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Abstract: Abstract For any fixed real \(a > 0\) and \(x \in {\mathbb {R}}^d, d \ge 1\) , we consider the real-valued random process \((S_n)_{n \ge 0}\) defined by \( S_0= a, S_n= a+\ln \vert g_n\cdots g_1x\vert , n \ge 1\) , where the \(g_k, k \ge 1, \) are i.i.d. nonnegative random matrices. By using the strategy initiated by Denisov and Wachtel to control fluctuations in cones of d-dimensional random walks, we obtain an asymptotic estimate and bounds on the probability that the process \((S_n)_{n \ge 0}\) remains nonnegative up to time n and simultaneously belongs to some compact set \([b, b+\ell ]\subset {\mathbb {R}}^+_*\) at time n. PubDate: 2024-05-13

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Abstract: This paper gives a new property for stochastic processes, called square-mean \(\mu -\) pseudo-S-asymptotically Bloch-type periodicity. We show how this property is preserved under some operations, such as composition and convolution, for stochastic processes. Our main results extend the classical results on S-asymptotically Bloch-type periodic functions. We also apply our results to some problems involving semilinear stochastic integrodifferential equations in abstract spaces PubDate: 2024-05-03

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Abstract: Abstract We consider the well-posedness problem of multi-dimensional reflected backward stochastic differential equations driven by G-Brownian motion (G-BSDEs) with diagonal generators. Two methods, including the penalization method and the Picard iteration argument, are provided to prove the existence and uniqueness of the solutions. We also study its connection with the obstacle problem of a system of fully nonlinear PDEs. PubDate: 2024-04-26

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Abstract: Abstract The paper discusses the Galton–Watson stochastic branching system. We deal only with the noncritical case. Our task in the work is to improve our recent result which explicitly calculated the famous constant in the theory of subcritical Galton–Watson branching systems, announced by Kolmogorov in 1938. We demonstrate the rate of convergence to the Kolmogorov constant. This improvement contributes to determining the speed of approximation rate in a number of classical limit theorems of the theory of Galton–Watson branching systems. PubDate: 2024-04-24

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Abstract: Abstract The present article describes the precise structure of the \(L^{p}\) -spaces of projective limit measures by introducing a category-theoretical perspective. This analysis is applied to measures on vector spaces and in particular to Gaussian measures on nuclear topological vector spaces. A simple application to constructive quantum field theory (QFT) is given through the Osterwalder–Schrader axioms. PubDate: 2024-04-20

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Abstract: Abstract In this paper, we study the weak differentiability of global strong solution of stochastic differential equations, the strong Feller property of the associated diffusion semigroups and the global stochastic flow property in which the singular drift b and the weak gradient of Sobolev diffusion \(\sigma \) are supposed to satisfy \(\left\ \left b\right \cdot \mathbbm {1}_{B(R)}\right\ _{p_1}\le O((\log R)^{{(p_1-d)^2}/{2p^2_1}})\) and \(\left\ \left\ \nabla \sigma \right\ \cdot \mathbbm {1}_{B(R)}\right\ _{p_1}\le O((\log ({R}/{3}))^{{(p_1-d)^2}/{2p^2_1}})\) , respectively. The main tools for these results are the decomposition of global two-point motions in Fang et al. (Ann Probab 35(1):180–205, 2007), Krylov’s estimate, Khasminskii’s estimate, Zvonkin’s transformation and the characterization for Sobolev differentiability of random fields in Xie and Zhang (Ann Probab 44(6):3661–3687, 2016). PubDate: 2024-04-19

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Abstract: Abstract We introduce a new formulation of reflected backward stochastic differential equations (BSDEs) and doubly reflected BSDEs associated with irregular obstacles. In the first part of the paper, we consider an extension of the classical optimal stopping problem over a larger set of stopping systems than the set of stopping times (namely, the set of split stopping times), where the payoff process \(\xi \) is irregular and in the case of a general filtration. Split stopping times are a powerful tool for modeling financial contracts and derivatives that depend on multiple conditions or triggers, and for incorporating stochastic processes with jumps and other types of discontinuities. We show that the value family can be aggregated by an optional process v, which is characterized as the Snell envelope of the reward process \(\xi \) over split stopping times. Using this, we prove the existence and uniqueness of a solution Y to irregular reflected BSDEs. In the second part of the paper, motivated by the classical Dynkin game with completely irregular rewards considered by Grigorova et al. (Electron J Probab 23:1–38, 2018), we generalize the previous equations to the case of two reflecting barrier processes. PubDate: 2024-04-11

