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

• 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

Conference Paper

• On Random Walks in Random Environment on Trees and Their Relationship with Multiplicative Chaos
  
  Menshikov, M., & Petritis, D. (2002). On Random Walks in Random Environment on Trees and Their Relationship with Multiplicative Chaos. In Mathematics and Computer Science II Algorithms, Trees, Combinatorics and Probabilities. https://doi.org/10.1007/978-3-0348-8211-8_25

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
• 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
• 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
• 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
• Localisation in a growth model with interaction. Arbitrary graphs
  
  Menshikov, M., & Shcherbakov, V. (2020). Localisation in a growth model with interaction. Arbitrary graphs. Alea (2006), 17(1), 473-489. https://doi.org/10.30757/alea.v17-19
• 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
• 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
• Localisation in a growth model with interaction
  
  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
• Long term behaviour of two interacting birth-and-death processes
  
  Menshikov, M., & Shcherbakov, V. (2018). Long term behaviour of two interacting birth-and-death processes. Markov Processes and Related Fields., 24(1), 85-106.
• 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
• Random dynamical systems with systematic drift competing with heavy-tailed randomness
  
  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.
• Explosion, implosion, and moments of passage times for continuous-time Markov chains: A semimartingale approach
  
  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
• On range and local time of many-dimensional submartingales
  
  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
• 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
• Introduction to shape stability for a storage model
  
  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
• Dynamics of the non-homogeneous supermarket model
  
  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
• On a general many-dimensional excited random walk
  
  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
• 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
• 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
• 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.
• 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
• Polling systems with parameter regeneration, the general case.
  
  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
• Urn-related random walk with drift $\rho x^\alpha/t^\beta$
  
  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
• Periodicity in the transient regime of exhaustive polling systems
  
  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
• Positive recurrence of processes associated to crystal growth models
  
  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
• 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
• On a many-dimensional random walk in a rarefied random environment.
  
  Menshikov, M., Popov, S., Sisko, V., & Vachkovskaia, M. (2004). On a many-dimensional random walk in a rarefied random environment. ’Markov Process Related Fields, 10, 137-160.
• Critical random walks on two-dimensional complexes with applications to polling systems
  
  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
• The loss of tension in an infinite membrane with holes distributed according to a Poisson law
  
  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
• On the connectivity properties of the complementary set in fractal percolation models
  
  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
• A mixture of the exclusion process and the voter model
  
  Belitsky, V., Ferrari, P., Menshikov, M., & Popov, S. Y. (2001). A mixture of the exclusion process and the voter model. Bernoulli - Journal of the Bernoulli Society, 7(1), 119-144. https://doi.org/10.2307/3318605
• Polling systems in the critical regime
  
  Menshikov, M., & Zuyev, S. (2001). Polling systems in the critical regime. Stochastic Process Appl, 92, 2001-2018.
• Polling systems in the critical rejime
  
  Menshikov, M., & Zuyev, S. (2001). Polling systems in the critical rejime. Stochastic Processes and Their Applications, 92(2), 201-218.
• A note on transience versus recurrence for a Branching random walk in random environment
  
  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.
• Lyapunov functions for random walks and strings in random environment
  
  Menshikov, M., Comets, F., & Popov, S. (1998). Lyapunov functions for random walks and strings in random environment. Annals of Probability, 26, 1433-1445.
• Passage-time moments for non-negative stochastic processes and an application to reflected random walks in a quadrant
  
  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
• Random walks in random labyrinths
  
  Menshikov, M., Grimmett, G., & Volkov, S. (1996). Random walks in random labyrinths. Markov Processes and Related Fields., 2, 69-86.