arXiv subjects

Mikhail V. Menshikov

Publications and source records attributed to Mikhail V. Menshikov.

At least 19 recordsLinked to original sources

arXiv2023

Strong transience for one-dimensional Markov chains with asymptotically zero drifts

For near-critical, transient Markov chains on the non-negative integers in the Lamperti regime, where the mean drift at $x$ decays as $1/x$ as $x \to \infty$, we quantify degree of transience via existence of moments for conditional return times and for last exit times, assuming increments are uniformly bounded. Our proof uses a Doob $h$-transform, for the transient process conditioned to return, and we show that the conditioned process is also of Lamperti type with appropriately transformed parameters. To do so, we obtain an asymptotic expansion for the ratio of two return probabilities, evaluated at two nearby starting points; a consequence of this is that the return probability for the transient Lamperti process is a regularly-varying function of the starting point.

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arXiv2023

Stochastic billiards with Markovian reflections in generalized parabolic domains

We study recurrence and transience for a particle that moves at constant velocity in the interior of an unbounded planar domain, with random reflections at the boundary governed by a Markov kernel producing outgoing angles from incoming angles. Our domains have a single unbounded direction and sublinear growth. We characterize recurrence in terms of the reflection kernel and growth rate of the domain. The results are obtained by transforming the stochastic billiards model to a Markov chain on a half-strip $\mathbb{R}_+ \!\times S$ where $S$ is a compact set. We develop the recurrence classification for such processes in the near-critical regime in which drifts of the $\mathbb{R}_+$ component are of generalized Lamperti type, and the $S$ component is asymptotically Markov; this extends earlier work that dealt with finite $S$.

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arXiv2022

Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity

For a multidimensional driftless diffusion in an unbounded, smooth, sub-linear generalized parabolic domain, with oblique reflection from the boundary, we give natural conditions under which either explosion occurs, if the domain narrows sufficiently fast at infinity, or else there is superdiffusive transience, which we quantify with a strong law of large numbers. For example, in the case of a planar domain, explosion occurs if and only if the area of the domain is finite. We develop and apply novel semimartingale criteria for studying explosions and establishing strong laws, which are of independent interest.

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arXiv2021

Cutpoints of non-homogeneous random walks

We give conditions under which near-critical stochastic processes on the half-line have infinitely many or finitely many cutpoints, generalizing existing results on nearest-neighbour random walks to adapted processes with bounded increments satisfying appropriate conditional increment moments conditions. We apply one of these results to deduce that a class of transient zero-drift Markov chains in $\mathbb{R}^d$, $d \geq 2$, possess infinitely many separating annuli, generalizing previous results on spatially homogeneous random walks.

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arXiv2020

Reflecting random walks in curvilinear wedges

We study a random walk (Markov chain) in an unbounded planar domain whose boundary is described by two curves of the form $x_2 = a^+ x_1^{β^+}$ and $x_2 = -a^- x_1^{β^-}$, with $x_1 \geq 0$. In the interior of the domain, the random walk has zero drift and a given increment covariance matrix. From the vicinity of the upper and lower sections of the boundary, the walk drifts back into the interior at a given angle $α^+$ or $α^-$ to the relevant inwards-pointing normal vector. Here we focus on the case where $α^+$ and $α^-$ are equal but opposite, which includes the case of normal reflection. For $0 \leq β^+, β^- < 1$, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times.

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arXiv2019

Random walks avoiding their convex hull with a finite memory

Fix integers $d \geq 2$ and $k\geq d-1$. Consider a random walk $X_0, X_1, \ldots$ in $\mathbb{R}^d$ in which, given $X_0, X_1, \ldots, X_n$ ($n \geq k$), the next step $X_{n+1}$ is uniformly distributed on the unit ball centred at $X_n$, but conditioned that the line segment from $X_n$ to $X_{n+1}$ intersects the convex hull of $\{0, X_{n-k}, \ldots, X_n\}$ only at $X_n$. For $k = \infty$ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic, i.e., to have a limiting speed and a limiting direction. We establish ballisticity for the finite-$k$ model, and comment on some open problems. In the case where $d=2$ and $k=1$, we obtain the limiting speed explicitly: it is $8/(9π^2)$.

