Staff profile

Current funded research programme

Anomalous diffusion via self-interaction and reflection, funded by the EPSRC, runs 2022-2026. This is a joint project between Durham (EP/W00657X/1, £448,461) and Warwick (EP/W006227/1, £457,337).

Research Summary

My research is in probability theory and its applications. My interests include random spatial models and stochastic processes inspired by physics or other applications. Some of my recent work has been on random walks, random spatial graphs, processes in random media, and interacting particle systems. For some more details about some of my research and publications, see here.

• probability theory
• random structures
• stochastic processes

Authored book

• Non-Homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems
  
  Menshikov, M., Popov, S., & Wade, A. (2016). Non-Homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems. Cambridge University Press. https://doi.org/10.1017/9781139208468

Chapter in book

• Reflecting random walks in curvilinear wedges
  
  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. (pp. 637-675). Springer Verlag. https://doi.org/10.1007/978-3-030-60754-8_26
• Angular asymptotics for random walks
  
  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 (pp. 315-342). Springer Verlag. https://doi.org/10.1007/978-3-030-83309-1_17
• Random processes
  
  Cruise, R. J., Hryniv, O., & Wade, A. R. (2015). Random processes. In M. Grinfeld (Ed.), Mathematical Tools for Physicists (pp. 3-38). Wiley. https://doi.org/10.1002/3527600434.eap382.pub2
• Random directed and on-line networks
  
  Penrose, M. D., & Wade, A. R. (2009). Random directed and on-line networks. In W. Kendall & I. Molchanov (Eds.), New Perspectives in Stochastic Geometry (pp. 248-274). Oxford University Press.

Journal Article

• Semi-infinite particle systems with exclusion interaction and heterogeneous jump rates
  
  Menshikov, M. V., Popov, S., & Wade, A. R. (2025). Semi-infinite particle systems with exclusion interaction and heterogeneous jump rates. Mathematical Sciences. Advance online publication. https://doi.org/10.1007/s00440-024-01357-2
• Passage-times for partially-homogeneous reflected random walks on the quadrant
  
  da Costa, C., Menshikov, M. V., & Wade, A. R. (2025). Passage-times for partially-homogeneous reflected random walks on the quadrant. Electronic Journal of Probability, 30, Article 55. https://doi.org/10.1214/25-EJP1313
• Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability
  
  Brešar, M., Mijatović, A., & Wade, A. (2025). Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability. The Annals of Probability, 53(1), 175-222. https://doi.org/10.1214/24-AOP1703
• Superdiffusive planar random walks with polynomial space–time drifts
  
  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
• Iterated-logarithm laws for convex hulls of random walks with drift
  
  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.
• Strong transience for one-dimensional Markov chains with asymptotically zero drifts
  
  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
• Stochastic billiards with Markovian reflections in generalized parabolic domains
  
  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
• Dynamics of Finite Inhomogeneous Particle Systems with Exclusion Interaction
  
  Malyshev, V., Menshikov, M. V., Popov, S., & Wade, A. (2023). 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
• Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity
  
  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, 59(4), 1813-1843. https://doi.org/10.1214/22-AIHP1309
• Energy-Constrained Random Walk with Boundary Replenishment
  
  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
• Deposition, diffusion, and nucleation on an interval
  
  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
• Cutpoints of non-homogeneous random walks
  
  Lo, C. H., Menshikov, M. V., & Wade, A. R. (2022). Cutpoints of non-homogeneous random walks. ALEA - Latin American Journal of Probability and Mathematical Statistics, 19, 493-510. https://doi.org/10.30757/alea.v19-19
• Random walks avoiding their convex hull with a finite memory
  
  Comets, F., Menshikov, M. V., & Wade, A. R. (2020). Random walks avoiding their convex hull with a finite memory. Indagationes Mathematicae, 31(1), 117-146. https://doi.org/10.1016/j.indag.2019.11.002
• On the centre of mass of a random walk
  
  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
• Markov chains with heavy-tailed increments and asymptotically zero drift
  
  Georgiou, N., Menshikov, M. V., Petritis, D., & Wade, A. R. (2019). Markov chains with heavy-tailed increments and asymptotically zero drift. Electronic Journal of Probability, 24, Article 62. https://doi.org/10.1214/19-ejp322
• Invariance principle for non-homogeneous random walks
  
  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
• The critical greedy server on the integers is recurrent
  
  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
• The convex hull of a planar random walk: perimeter, diameter, and shape
  
