and the Poisson Voronoi triangulation in \(\mathbb {R}^2\) . More precisely, we prove that the probability that a simple random walk crosses a rectangle in the hard direction with uniformly positive probability is stretched exponentially likely in the size of the rectangle. As an application, we prove a near optimal decorrelation result for uniform spanning trees for such graphs. This is the key missing step in the application of the proof strategy of Berestycki et al. (Ann Probab 48(1):1–52, 2020) for such graphs [in Berestycki et al. (2020), random walk RSW was assumed to hold with probability 1]. Applications to almost sure Gaussian-free field scaling limit for dimers on Temperleyan-type modification on such graphs are also discussed. PubDate: 2023-12-01
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Abstract: Abstract We introduce the notion of a conditional distribution to a zero-probability event in a given direction of approximation and prove that the conditional distribution of a family of independent Brownian particles to the event that their paths coalesce after the meeting coincides with the law of a modified massive Arratia flow, defined in Konarovskyi (Ann Probab 45(5):3293–3335, 2017. https://doi.org/10.1214/16-AOP1137). PubDate: 2023-12-01
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Abstract: Abstract We study planar random motions with finite velocities, of norm \(c>0\) , along orthogonal directions and changing at the instants of occurrence of a nonhomogeneous Poisson process with rate function \(\lambda = \lambda (t),\ t\ge 0\) . We focus on the distribution of the current position \(\bigl (X(t), Y(t)\bigr ),\ t\ge 0\) , in the case where the motion has orthogonal deviations and where also reflection is admitted. In all the cases, the process is located within the closed square \(S_{ct}=\{(x,y)\in {\mathbb {R}}^2\,:\, x + y \le ct\}\) and we obtain the probability law inside \(S_{ct}\) , on the edge \(\partial S_{ct}\) and on the other possible singularities, by studying the partial differential equations governing all the distributions examined. A fundamental result is that the vector process (X, Y) is probabilistically equivalent to a linear transformation of two (independent or dependent) one-dimensional symmetric telegraph processes with rate function proportional to \(\lambda \) and velocity c/2. Finally, we extend the results to a wider class of orthogonal-type evolutions. PubDate: 2023-12-01
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Abstract: Abstract The paper is concerned with a new approach for the recurrence properties of the oscillating random walk on \(\mathbb {Z}\) in the sense of Kemperman. In the case when the random walk is ascending on \(\mathbb {Z}^-\) and descending on \(\mathbb {Z}^+\) , the invariant measure of the embedded process of successive crossing times is explicitly determined, which yields a sufficient condition for recurrence. Finally, we make use of this result to show that the general oscillating random walk is recurrent under some moment assumptions. PubDate: 2023-12-01
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Abstract: Abstract Let \(R_n\) be an \(n \times n\) random matrix with i.i.d. subgaussian entries. Let M be an \(n \times n\) deterministic matrix with norm \(\Vert M \Vert \le n^\gamma \) where \(1/2<\gamma <1\) . The goal of this paper is to give a general estimate of the smallest singular value of the sum \(R_n + M\) , which improves an earlier result of Tao and Vu. PubDate: 2023-12-01
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Abstract: In the language of random counting measures, many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition of these objects, Gibbs processes can be introduced in abstract spaces without any topological structure. In this general setting, partition functions, Janossy densities, and correlation functions are studied. While the definition covers finite and infinite Gibbs processes alike, the finite case allows, even in abstract spaces, for an equivalent and more explicit characterization via a familiar series expansion. Recent generalizations of factorial measures to arbitrary measurable spaces, where counting measures cannot be written as sums of Dirac measures, likewise allow to generalize the concept of Hamiltonians. The DLR equations, which completely characterize a Gibbs process, as well as basic results for the local convergence topology, are also formulated in full generality. We prove a new theorem on the extraction of locally convergent subsequences from a sequence of point processes and use this statement to provide existence results for Gibbs processes in general spaces with potentially infinite range of interaction. These results are used to guarantee the existence of Gibbs processes with cluster-dependent interactions and to prove a recent conjecture concerning the existence of Gibbsian particle processes. PubDate: 2023-12-01
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Abstract: Abstract This paper is concerned with sample path properties of real-valued isotropic Gaussian fields on compact two-point homogeneous spaces. In particular, we establish the property of strong local nondeterminism of an isotropic Gaussian field and then exploit this result to establish an exact uniform modulus of continuity for its sample paths. PubDate: 2023-12-01