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Abstract: Abstract We study Jacobi processes \((X_{t})_{t\ge 0}\) on \([-1,1]^N\) and \([1,\infty [^N\) which are motivated by the Heckman–Opdam theory and associated integrable particle systems. These processes depend on three positive parameters and degenerate in the freezing limit to solutions of deterministic dynamical systems. In the compact case, these models tend for \(t\rightarrow \infty \) to the distributions of the \(\beta \) -Jacobi ensembles and, in the freezing case, to vectors consisting of ordered zeros of one-dimensional Jacobi polynomials. We derive almost sure analogues of Wigner’s semicircle and Marchenko–Pastur limit laws for \(N\rightarrow \infty \) for the empirical distributions of the N particles on some local scale. We there allow for arbitrary initial conditions, which enter the limiting distributions via free convolutions. These results generalize corresponding stationary limit results in the compact case for \(\beta \) -Jacobi ensembles and, in the deterministic case, for the empirical distributions of the ordered zeros of Jacobi polynomials. The results are also related to free limit theorems for multivariate Bessel processes, \(\beta \) -Hermite and \(\beta \) -Laguerre ensembles, and the asymptotic empirical distributions of the zeros of Hermite and Laguerre polynomials for \(N\rightarrow \infty \) . PubDate: 2024-04-11

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Abstract: Abstract We consider stochastic differential equations (SDEs) with (distributional) drift in negative Besov spaces and random initial condition and investigate them from two different viewpoints. In the first part we set up a martingale problem and show its well-posedness. We then prove further properties of the martingale problem, such as continuity with respect to the drift and the link with the Fokker–Planck equation. We also show that the solutions are weak Dirichlet processes for which we evaluate the quadratic variation of the martingale component. In the second part we identify the dynamics of the solution of the martingale problem by describing the proper associated SDE. Under suitable assumptions we show equivalence with the solution to the martingale problem. PubDate: 2024-04-06

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Abstract: Abstract We deal with McKean–Vlasov and Boltzmann-type jump equations. This means that the coefficients of the stochastic equation depend on the law of the solution, and the equation is driven by a Poisson point measure with intensity measure which depends on the law of the solution as well. Alfonsi and Bally (Construction of Boltzmann and McKean Vlasov type flows (the sewing lemma approach), 2021, arXiv:2105.12677) have proved that under some suitable conditions, the solution \(X_t\) of such equation exists and is unique. One also proves that \(X_t\) is the probabilistic interpretation of an analytical weak equation. Moreover, the Euler scheme \(X_t^{{\mathcal {P}}}\) of this equation converges to \(X_t\) in Wasserstein distance. In this paper, under more restrictive assumptions, we show that the Euler scheme \(X_t^{{\mathcal {P}}}\) converges to \(X_t\) in total variation distance and \(X_t\) has a smooth density (which is a function solution of the analytical weak equation). On the other hand, in view of simulation, we use a truncated Euler scheme \(X^{{\mathcal {P}},M}_t\) which has a finite numbers of jumps in any compact interval. We prove that \(X^{{\mathcal {P}},M}_{t}\) also converges to \(X_t\) in total variation distance. Finally, we give an algorithm based on a particle system associated with \(X^{{\mathcal {P}},M}_t\) in order to approximate the density of the law of \(X_t\) . Complete estimates of the error are obtained. PubDate: 2024-04-03

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Abstract: Abstract For a class of \(\Lambda \) -Fleming–Viot processes with Brownian spatial motion in \(\mathbb {R}^d\) whose associated \(\Lambda \) -coalescents come down from infinity, we obtain sharp global and local moduli of continuity for the ancestral processes recovered from the associated lookdown representations. As applications, we establish both global and local moduli of continuity for the \(\Lambda \) -Fleming–Viot support processes. In particular, if the \(\Lambda \) -coalescent is the Beta \((2-\beta ,\beta )\) coalescent for \(\beta \in (1,2]\) with \(\beta =2\) corresponding to Kingman’s coalescent, then for \(h(t)=\sqrt{t\log (1/t)}\) , the global modulus of continuity holds for the support process with modulus function \(\sqrt{2\beta /(\beta -1)}h(t)\) , and both the left and right local moduli of continuity hold for the support process with modulus function \(\sqrt{2/(\beta -1)}h(t)\) . PubDate: 2024-04-03

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Abstract: Abstract We analyze the p-random Wishart tensor ( \(p\ge 2\) ) associated with an initial \(n\times d\) random matrix \(\mathcal {X}_{n,d}\) whose entries are independent and belong to the Wiener chaos. The order of the Wiener chaos is constant in each row of the matrix, and it may change from one row to another. Therefore, the entries of our starting matrix are not identically distributed (we only assume that their second moments coincide). We prove that, in high-dimensional regime (i.e. when n and d are large enough), the multidimensional random vector corresponding to the p-Wishart tensor is close in distribution to a standard multidimensional Gaussian vector. We also evaluate the Wasserstein distance between the probability distributions of these two random vectors. By using the techniques of the Stein–Malliavin calculus, we show that when the dimensions n and d are large enough and p is fixed, this Wasserstein distance is of order less than \(\sqrt{\frac{n^{2p-1}}{d}}\) . PubDate: 2024-03-30

