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Metastability for telegraph processes in a double-well potential

Published online by Cambridge University Press:  19 September 2025

Nikita Ratanov Open the ORCID record for Nikita Ratanov [Opens in a new window]
Nikita Ratanov*
Affiliation:
Chelyabinsk State University, Russia
*
*Postal address: Br. Kashirinykh, 129, Chelyabinsk, 454000, Russia. Email: rtnnkt@gmail.com
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Abstract

We study equations with nonlinearity in the form of a double-well potential, randomised by a velocity-switching (telegraph) stochastic process. If the speed parameters of the randomisation are small, then this dynamics has one metastable uncertainty interval and two invariant attractors. The probabilities of leaving the metastable interval through the upper boundary are determined, as well as characteristics of the first crossing times. Invariant measures are also found. When and if the direction of the telegraph process velocity coincides with the direction of the periodic change in potential, the system can go into a metastable state, having received a time window for the interwell transition. The obtained results can be used as an alternative to stochastic resonance models.

Keywords

Piecewise deterministic processfirst-passage timeinvariant measurestochastic resonanceclimate models

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Original Article
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Journal of Applied Probability , First View , pp. 1 - 21
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© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Benzi, R., Parisi, G., Sutera, A. and Vulpiani, A. (1983). A theory of stochastic resonance in climatic change. SIAM J. Appl. Math. 43, 565. DOI: 10.1137/0143037.10.1137/0143037CrossRefGoogle Scholar
Borodin, A. and Gorin, V. (2019). A stochastic telegraph equation from the six-vertex model. Ann. Prob. 47, 4137–4194.10.1214/19-AOP1356CrossRefGoogle Scholar
Bovier, A. and den Hollander, F. (2015). Metastability: A Potential-Theoretic Approach. Springer, Berlin.10.1007/978-3-319-24777-9CrossRefGoogle Scholar
Brémaud, P. (1999). Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, Berlin.10.1007/978-1-4757-3124-8CrossRefGoogle Scholar
Bulsara, A., Jacobs, E. W., Zhou, T., Moss, F. and Kiss, L. (1991). Stochastic resonance in a single neuron model: Theory and analog simulation. J. Theoret. Biol. 152, 531–555. DOI: 10.1016/s0022-5193(05)80396-0.10.1016/S0022-5193(05)80396-0CrossRefGoogle Scholar
Carrillo, J. A. and Roux, P. (2025). Nonlinear partial differential equations in neuroscience: From modeling to mathematical theory. Math. Mod. Meth. Appl. Sci. 35, 403–584. DOI: 10.1142/S0218202525400044.10.1142/S0218202525400044CrossRefGoogle Scholar
Das, M. and Kantz, H. (2020). Stochastic resonance and hysteresis in climate with state-dependent fluctuations. Phys. Rev. E 101, 062145.10.1103/PhysRevE.101.062145CrossRefGoogle ScholarPubMed
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.10.1111/j.2517-6161.1984.tb01308.xCrossRefGoogle Scholar
Donahue, J. C. and Dubovsky, S. (2022). Quantization of the zigzag model. J. High Energy Phys. 2002, 47. DOI: 10.1007/JHEP08(2022)047.10.1007/JHEP08(2022)047CrossRefGoogle Scholar
Dybiec, B. and Gudowska-Nowak, E. (2009). Lévy stable noise-induced transitions: Stochastic resonance, resonant activation and dynamic hysteresis. J. Statist. Mech. 2009, P05004.10.1088/1742-5468/2009/05/P05004CrossRefGoogle Scholar
Freidlin, M. I. and Koralov, L. (2023). Perturbations of parabolic equations and diffusion processes with degeneration: Boundary problems, metastability, and homogenization, Ann. Prob. 51, 1752–1784.10.1214/23-AOP1631CrossRefGoogle Scholar
Freidlin, M. I. and Koralov, L. (2024). Metastability in parabolic equations and diffusion processes with a small parameter. Preprint, arXiv:2403.12333v1.Google Scholar
Gammaitoni, L., Hänggi, P., Jung, P. and Marchesoni, F. (1998). Stochastic resonance. Rev. Mod. Phys. 70, 223–287. DOI: 10.1103/RevModPhys.70.223.10.1103/RevModPhys.70.223CrossRefGoogle Scholar
Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Q. J. Mech. Appl. Math., 4, 129–156.10.1093/qjmam/4.2.129CrossRefGoogle Scholar