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Abstract: Abstract Independent Haar-unitary random matrices and independent Haar-orthogonal random matrices are known to be asymptotically liberating ensembles, and they give rise to asymptotic free independence when used for conjugation of constant matrices. G. Anderson and B. Farrel showed that a certain family of discrete random unitary matrices can actually be used to the same end. In this paper, we investigate fluctuation moments and higher-order moments induced on constant matrices by conjugation with asymptotically liberating ensembles. We show for the first time that the fluctuation moments associated with second-order free independence can be obtained from conjugation with an ensemble consisting of signed permutation matrices and the discrete Fourier transform matrix. We also determine fluctuation moments induced by various related ensembles where we do not get known expressions but others related to traffic free independence. PubDate: 2023-12-01
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Abstract: Abstract In this paper, we study the discretization of the ergodic Functional Central Limit Theorem (CLT) established by Bhattacharya (see Bhattacharya in Z Wahrscheinlichkeitstheorie Verwandte Geb 60:185–201, 1982) which states the following: Given a stationary and ergodic Markov process \((X_t)_{t \geqslant 0}\) with unique invariant measure \(\nu \) and infinitesimal generator A, then, for every smooth enough function f, \((n^{1/2} \frac{1}{n}\int _0^{nt} Af(X_s){\textrm{d}}s)_{t \geqslant 0}\) converges in distribution towards the distribution of the process \((\sqrt{-2 \langle f, Af \rangle _{\nu }} W_{t})_{t \geqslant 0}\) with \((W_{t})_{t \geqslant 0}\) a Wiener process. In particular, we consider the marginal distribution at fixed \(t=1\) , and we show that when \(\int _0^{n} Af(X_s)ds\) is replaced by a well chosen discretization of the time integral with order q (e.g. Riemann discretization in the case \(q=1\) ), then the CLT still holds but with rate \(n^{q/(2q+1)}\) instead of \(n^{1/2}\) . Moreover, our results remain valid when \((X_t)_{t \geqslant 0}\) is replaced by a q-weak order approximation (not necessarily stationary). This paper presents both the discretization method of order q for the time integral and the q-order ergodic CLT we derive from them. We finally propose applications concerning the first order CLT for the approximation of Markov Brownian diffusion stationary regimes with Euler scheme (where we recover existing results from the literature) and the second order CLT for the approximation of Brownian diffusion stationary regimes using Talay’s scheme (Talay in Stoch Stoch Rep 29:13–36, 1990) of weak order two. PubDate: 2023-12-01
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Abstract: Abstract Let \(C\ge 2\) be a positive integer. Consider the set of \(n\times n\) non-negative integer matrices whose row sums and column sums are all equal to Cn and let \(X=(X_{ij})_{1\le i,j\le n}\) be uniformly distributed on this set. This X is called the random contingency table with uniform margin. In this paper, we study various asymptotic properties of \(X=(X_{ij})_{1\le i,j\le n}\) as \(n\rightarrow \infty \) . PubDate: 2023-12-01
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Abstract: Abstract The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly appears quite naturally in ergodic theory in the study of measure-preserving transformations arising from various group actions or, equivalently, the study of stationary sequences when adopting a probabilistic perspective as in this paper. Our purpose is to give a new and relatively short proof of the coboundary theorem due to Schmidt (Cocycles on ergodic transformation groups. Macmillan lectures in mathematics, vol 1, Macmillan Company of India, Ltd., Delhi, 1977) which provides a sharp criterion that determines (and rules out) when two stationary processes belong to the same null-homology equivalence class. We also discuss various aspects of null-homology within the class of Markov random walks and compare null-homology with a formally stronger notion which we call strict-sense null-homology. Finally, we also discuss some concrete cases where the notion of null-homology turns up in a relevant manner. PubDate: 2023-12-01
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Abstract: Abstract In this article, we introduce fractional Poisson fields of order k in n-dimensional Euclidean space of positive real valued vectors. We also work on time-fractional Poisson process of order k, space-fractional Poisson processes of order k and a tempered version of time-space fractional Poisson processes of order k. We discuss generalized fractional Poisson processes of order k in terms of Bernstein functions. These processes are defined in terms of fractional compound Poisson processes. The time-fractional Poisson process of order k naturally generalizes the Poisson process and the Poisson process of order k to a heavy-tailed waiting-times counting process. The space-fractional Poisson process of order k allows on average an infinite number of arrivals in any interval. We derive the marginal probabilities governing difference–differential equations of the introduced processes. We also provide the Watanabe martingale characterization for some time-changed Poisson processes. PubDate: 2023-12-01