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arXiv2018

Markov chains with heavy-tailed increments and asymptotically zero drift

We 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 $c y^{-α}$ for an exponent $α\in (1,2)$ and if the drift at $x$ is about $b x^{-γ}$, then the critical regime has $γ= α-1$ and recurrence/transience is determined by the sign of $b + cπ\textrm{cosec} (πα)$. 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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arXiv2016

Heavy-tailed random walks on complexes of half-lines

We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed by an irreducible Markov transition matrix, with associated stationary distribution $μ_k$. If $χ_k$ is $1$ for one-sided half-lines $k$ and $1/2$ for two-sided half-lines, and $α_k$ is the tail exponent of the jumps on half-line $k$, we show that the recurrence classification for the case where all $α_k χ_k \in (0,1)$ is determined by the sign of $\sum_k μ_k \cot ( χ_k πα_k )$. In the case of two half-lines, the model fits naturally on $\mathbb{R}$ and is a version of the oscillating random walk of Kemperman. In that case, the cotangent criterion for recurrence becomes linear in $α_1$ and $α_2$; our general setting exhibits the essential non-linearity in the cotangent criterion. For the general model, we also show existence and non-existence of polynomial moments of return times. Our moments results are sharp (and new) for several cases of the oscillating random walk; they are apparently even new for the case of a homogeneous random walk on $\mathbb{R}$ with symmetric increments of tail exponent $α\in (1,2)$.

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arXiv2015

Anomalous recurrence properties of many-dimensional zero-drift random walks

Famously, 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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arXiv2012

Random walk in mixed random environment without uniform ellipticity

We study a random walk in random environment on the non-negative integers. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i) points endowed with probabilities drawn from a symmetric distribution with heavy tails at 0 and 1, and (ii) `fast points' with a fixed systematic drift. Without these fast points, the model is related to the diffusion in heavy-tailed (`stable') random potential studied by Schumacher and Singh; the fast points perturb that model. The two components compete to determine the behaviour of the random walk; we identify phase transitions in terms of the model parameters. We give conditions for recurrence and transience, and prove almost-sure bounds for the trajectories of the walk.

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arXiv2012

Excursions and path functionals for stochastic processes with asymptotically zero drifts

We study discrete-time stochastic processes $(X_t)$ on $[0,\infty)$ with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at $x$ is about $c/x$. Our focus is the recurrent case (when $c$ is not too large). We give sharp asymptotics for various functionals associated with the process and its excursions, including results on maxima and return times. These results include improvements on existing results in the literature in several respects, and also include new results on excursion sums and additive functionals of the form $\sum_{s \leq t} X_s^α$, $α>0$. We make minimal moments assumptions on the increments of the process. Recently there has been renewed interest in Lamperti-type process in the context of random polymers and interfaces, particularly nearest-neighbour random walks on the integers; some of our results are new even in that setting. We give applications of our results to processes on the whole of $\R$ and to a class of multidimensional `centrally biased' random walks on $\R^d$; we also apply our results to the simple harmonic urn, allowing us to sharpen existing results and to verify a conjecture of Crane et al.

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arXiv2012

Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non-existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are more general than existing results in the literature, which consider only the case of sums of independent (typically, identically distributed) random variables. We do not assume the Markov property. Existing results that we generalize include a circle of ideas related to the Marcinkiewicz-Zygmund strong law of large numbers, as well as more recent work of Kesten and Maller. Our proofs are robust and use martingale methods. We demonstrate the benefit of the generality of our results by applications to some non-classical models, including random walks with heavy-tailed increments on two-dimensional strips, which include, for instance, certain generalized risk processes.

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arXiv2010

Random walk with barycentric self-interaction

We study the asymptotic behaviour of a $d$-dimensional self-interacting random walk $X_n$ ($n = 1,2,...$) which is repelled or attracted by the centre of mass $G_n = n^{-1} \sum_{i=1}^n X_i$ of its previous trajectory. The walk's trajectory $(X_1,...,X_n)$ models a random polymer chain in either poor or good solvent. In addition to some natural regularity conditions, we assume that the walk has one-step mean drift directed either towards or away from its current centre of mass $G_n$ and of magnitude $\| X_n - G_n \|^{-β}$ for $β\geq 0$. When $β<1$ and the radial drift is outwards, we show that $X_n$ is transient with a limiting (random) direction and satisfies a super-diffusive law of large numbers: $n^{-1/(1+β)} X_n$ converges almost surely to some random vector. When $β\in (0,1)$ there is sub-ballistic rate of escape. For $β\geq 0$ we give almost-sure bounds on the norms $\|X_n\|$, which in the context of the polymer model reveal extended and collapsed phases. Analysis of the random walk, and in particular of $X_n - G_n$, leads to the study of real-valued time-inhomogeneous non-Markov processes $Z_n$ on $[0,\infty)$ with mean drifts at $x$ given approximately by $ρx^{-β} - (x/n)$, where $β\geq 0$ and $ρ\in \R$. The study of such processes is a time-dependent variation on a classical problem of Lamperti; moreover, they arise naturally in the context of the distance of simple random walk on $\Z^d$ from its centre of mass, for which we also give an apparently new result. We give a recurrence classification and asymptotic theory for processes $Z_n$ just described, which enables us to deduce the complete recurrence classification (for any $β\geq 0$) of $X_n - G_n$ for our self-interacting walk.

