Search ResultsShowing 1-11 of 11
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arXiv:2406.12562 (Published 2024-06-18)
Censored fractional Bernstein derivatives and stochastic processes
Categories: math.PRIn this paper, we define the censored fractional Bernstein derivative on the positive half line $(0, \infty)$ based on the Bernstein Riemann--Liouville fractional derivative. This derivative can be shown to be the generator of the censored subordinator by solving a resolvent equation. We also show that the censored subordinator hits the boundary in finite time under certain conditions.
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arXiv:2306.14376 (Published 2023-06-26)
Local time of transient random walk on the lattice of positive half line
Comments: 25 pagesCategories: math.PRIn this paper, we study spatially inhomogeneous random walks on the lattice of positive half line. Fix a positive integer $a.$ Let $C$ be the collection of points at which the local times of the random walk are exactly $a.$ We give a criterion to tell whether $C$ is a finite set or not. Our result answer an open problem proposed by E. Cs\'aki, A. F\"oldes, P. R\'ev\'esz [J. Theoret. Probab. 23 (2) (2010) 624-638.]. When $a=1,$ $C$ is just the set of strong cutpoints studied in the above mentioned paper. By a moment method, when the local drift of the walk at $i\ge 1$ is $1/i,$ we show that $2|C\cap [0,n]|/\log n\overset{\mathcal D}{\rightarrow} S$ as $n\rightarrow\infty,$ where $S$ is an exponentially distributed random variable with $P(S>t)=e^{-t},\ t>0.$
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arXiv:2303.08961 (Published 2023-03-15)
Effusion of stochastic processes on a line
Comments: 19 pages, 2 figuesCategories: cond-mat.stat-mech, math.PRWe consider the problem of leakage or effusion of an ensemble of independent stochastic processes from a region where they are initially randomly distributed. The case of Brownian motion, initially confined to the left half line with uniform density and leaking into the positive half line is an example which has been extensively studied in the literature. Here we derive new results for the average number and variance of the number of leaked particles for arbitrary Gaussian processes initially confined to the negative half line and also derive its joint two-time probability distribution, both for the annealed and the quenched initial conditions. For the annealed case, we show that the two-time joint distribution is a bivariate Poisson distribution. We also discuss the role of correlations in the initial particle positions on the statistics of the number of particles on the positive half line. We show that the strong memory effects in the variance of the particle number on the positive real axis for Brownian particles, seen in recent studies, persist for arbitrary Gaussian processes and also at the level of two-time correlation functions.
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arXiv:2206.09402 (Published 2022-06-19)
Cutpoints of (1,2) and (2,1) random walks on the lattice of positive half line
Comments: 31 pagesIn this paper, we study (1,2) and (2,1) random walks in varying environments on the lattice of positive half line. We assume that the transition probabilities at site $n$ are asymptotically constants as $n\rightarrow\infty.$ For (1,2) random walk, we get some elaborate asymptotic behaviours of various escape probabilities and hitting probabilities of the walk. Such observations and some delicate analysis of continued fractions and the product of nonnegative matrices enable us to give criteria for finiteness of the number of cutpoints of both (1,2) and (2,1) random walks, which generalize E. Cs\'aki, A. F\"oldes and P. R\'ev\'esz [J. Theor. Probab. 23: 624-638 (2010)] and H.-M. Wang [Markov Processes Relat. Fields 25: 125-148 (2019)]. For near-recurrent random walks, whenever there are infinitely many cutpoints, we also study the asymptotics of the number of cutpoints in $[0,n].$
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arXiv:2004.12422 (Published 2020-04-26)
On a maximum of nearest-neighbor random walk with asymptotically zero drift on lattice of positive half line
Comments: 10 pagesCategories: math.PRConsider a nearest-neighbor random walk with certain asymptotically zero drift on the positive half line. Let $M$ be the maximum of an excursion starting from $1$ and ending at $0.$ We study the distribution of $M$ and characterize its asymptotics, which is quite different from those of simple random walks.
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arXiv:1707.06423 (Published 2017-07-20)
On the number of points skipped by a transient (1,2) random walk on the line
Categories: math.PRConsider a transient near-critical (1,2) random walk on the positive half line. We give a criteria for the finiteness of the number of the skipped points (the points never visited) by the random walk. This result generalizes (partially) the criteria for the finiteness of the number of cutpoingts of the nearest neighbor random walk on the line by Cs\'aki, F\"olders, R\'ev\'esz [J Theor Probab (2010) 23: 624-638].
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arXiv:1604.06839 (Published 2016-04-23)
Stability problems for Cantor stochastic differential equations
We consider driftless stochastic differential equations and the diffusions starting from the positive half line. It is shown that the Feller test for explosions gives a necessary and sufficient condition to hold pathwise uniqueness for diffusion coefficients that are positive and monotonically increasing or decreasing on the positive half line and the value at the origin is zero. Then, stability problems are studied from the aspect of H\"older-continuity and a generalized Nakao-Le Gall condition. Comparing the convergence rate of H\"older-continuous case, the sharpness and stability of the Nakao-Le Gall condition on Cantor stochastic differential equations is confirmed.
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arXiv:1602.03107 (Published 2016-02-09)
Range of (1,2) random walk in random environment
Comments: 14 pagesCategories: math.PRConsider $(1,2)$ random walk in random environment $\{X_n\}_{n\ge0}.$ In each step, the walk jumps at most a distance $2$ to the right or a distance $1$ to the left. For the walk transient to the right, it is proved that almost surely $\lim_{x\rightarrow\infty}\frac{\#\{X_n:\ 0\le X_n\le x,\ n\ge0\}}{x}=\theta$ for some $0<\theta<1.$ The result shows that the range of the walk covers only a linear proportion of the lattice of the positive half line. For the nearest neighbor random walk in random or non-random environment, this phenomenon could not appear in any circumstance.
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arXiv:1204.2755 (Published 2012-04-12)
Limit theorems for continuous time branching flows
Comments: 18 pagesCategories: math.PRWe construct a flow of continuous time and discrete state branching processes. Some scaling limit theorems for the flow are proved, which lead to the path-valued branching processes and nonlocal branching superprocesses over the positive half line studied in Li (2012).
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arXiv:1204.1248 (Published 2012-04-05)
Some limit theorems for flows of branching processes
Comments: 14 pagesCategories: math.PRWe construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.
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Asymptotic eigenvalue distributions of block-transposed Wishart matrices
Journal: J. Theoret. Probab. 26 (2013), 855-869Keywords: asymptotic eigenvalue distributions, block-transposed wishart matrices, quantum information theory, free poisson laws, positive half lineTags: journal articleWe study the partial transposition ${W}^\Gamma=(\mathrm{id}\otimes \mathrm{t})W\in M_{dn}(\mathbb C)$ of a Wishart matrix $W\in M_{dn}(\mathbb C)$ of parameters $(dn,dm)$. Our main result is that, with $d\to\infty$, the law of $m{W}^\Gamma$ is a free difference of free Poisson laws of parameters $m(n\pm 1)/2$. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half line.