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Abstract: Abstract We study the ergodic behaviour of a discrete-time process X which is a Markov chain in a stationary random environment. The laws of \(X_t\) are shown to converge to a limiting law in (weighted) total variation distance as \(t\rightarrow \infty \) . Convergence speed is estimated, and an ergodic theorem is established for functionals of X. Our hypotheses on X combine the standard “drift” and “small set” conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain “maximal process” of the random environment. We are able to cover a wide range of models that have heretofore been intractable. In particular, our results are pertinent to difference equations modulated by a stationary (Gaussian) process. Such equations arise in applications such as discretized stochastic volatility models of mathematical finance. PubDate: 2023-12-01
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Abstract: Abstract We completely characterize when the free effective resistance of an infinite graph whose vertices have finite degrees can be expressed in terms of simple hitting probabilities of the random walk on the graph. PubDate: 2023-12-01
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Abstract: Abstract This paper deals with generalized backward doubly stochastic differential equations driven by a Lévy process (GBDSDEL, in short). Under left or right continuous and linear growth conditions, we prove the existence of minimal (resp. maximal) solutions. PubDate: 2023-12-01
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Abstract: Abstract In this paper, we study the strong convergence rate of the averaging principle of two-time-scale forward-backward stochastic differential equations (FBSDEs, for short). First, we present the well-posedness of the objective equations and then we give some a priori estimates for FBSDEs, backward stochastic auxiliary equations and backward stochastic averaged equations. Second, a strong convergence rate of the averaging principle for two-time-scale FBSDEs is derived. As far as we know, this is the first result on the strong convergence rate of the averaging principle of two-time-scale backward stochastic differential equations (BSDEs, for short). PubDate: 2023-12-01
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Abstract: Abstract We consider Galton–Watson trees conditioned on both the total number of vertices n and the number of leaves k. The focus is on the case in which both k and n grow to infinity and \(k = \alpha n + O(1)\) , with \(\alpha \in (0, 1)\) . Assuming exponential decay of the offspring distribution, we show that the rescaled random tree converges in distribution to Aldous’s Continuum Random Tree with respect to the Gromov–Hausdorff topology. The scaling depends on a parameter \(\sigma ^*\) which we calculate explicitly. Additionally, we compute the limit for the degree sequences of these random trees. PubDate: 2023-12-01
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Abstract: Abstract It is well known that the law of a one-dimensional diffusion on natural scale is fully characterized by its speed measure. Stone proved a continuous dependence of such diffusions on their speed measures. In this paper we establish the converse direction, i.e., we prove a continuous dependence of the speed measures on their diffusions. Furthermore, we take a topological point of view on the relation. More precisely, for suitable topologies, we establish a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the set of locally finite speed measures. PubDate: 2023-12-01
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Abstract: Abstract This paper aims to investigate stochastic fractional Schrödinger evolution equations with potential and Poisson jumps in Hilbert space. The solvability of the proposed system is established by using fractional calculus, semigroup theory, Krasnoselskii’s fixed point theorems and stochastic analysis. Furthermore, sufficient conditions are formulated and proved to assure that the mild solution decays exponentially to zero in the square mean. Lastly, an application is given to demonstrate the developed theory. PubDate: 2023-12-01
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Abstract: Abstract This paper is devoted to the analysis of the finite-dimensional distributions and asymptotic behavior of extremal Markov processes connected with the Kendall convolution. In particular, we provide general formulas for the finite dimensional distributions of the random walk driven by the Kendall convolution for a large class of step size distributions. Moreover, we prove limit theorems for random walks and associated continuous-time stochastic processes. PubDate: 2023-09-05 DOI: 10.1007/s10959-023-01285-2
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