Staff profile

Research Summary

My research is in stochastic processes, with emphasis on the Lyapunov function method, including random walks, processes in random media, and interacting particle systems. Other interests include percolation theory.

• Markov chains
• probability

• 2000: Editorial board service: Editor of Markov Process Related Fields
• 2000: Plenary and invited talks: Invited talk at Random Media in Atacama (Chile, December 2016).

Authored book

• 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

• 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

Conference Paper

• Menshikov, M., & Petritis, D. (2002, September). On Random Walks in Random Environment on Trees and Their Relationship with Multiplicative Chaos. Presented at International Colloquium of Mathematics and Computer Science, University of Versailles-St-Quentin

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
• 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
• 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
• 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
• 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
• 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
• 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
• Menshikov, M., & Shcherbakov, V. (2020). Localisation in a growth model with interaction. Arbitrary graphs. Alea (2006. Online), 17(1), 473-489. https://doi.org/10.30757/alea.v17-19
• 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
• 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
• Costa, M., Menshikov, M., Shcherbakov, V., & Vachkovskaia, M. (2018). Localisation in a growth model with interaction. Journal of Statistical Physics, 171(6), 1150-1175. https://doi.org/10.1007/s10955-018-2055-4
• Menshikov, M., & Shcherbakov, V. (2018). Long term behaviour of two interacting birth-and-death processes. Markov processes and related fields, 24(1), 85-106
• 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
• Belitsky, V., Menshikov, M., Petritis, D., & Vachkovskaia, M. (2016). Random dynamical systems with systematic drift competing with heavy-tailed randomness. Markov processes and related fields, 22(4), 629-652
• Menshikov, M., & Petritis, D. (2014). Explosion, implosion, and moments of passage times for continuous-time Markov chains: A semimartingale approach. Stochastic Processes and their Applications, 124(7), 2388-2414. https://doi.org/10.1016/j.spa.2014.03.001
• Menshikov, M., & Popov, S. (2014). On range and local time of many-dimensional submartingales. Journal of Theoretical Probability, 27(2), 601-617. https://doi.org/10.1007/s10959-012-0431-6
• 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
• 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
• 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
• Menshikov, M., Sisko, V., & Vachkovskaia, M. (2013). Introduction to shape stability for a storage model. Methodology and Computing in Applied Probability, 15(1), 125-146. https://doi.org/10.1007/s11009-011-9229-8
• MacPhee, I., Menshikov, M., & Vachkovskaia, M. (2012). Dynamics of the non-homogeneous supermarket model. Stochastic Models, 28(4), 533-556. https://doi.org/10.1080/15326349.2012.726031
• Menshikov, M., Popov, S., Ramírez, A. F., & Vachkovskaia, M. (2012). On a general many-dimensional excited random walk. Annals of Probability, 40(5), 2106-2130. https://doi.org/10.1214/11-aop678
• 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
• 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
• 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
• 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
• 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
• 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
• Menshikov, M., & Volkov, S. (2008). Urn-related random walk with drift $\rho x^\alpha/t^\beta$. Electronic Journal of Probability, 13, 944-960. https://doi.org/10.1214/ejp.v13-508
• MacPhee, I., Menshikov, M., Petritis, D., & Popov, S. (2008). Polling systems with parameter regeneration, the general case. Annals of Applied Probability, 18(6), https://doi.org/10.1214/08-aap519
• MacPhee, I., Menshikov, M., Popov, S., & Volkov, S. (2006). Periodicity in the transient regime of exhaustive polling systems. Annals of Applied Probability, 16(4), 1816-1850. https://doi.org/10.1214/105051606000000376
• Andjel, A., Menshikov, M., & Sisko, V. (2006). Positive recurrence of processes associated to crystal growth models. Annals of Applied Probability, 16(3), 1059-1085. https://doi.org/10.1214/105051606000000079
• 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
• Menshikov, M., Popov, S., Sisko, V., & Vachkovskaia, M. (2004). On a many-dimensional random walk in a rarefied random environment
• MacPhee, I., & Menshikov, M. (2003). Critical random walks on two-dimensional complexes with applications to polling systems. Annals of Applied Probability, 13(4), 1399-1422. https://doi.org/10.1214/aoap/1069786503
• Menshikov, M., Rybnikov, K., & Volkov, S. (2002). The loss of tension in an infinite membrane with holes distributed according to a Poisson law. Advances in Applied Probability, 34(2), https://doi.org/10.1239/aap/1025131219
• Menshikov, M., Yu, P. S., & Vachkovskaia, M. (2001). On the connectivity properties of the complementary set in fractal percolation models. Probability Theory and Related Fields, 119(2), 176-186. https://doi.org/10.1007/pl00008757
• Belitsky, V., Ferrari, P., Menshikov, M., & Popov, S. Y. (2001). A mixture of the exclusion process and the voter model. Bernoulli (Andover), 7(1), 119-144. https://doi.org/10.2307/3318605
• Menshikov, M., & Zuyev, S. (2001). Polling systems in the critical regime
• Menshikov, M., & Zuyev, S. (2001). Polling systems in the critical rejime. Stochastic Processes and their Applications, 92(2), 201-218
• Menshikov, M., den Hollander, F., & Popov, S. (1999). A note on transience versus recurrence for a Branching random walk in random environment. Journal of Statistical Physics, 95, 587-614
• Menshikov, M., Comets, F., & Popov, S. (1998). Lyapunov functions for random walks and strings in random environment. Annals of Probability, 26, 1433-1445
• Menshikov, M., Aspandiiarov, S., & Iasnogorodski, R. (1996). Passage-time moments for non-negative stochastic processes and an application to reflected random walks in a quadrant. Annals of Probability, 24, 932-960. https://doi.org/10.1214/aop/1039639371
• Menshikov, M., Grimmett, G., & Volkov, S. (1996). Random walks in random labyrinths. Markov processes and related fields, 2, 69-86