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arXiv2010

Passage-time moments and hybrid zones for the exclusion-voter model

We study the non-equilibrium dynamics of a one-dimensional interacting particle system that is a mixture of the voter model and the exclusion process. With the process started from a finite perturbation of the ground state Heaviside configuration consisting of 1's to the left of the origin and 0's elsewhere, we study the relaxation time $τ$, that is, the first hitting time of the ground state configuration (up to translation). We give conditions for $τ$ to be finite and for certain moments of $τ$ to be finite or infinite, and prove a result that approaches a conjecture of Belitsky et al. (Bernoulli 7 (2001) 119--144). Ours are the first non-existence-of-moments results for $τ$ for the mixture model. Moreover, we give almost sure asymptotics for the evolution of the size of the hybrid (disordered) region. Most of our results pertain to the discrete-time setting, but several transfer to continuous-time. As well as the mixture process, some of our results also cover pure exclusion. We state several significant open problems.

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arXiv2010

Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts

We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time $τ_α$ from a wedge with apex at the origin and interior half-angle $α$ by a non-homogeneous random walk on the square lattice with mean drift at $x$ of magnitude $O(1/|x|)$ as $|x| \to \infty$. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors (see arXiv:0910.1772) stated that $τ_α< \infty$ a.s. for any $α$ (while for a stronger drift field $τ_α$ is infinite with positive probability). Here we study the more difficult problem of the existence and non-existence of moments $E[τ_α^s]$, $s>0$. Assuming (in common with much of the literature) a uniform bound on the walk's increments, we show that for $α< π/2$ there exists $s_0 \in (0,\infty)$ such that $E[τ_α^s]$ is finite for $s < s_0$ but infinite for $s > s_0$; under specific assumptions on the drift field we show that we can attain $E[τ_α^s] = \infty$ for any $s > 1/2$. We show that for $α\leq π$ there is a phase transition between drifts of magnitude $O(1/|x|)$ (the critical regime) and $o(1/|x|)$ (the subcritical regime). In the subcritical regime we obtain a non-homogeneous random walk analogue of a theorem for Brownian motion due to Spitzer, under considerably weaker conditions than those previously given (including work by Varopoulos) that assumed zero drift.

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arXiv2010

Rate of escape and central limit theorem for the supercritical Lamperti problem

The study of discrete-time stochastic processes on the half-line with mean drift at $x$ given by $μ_1 (x) \to 0$ as $x \to \infty$ is known as Lamperti's problem. We give sharp almost-sure bounds for processes of this type in the case where $μ_1 (x)$ is of order $x^{-β}$ for some $β\in (0,1)$. The bounds are of order $t^{1/(1+β)}$, so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of $(2+2β+\varepsilon)$-moments for our main results, so 4th moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where $x^βμ_1 (x)$ has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where $β=0$. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.

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arXiv2009

Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift

We study the first exit time $τ$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on $\Z^d$ ($d \geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at $\bx \in \Z^d$ is of magnitude $O(\| \bx\|^{-1})$, we show that $τ<\infty$ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude $\| \bx\|^{-β}$, $β\in (0,1)$, we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on 2nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model.

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arXiv2008

Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains

We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary. Physical motivation for the process originates with ideal gas models in the Knudsen regime, with particles reflecting off microscopically rough surfaces. We classify the process into recurrent and transient cases. We also give almost-sure results on the long-term behaviour of the location of the particle, including a super-diffusive rate of escape in the transient case. A key step in obtaining our results is to relate our process to an instance of a one-dimensional stochastic process with asymptotically zero drift, for which we prove some new almost-sure bounds of independent interest. We obtain some of these bounds via an application of general semimartingale criteria, also of some independent interest.

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