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Abstract: Abstract In this paper, we consider asymptotic behaviors for random geometric series. We first study the convergence rates in the central limit theorem, i.e., the Berry–Esseen bound and Edgeworth expansions, and precise deviations. Then we define a bounded linear operator from the path space of random walk to the path space of the random geometric series and establish the functional central limit theorem, the functional law of iterated logarithm, and functional large deviation principles for the random geometric series. PubDate: 2024-03-29

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Abstract: Abstract We consider a Markovian model of an SIR epidemic spreading on a contact graph that is drawn uniformly at random from the set of all graphs with n vertices and given vertex degrees. Janson, Luczak and Windridge (Random Struct Alg 45(4):724–761, 2014) prove that the evolution of such an epidemic is well approximated by the solution to a simple set of differential equations, thus providing probabilistic underpinnings to the works of Miller (J Math Biol 62(3):349–358, 2011) and Volz (J Math Biol 56(3):293–310, 2008). The present paper provides an additional probabilistic interpretation of the limiting deterministic functions in Janson, Luczak and Windridge (Random Struct Alg 45(4):724–761, 2014), thus clarifying further the connection between their results and the results of Miller and Volz. PubDate: 2024-03-24

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Abstract: Abstract In this paper, we consider (upward skip-free) discrete-time and discrete-space Markov additive chains (MACs) and develop the theory for the so-called \(\widetilde{{\textbf {W}}}\) and \(\widetilde{{\textbf {Z}}}\) scale matrices, which are shown to play a vital role in the determination of a number of exit problems and related fluctuation identities. The theory developed in this fully discrete set-up follows similar lines of reasoning as the analogous theory for Markov additive processes in continuous time and is exploited to obtain the probabilistic construction of the scale matrices, identify the form of the generating function and produce a simple recursion relation for \(\widetilde{{\textbf {W}}}\) , as well as its connection with the so-called occupation mass formula. In addition to the standard one- and two-sided exit problems (upwards and downwards), we also derive distributional characteristics for a number of quantities related to the one- and two-sided ‘reflected’ processes. PubDate: 2024-03-17

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Abstract: Abstract Let \(X=\{ X(t), t\in \mathbb {R}^{N}\} \) be a centered space-time anisotropic Gaussian random field in \(\mathbb {R}^d\) with stationary increments, where the components \(X_{i}(i=1,\ldots ,d)\) are independent but distributed differently. Under certain conditions, we not only give the Hausdorff dimension of the graph sets of X in the asymmetric metric in the recurrent case, but also determine the exact Hausdorff measure functions of the graph sets of X in the transient and recurrent cases, respectively. Moreover, we establish a uniform Hausdorff dimension result for the image sets of X. Our results extend the corresponding results on fractional Brownian motion and space or time anisotropic Gaussian random fields. PubDate: 2024-03-15 DOI: 10.1007/s10959-024-01323-7

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Abstract: Abstract We introduce the voter model on the infinite integer lattice with a slow membrane and investigate its hydrodynamic behavior and nonequilibrium fluctuations. The voter model is one of the classical interacting particle systems with state space \(\{0,1\}^{\mathbb Z^d}\) . In our model, a voter adopts one of its neighbors’ opinion at rate one except for neighbors crossing the hyperplane \(\{x:x_1 = 1/2\}\) , where the rate is \(\alpha N^{-\beta }\) and thus is called a slow membrane. Above, \(\alpha >0 \ \textrm{and} \ \beta \ge 0\) are given parameters and the positive integer N is a scaling parameter. We consider the limit \(N \rightarrow \infty \) and prove that the hydrodynamic limits are given by the heat equation without or with Robin/Neumann conditions depending on the values of \(\beta \) . We also consider the nonequilibrium fluctuations, where the limit is described by generalized Ornstein–Uhlenbeck processes with certain boundary conditions corresponding to the hydrodynamic equation. PubDate: 2024-03-13 DOI: 10.1007/s10959-024-01321-9

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Abstract: Abstract We consider rough differential equations whose coefficients contain path-dependent bounded variation terms and prove the existence and a priori estimate of solutions. These equations include classical path-dependent stochastic differential equations containing running maximum processes and normal reflection terms. We apply these results to determine the topological support of the solution processes. PubDate: 2024-03-08 DOI: 10.1007/s10959-024-01319-3

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Abstract: Abstract We study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift. PubDate: 2024-03-07 DOI: 10.1007/s10959-024-01317-5

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Abstract: Abstract Let \(\eta _1\) , \(\eta _2,\ldots \) be independent copies of a random variable \(\eta \) with zero mean and finite variance which is bounded from the right, that is, \(\eta \le b\) almost surely for some \(b>0\) . Considering different types of the asymptotic behaviour of the probability \(\mathbb {P}\{\eta \in [b-x,b]\}\) as \(x\rightarrow 0+\) , we derive precise tail asymptotics of the random Dirichlet series \(\sum _{k\ge 1}k^{-\alpha }\eta _k\) for \(\alpha \in (1/2, 1]\) . PubDate: 2024-03-05 DOI: 10.1007/s10959-024-01318-4