  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
• A radial invariance principle for non-homogeneous random walks
  
  Georgiou, N., Mijatović, A., & Wade, A. R. (2018). A radial invariance principle for non-homogeneous random walks. Electronic Communications in Probability, 23, Article 56. https://doi.org/10.1214/18-ecp159
• Heavy-tailed random walks on complexes of half-lines
  
  Menshikov, M. V., Petritis, D., & Wade, A. R. (2018). Heavy-tailed random walks on complexes of half-lines. Journal of Theoretical Probability, 31(3), 1819-1859. https://doi.org/10.1007/s10959-017-0753-5
• Non-homogeneous random walks on a half strip with generalized Lamperti drifts
  
  Lo, C. H., & Wade, A. R. (2017). Non-homogeneous random walks on a half strip with generalized Lamperti drifts. Markov Processes and Related Fields., 23(1), 125-146.
• Anomalous recurrence properties of many-dimensional zero-drift random walks
  
  Georgiou, N., Menshikov, M. V., Mijatovic, A., & Wade, A. R. (2016). Anomalous recurrence properties of many-dimensional zero-drift random walks. Advances in Applied Probability, 48(Issue A), 99-118. https://doi.org/10.1017/apr.2016.44
• Convex hulls of random walks and their scaling limits
  
  Wade, A. R., & Xu, C. (2015). Convex hulls of random walks and their scaling limits. Stochastic Processes and Their Applications, 125(11), 4300-4320. https://doi.org/10.1016/j.spa.2015.06.008
• Phase transitions for random geometric preferential attachment graphs
  
  Jordan, J., & Wade, A. R. (2015). Phase transitions for random geometric preferential attachment graphs. Advances in Applied Probability, 47(2), 565-588. https://doi.org/10.1239/aap/1435236988
• Convergence in a multidimensional randomized Keynesian beauty contest
  
  Grinfeld, M., Volkov, S., & Wade, A. R. (2015). Convergence in a multidimensional randomized Keynesian beauty contest. Advances in Applied Probability, 47(1), 57-82. https://doi.org/10.1239/aap/1427814581
• Convex hulls of planar random walks with drift
  
  Wade, A. R., & Xu, C. (2015). Convex hulls of planar random walks with drift. Proceedings of the American Mathematical Society, 143(1), 433-445. https://doi.org/10.1090/s0002-9939-2014-12239-8
• Non-homogeneous random walks on a semi-infinite strip
  
  Georgiou, N., & Wade, A. R. (2014). Non-homogeneous random walks on a semi-infinite strip. Stochastic Processes and Their Applications, 124(10), 3179-3205. https://doi.org/10.1016/j.spa.2014.05.005
• Rank deficiency in sparse random GF[2] matrices
  
  Darling, R. W., Penrose, M. D., Wade, A. R., & Zabell, S. L. (2014). Rank deficiency in sparse random GF[2] matrices. Electronic Journal of Probability, 19, Article 83. https://doi.org/10.1214/ejp.v19-2458
• Random walk in mixed random environment without uniform ellipticity
  
  Hryniv, O., Menshikov, M. V., & Wade, A. R. (2013). Random walk in mixed random environment without uniform ellipticity. Proceedings of the Steklov Institute of Mathematics, 282(1), 106-123. https://doi.org/10.1134/s0081543813060102
• Excursions and path functionals for stochastic processes with asymptotically zero drifts
  
  Hryniv, O., Menshikov, M. V., & Wade, A. R. (2013). Excursions and path functionals for stochastic processes with asymptotically zero drifts. Stochastic Processes and Their Applications, 123(6), 1891-1921. https://doi.org/10.1016/j.spa.2013.02.001
• Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts
  
  MacPhee, I., Menshikov, M., & Wade, A. (2013). Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts. Journal of Theoretical Probability, 26(1), 1-30. https://doi.org/10.1007/s10959-012-0411-x
• Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips
  
  Hryniv, O., MacPhee, I. M., Menshikov, M. V., & Wade, A. R. (2012). Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips. Electronic Journal of Probability, 17, Article 59. https://doi.org/10.1214/ejp.v17-2216
• Rank-driven Markov processes
  
  Grinfeld, M., Knight, P. A., & Wade, A. R. (2012). Rank-driven Markov processes. Journal of Statistical Physics, 146(2), 378-407. https://doi.org/10.1007/s10955-011-0368-7
• Bak-Sneppen-type models and rank-driven processes
  
