Search ResultsShowing 1-7 of 7
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arXiv:2010.00671 (Published 2020-10-01)
Deposition, diffusion, and nucleation on an interval
Comments: 47 pages, 3 figuresCategories: math.PRMotivated by nanoscale growth of ultra-thin films, we study a model of deposition, on an interval substrate, of particles that perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model popular in the applied literature. We show that the associated interval-splitting process converges in the sparse deposition limit to a Markovian process (in the vein of Brennan and Durrett) governed by a splitting density with a compact Fourier series expansion but, apparently, no simple closed form. We show that the same splitting density governs the fixed deposition rate, large time asymptotics of the normalized gap distribution, so these asymptotics are independent of deposition rate. The splitting density is derived by solving an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson.
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arXiv:1806.07166 (Published 2018-06-19)
Markov chains with heavy-tailed increments and asymptotically zero drift
Categories: math.PRWe study the recurrence/transience phase transition for Markov chains on $\mathbb{R}_+$, $\mathbb{R}$, and $\mathbb{R}^2$ whose increments have heavy tails with exponent in $(1,2)$ and asymptotically zero mean. This is the infinite-variance analogue of the classical Lamperti problem. On $\mathbb{R}_+$, for example, we show that if the tail of the positive increments is about $c y^{-\alpha}$ for an exponent $\alpha \in (1,2)$ and if the drift at $x$ is about $b x^{-\gamma}$, then the critical regime has $\gamma = \alpha -1$ and recurrence/transience is determined by the sign of $b + c\pi \textrm{cosec} (\pi \alpha)$. On $\mathbb{R}$ we classify whether transience is directional or oscillatory, and extend an example of Rogozin \& Foss to a class of transient martingales which oscillate between $\pm \infty$. In addition to our recurrence/transience results, we also give sharp results on the existence/non-existence of moments of passage times.
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arXiv:1801.07882 (Published 2018-01-24)
Invariance principle for non-homogeneous random walks
Comments: 36 pagesCategories: math.PRWe prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in $\mathbb{R}^d$, which may be recurrent in any dimension. The limit $\mathcal{X}$ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\geq2$. To characterise $\mathcal{X}$, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in $\mathbb{R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of $\mathcal{X}$ and thus develop the excursion theory of $\mathcal{X}$ without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for $\mathcal{X}$ in $\mathbb{R}^d$, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of $\mathcal{X}$ is time-reversible. If so, the excursions of $\mathcal{X}$ in $\mathbb{R}^d$ generalise the classical Pitman-Yor splitting-at-the-maximum property of Bessel excursions.
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arXiv:1708.07683 (Published 2017-08-25)
A radial invariance principle for non-homogeneous random walks
Comments: 10 pagesCategories: math.PRConsider non-homogeneous zero-drift random walks in $\mathbb{R}^d$, $d \geq 2$, with the asymptotic increment covariance matrix $\sigma^2 (\mathbf{u})$ satisfying $\mathbf{u}^\top \sigma^2 (\mathbf{u}) \mathbf{u} = U$ and $\mathrm{tr}\ \sigma^2 (\mathbf{u}) = V$ in all in directions $\mathbf{u}\in\mathbb{S}^{d-1}$ for some positive constants $U<V$. In this paper we establish weak convergence of the radial component of the walk to a Bessel process with dimension $V/U$. This can be viewed as an extension of an invariance principle of Lamperti.
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arXiv:1506.08541 (Published 2015-06-29)
Anomalous recurrence properties of many-dimensional zero-drift random walks
Comments: 22 pages, 4 figuresCategories: math.PRFamously, a $d$-dimensional, spatially homogeneous random walk whose increments are non-degenerate, have finite second moments, and have zero mean is recurrent if $d \in \{1,2\}$ but transient if $d \geq 3$. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially non-homogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension $d \geq 2$, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk's current position; these \emph{elliptic random walks} generalize the classical homogeneous Pearson--Rayleigh walk (the spherical case). Our proof of the recurrence classification is based on fundamental work of Lamperti.
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arXiv:1402.2558 (Published 2014-02-11)
Non-homogeneous random walks on a semi-infinite strip
Comments: 27 pagesJournal: Stochastic Processes and their Applications, Vol. 124 (2014), no. 10, p. 3179-3205Categories: math.PRKeywords: non-homogeneous random walks, semi-infinite strip, markov chain, recurrence phase transitions, increment moment parametersTags: journal articleWe study the asymptotic behaviour of Markov chains $(X_n,\eta_n)$ on $\mathbb{Z}_+ \times S$, where $\mathbb{Z}_+$ is the non-negative integers and $S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of $X_n$, and that, roughly speaking, $\eta_n$ is close to being Markov when $X_n$ is large. This departure from much of the literature, which assumes that $\eta_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for $X_n$ given $\eta_n$. We give a recurrence classification in terms of increment moment parameters for $X_n$ and the stationary distribution for the large-$X$ limit of $\eta_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between $X_n$ (rescaled) and $\eta_n$. Our results can be seen as generalizations of Lamperti's results for non-homogeneous random walks on $\mathbb{Z}_+$ (the case where $S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where $\eta_n$ tracks an internal state of the system.
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The simple harmonic urn
Comments: Published in at http://dx.doi.org/10.1214/10-AOP605 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)Journal: Annals of Probability 2011, Vol. 39, No. 6, 2119-2177DOI: 10.1214/10-AOP605Categories: math.PRKeywords: simple harmonic urn, uniform renewal process, urn process, independent interest, markov chainTags: journal articleWe study a generalized P\'{o}lya urn model with two types of ball. If the drawn ball is red, it is replaced together with a black ball, but if the drawn ball is black it is replaced and a red ball is thrown out of the urn. When only black balls remain, the roles of the colors are swapped and the process restarts. We prove that the resulting Markov chain is transient but that if we throw out a ball every time the colors swap, the process is recurrent. We show that the embedded process obtained by observing the number of balls in the urn at the swapping times has a scaling limit that is essentially the square of a Bessel diffusion. We consider an oriented percolation model naturally associated with the urn process, and obtain detailed information about its structure, showing that the open subgraph is an infinite tree with a single end. We also study a natural continuous-time embedding of the urn process that demonstrates the relation to the simple harmonic oscillator; in this setting, our transience result addresses an open problem in the recurrence theory of two-dimensional linear birth and death processes due to Kesten and Hutton. We obtain results on the area swept out by the process. We make use of connections between the urn process and birth--death processes, a uniform renewal process, the Eulerian numbers, and Lamperti's problem on processes with asymptotically small drifts; we prove some new results on some of these classical objects that may be of independent interest. For instance, we give sharp new asymptotics for the first two moments of the counting function of the uniform renewal process. Finally, we discuss some related models of independent interest, including a "Poisson earthquakes" Markov chain on the homeomorphisms of the plane.