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Iterated-logarithm laws for convex hulls of random walks with drift (2024)
Journal Article
Cygan, W., Sandrić, N., Šebek, S., & Wade, A. R. (2024). Iterated-logarithm laws for convex hulls of random walks with drift. Transactions of the American Mathematical Society, 377(9), 6695-6724

We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is... Read More about Iterated-logarithm laws for convex hulls of random walks with drift.

Superdiffusive planar random walks with polynomial space–time drifts (2024)
Journal Article
da Costa, C., Menshikov, M., Shcherbakov, V., & Wade, A. (2024). Superdiffusive planar random walks with polynomial space–time drifts. Stochastic Processes and their Applications, 176, Article 104420. https://doi.org/10.1016/j.spa.2024.104420

We quantify superdiffusive transience for a two-dimensional random walk in which the vertical coordinate is a martingale and the horizontal coordinate has a positive drift that is a polynomial function of the individual coordinates... Read More about Superdiffusive planar random walks with polynomial space–time drifts.

Stochastic billiards with Markovian reflections in generalized parabolic domains (2023)
Journal Article
da Costa, C., Menshikov, M. V., & Wade, A. R. (2023). Stochastic billiards with Markovian reflections in generalized parabolic domains. Annals of Applied Probability, 33(6B), 5459-5496. https://doi.org/10.1214/23-AAP1952

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 d... Read More about Stochastic billiards with Markovian reflections in generalized parabolic domains.

Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity (2023)
Journal Article
Menshikov, M. V., Mijatović, A., & Wade, A. R. (2023). Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 59(4), 1813-1843. https://doi.org/10.1214/22-AIHP1309

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... Read More about Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity.

Dynamics of Finite Inhomogeneous Particle Systems with Exclusion Interaction (2023)
Journal Article
Malyshev, V., Menshikov, M. V., Popov, S., & Wade, A. (in press). Dynamics of Finite Inhomogeneous Particle Systems with Exclusion Interaction. Journal of Statistical Physics, 190(11), Article 184. https://doi.org/10.1007/s10955-023-03190-8

We study finite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses ju... Read More about Dynamics of Finite Inhomogeneous Particle Systems with Exclusion Interaction.

Strong transience for one-dimensional Markov chains with asymptotically zero drifts (2023)
Journal Article
Lo, C. H., Menshikov, M. V., & Wade, A. R. (2024). Strong transience for one-dimensional Markov chains with asymptotically zero drifts. Stochastic Processes and their Applications, 170, Article 104260. https://doi.org/10.1016/j.spa.2023.104260

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 → ∞ , we quantify degree of transience via existence of moments for conditional retu... Read More about Strong transience for one-dimensional Markov chains with asymptotically zero drifts.

Energy-Constrained Random Walk with Boundary Replenishment (2023)
Journal Article
Wade, A. R., & Grinfeld, M. (2023). Energy-Constrained Random Walk with Boundary Replenishment. Journal of Statistical Physics, 190(10), Article 155. https://doi.org/10.1007/s10955-023-03165-9

We study an energy-constrained random walker on a length-N interval of the one-dimensional integer lattice, with boundary reflection. The walker consumes one unit of energy for every step taken in the interior, and energy is replenished up to a capac... Read More about Energy-Constrained Random Walk with Boundary Replenishment.

Deposition, diffusion, and nucleation on an interval (2022)
Journal Article
Georgiou, N., & Wade, A. R. (2022). Deposition, diffusion, and nucleation on an interval. Annals of Applied Probability, 32(6), 4849-4892. https://doi.org/10.1214/22-aap1804

Motivated 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... Read More about Deposition, diffusion, and nucleation on an interval.

Cutpoints of non-homogeneous random walks (2022)
Journal Article
Lo, C. H., Menshikov, M. V., & Wade, A. R. (2022). Cutpoints of non-homogeneous random walks. Alea (2006. Online), 19, 493-510. https://doi.org/10.30757/alea.v19-19

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 satisfyin... Read More about Cutpoints of non-homogeneous random walks.

Reflecting random walks in curvilinear wedges (2021)
Book Chapter
Menshikov, M. V., Mijatović, A., & Wade, A. R. (2021). Reflecting random walks in curvilinear wedges. In M. Vares, R. Fernández, L. Fontes, & C. Newman (Eds.), In and out of equilibrium 3: celebrating Vladas Sidoarvicius (637-675). Springer Verlag. https://doi.org/10.1007/978-3-030-60754-8_26

We study a random walk (Markov chain) in an unbounded planar domain bounded by two curves of the form x2=a+xβ+1 and x2=−a−xβ−1 , with x1 ≥ 0. In the interior of the domain, the random walk has zero drift and a given increment covariance matrix. From... Read More about Reflecting random walks in curvilinear wedges.

Angular asymptotics for random walks (2021)
Book Chapter
López Hernández, A., & Wade, A. R. (2021). Angular asymptotics for random walks. In L. Chaumont, & A. E. Kyprianou (Eds.), A Lifetime of Excursions Through Random Walks and Lévy Processes (315-342). Springer Verlag. https://doi.org/10.1007/978-3-030-83309-1_17

We study the set of directions asymptotically explored by a spatially homogeneous random walk in d-dimensional Euclidean space. We survey some pertinent results of Kesten and Erickson, make some further observations, and present some examples. We als... Read More about Angular asymptotics for random walks.

Markov Chains (2020)
Journal Article
Wade, A. R. (online). Markov Chains

Invariance principle for non-homogeneous random walks (2019)
Journal Article
Georgiou, N., Mijatović, A., & Wade, A. R. (2019). Invariance principle for non-homogeneous random walks. Electronic Journal of Probability, 24, https://doi.org/10.1214/19-ejp302

We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in Rd, which may be recurrent in any dimension. The limit X is an elliptic martingale diffusion, which may be point-recurrent at the origin for any d 2.... Read More about Invariance principle for non-homogeneous random walks.

The critical greedy server on the integers is recurrent (2019)
Journal Article
Cruise, J. R., & Wade, A. R. (2019). The critical greedy server on the integers is recurrent. Annals of Applied Probability, 29(2), 1233-1261. https://doi.org/10.1214/18-aap1434

Each site of Z hosts a queue with arrival rate λ. A single server, starting at the origin, serves its current queue at rate μ until that queue is empty, and then moves to the longest neighbouring queue. In the critical case λ=μ, we show that the serv... Read More about The critical greedy server on the integers is recurrent.

The convex hull of a planar random walk: perimeter, diameter, and shape (2018)
Journal Article
McRedmond, J., & Wade, A. R. (2018). The convex hull of a planar random walk: perimeter, diameter, and shape. Electronic Journal of Probability, 23, Article 131. https://doi.org/10.1214/18-ejp257

We study the convex hull of the first n steps of a planar random walk, and present large-n asymptotic results on its perimeter length Ln, diameter Dn, and shape. In the case where the walk has a non-zero mean drift, we show that Ln=Dn ! 2 a.s., and g... Read More about The convex hull of a planar random walk: perimeter, diameter, and shape.

On the centre of mass of a random walk (2018)
Journal Article
Lo, C. H., & Wade, A. R. (2019). On the centre of mass of a random walk. Stochastic Processes and their Applications, 129(11), 4663-4686. https://doi.org/10.1016/j.spa.2018.12.007

For a random walk Sn on Rd we study the asymptotic behaviour of the associated centre of mass process Gn=n−1∑ni=1Si. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero... Read More about On the centre of mass of a random walk.