  Grinfeld, M., Knight, P. A., & Wade, A. R. (2011). Bak-Sneppen-type models and rank-driven processes. Physical Review . E, Statistical, Nonlinear, and Soft Matter Physics, 84(4), Article 041124. https://doi.org/10.1103/physreve.84.041124
• Random walk with barycentric self-interaction
  
  Comets, F., Menshikov, M. V., Volkov, S., & Wade, A. R. (2011). Random walk with barycentric self-interaction. Journal of Statistical Physics, 143(5), 855-888. https://doi.org/10.1007/s10955-011-0218-7
• The simple harmonic urn
  
  Crane, E., Georgiou, N., Volkov, S., Wade, A. R., & Waters, R. J. (2011). The simple harmonic urn. Annals of Probability, 39(6), 2119-2177. https://doi.org/10.1214/10-aop605
• Passage-time moments and hybrid zones for the exclusion-voter model
  
  MacPhee, I. M., Menshikov, M. V., Volkov, S., & Wade, A. R. (2010). Passage-time moments and hybrid zones for the exclusion-voter model. Bernoulli - Journal of the Bernoulli Society, 16(4), 1312-1342. https://doi.org/10.3150/09-bej243
• Limit theorems for random spatial drainage networks
  
  Penrose, M. D., & Wade, A. R. (2010). Limit theorems for random spatial drainage networks. Advances in Applied Probability, 42(3), 659-688. https://doi.org/10.1239/aap/1282924058
• Rate of escape and central limit theorem for the supercritical Lamperti problem
  
  Menshikov, M., & Wade, A. R. (2010). Rate of escape and central limit theorem for the supercritical Lamperti problem. Stochastic Processes and Their Applications, 120(10), 2078-2099. https://doi.org/10.1016/j.spa.2010.06.004
• Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift
  
  MacPhee, I. M., Menshikov, M. V., & Wade, A. R. (2010). Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift. Markov Processes and Related Fields., 16(2), 351-388.
• Asymptotic theory for the multidimensional random on-line nearest-neighbour graph
  
  Wade, A. R. (2009). Asymptotic theory for the multidimensional random on-line nearest-neighbour graph. Stochastic Processes and Their Applications, 119(6), 1889-1911. https://doi.org/10.1016/j.spa.2008.09.006
• Multivariate normal approximation in geometric probability
  
  Penrose, M. D., & Wade, A. R. (2008). Multivariate normal approximation in geometric probability. Journal of Statistical Theory and Practice, 2(2), 293-326. https://doi.org/10.1080/15598608.2008.10411876
• Limit theory for the random on-line nearest-neighbor graph
  
  Penrose, M. D., & Wade, A. R. (2008). Limit theory for the random on-line nearest-neighbor graph. Random Structures and Algorithms, 32(2), 125-156. https://doi.org/10.1002/rsa.20185
• Logarithmic speeds for one-dimensional perturbed random walks in random environments
  
  Menshikov, M., & Wade, A. R. (2008). Logarithmic speeds for one-dimensional perturbed random walks in random environments. Stochastic Processes and Their Applications, 118(3), 389-416. https://doi.org/10.1016/j.spa.2007.04.011
• Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains
  
  Menshikov, M., Vachkovskaia, M., & Wade, A. (2008). Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains. Journal of Statistical Physics, 132(6), 1097-1133. https://doi.org/10.1007/s10955-008-9578-z
• Explicit laws of large numbers for random nearest-neighbour-type graphs
  
  Wade, A. R. (2007). Explicit laws of large numbers for random nearest-neighbour-type graphs. Advances in Applied Probability, 39(2), 326-342. https://doi.org/10.1239/aap/1183667613
• On the total length of the random minimal directed spanning tree
  
  Penrose, M. D., & Wade, A. R. (2006). On the total length of the random minimal directed spanning tree. Advances in Applied Probability, 38(2), 336-372. https://doi.org/10.1239/aap/1151337075
• Random walk in random environment with asymptotically zero perturbation
  
  Menshikov, M., & Wade, A. (2006). Random walk in random environment with asymptotically zero perturbation. Journal of the European Mathematical Society, 8(3), 491-513. https://doi.org/10.4171/jems/64
• Random minimal directed spanning trees and Dickman-type distributions
  
  Penrose, M. D., & Wade, A. R. (2004). Random minimal directed spanning trees and Dickman-type distributions. Advances in Applied Probability, 36(3), 691-714. https://doi.org/10.1239/aap/1093962229