Goychuk, I. and Kharchenko, V. (2012). Fractional Brownian motors and stochastic resonance. Phys. Rev. E 85, 051131.10.1103/PhysRevE.85.051131CrossRefGoogle ScholarPubMed
Gutkin, B. S., Jost, J. and Tuckwell, H. C. (2008). The silencing of neuronal activity by noise and the phenomenon of inverse stochastic resonance. Preprint, arXiv:0805.0592 [q-bio.NC].Google Scholar
Gzyl, H. (2025). Lorentz covariant physical Brownian motion: Classical and quantum. Ann. Phys. 472, 169857.10.1016/j.aop.2024.169857CrossRefGoogle Scholar
Herrmann, S., Imkeller, P., Pavlyukevich, I. and Peithmann, D. (2014). Stochastic Resonance: A Mathematical Approach in the Small Noise Limit (Math. Surv. Monographs 194). American Mathematical Society, Providence, RI.Google Scholar
Kac, M. (1974). A stochastic model related to the telegrapher’s equation Rocky Mountain J. Math. 4, 497–509.10.1216/RMJ-1974-4-3-497CrossRefGoogle Scholar
McDonnell, M. D., Stocks, N. G., Pearce, C. E. M. and Abbott, D. (2008). Stochastic Resonance: From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization. Cambridge University Press.10.1017/CBO9780511535239CrossRefGoogle Scholar
Milankovitch, M. (1930). Mathematische klimalehre und astronomische theorie der klimaschwankungen. In Handbuch der Klimatologie, Vol. I, eds W. Köppen and R. Geiger. Bornträger, Berlin.Google Scholar
Montroll, E. W. and Weiss, G. H. (1965). Random walks on lattices II. J. Math. Phys. 6, 167. DOI: 10.1063/1.1704269.10.1063/1.1704269CrossRefGoogle Scholar
Náprstek, J. and Fischer, C. (2017). Stochastic resonance and related topics. In Resonance, ed. Awrejcewicz, J.. IntechOpen, Rijeka, pp. 35–70. DOI: 10.5772/intechopen.70178.10.5772/intechopen.70178CrossRefGoogle Scholar
Nicolis, C. (1993). Long-term climatic transitions and stochastic resonance. J. Statist. Phys. 70, 3–14. DOI: 10.1007/bf01053950.10.1007/BF01053950CrossRefGoogle Scholar
Ratanov, N. (2010). Option pricing model based on a Markov-modulated diffusion with jumps. Braz. J. Prob. Statist. 24, 413–431.10.1214/09-BJPS037CrossRefGoogle Scholar
Ratanov, N. (2020). Mean-reverting neuronal model based on two alternating patterns. BioSystems 196, 104190. DOI: 10.1016/j.biosystems.2020.104190.10.1016/j.biosystems.2020.104190CrossRefGoogle ScholarPubMed
Ratanov, N. (2021). Ornstein–Uhlenbeck processes of bounded variation. Methodol. Comput. Appl. Prob. 23, 925–946. DOI: 10.1007/s11009-020-09794-x.10.1007/s11009-020-09794-xCrossRefGoogle Scholar
Ratanov, N. (2022). Kac–Ornstein–Uhlenbeck processes: Stationary distributions and exponential functionals. Methodol. Comput. Appl. Prob. 24, 2703–2721. DOI: 10.1007/s11009-022-09956-z.10.1007/s11009-022-09956-zCrossRefGoogle Scholar
Ratanov, N. (2022). On time-homogeneous piecewise deterministic Markov processes. Global Stoch. Anal. 9, 35–44.Google Scholar
Ratanov, N. (2023). Stochastically resetting Kac–Ornstein–Uhlenbeck processes. Global Stoch. Anal. 10, 93–111.Google Scholar
Ratanov, N. and Kolesnik, A. (2022). Telegraph Processes and Option Pricing, 2nd edn. Springer, Heidelberg.10.1007/978-3-662-65827-7CrossRefGoogle Scholar
Sandev, T., Kocarev, L., Metzler, R. and Chechkin, A. (2022). Stochastic dynamics with multiplicative dichotomic noise: Heterogeneous telegrapher’s equation, anomalous crossovers and resetting. Chaos, Solitons & Fractals 165, 112878.10.1016/j.chaos.2022.112878CrossRefGoogle Scholar
Simonotto, E., Riani, M., Seife, Ch, Roberts, M., Twitty, J. and Moss, F. (1997). Visual perception of stochastic resonance. Phys. Rev. Lett. 78, 1186.10.1103/PhysRevLett.78.1186CrossRefGoogle Scholar
Strogatz, S. H. (2018). Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, Boca Raton, FL.Google Scholar
Vasdekis, G. and Roberts, G. O. (2023). Speed up zig-zag. Ann. Appl. Prob. 33, 4693–4746.10.1214/23-AAP1930CrossRefGoogle Scholar
Vellela, M. and Qian, H. (2009). Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: The Schlögl model revisited. J. R. Soc. Interface, 6, 925–940.10.1098/rsif.2008.0476CrossRefGoogle ScholarPubMed
Verma, M. K., Kumar, A. and Pattanayak, A. (2018). Stochastic bistable systems: Competing hysteresis and phase coexistence. Statist. Nonlin. Soft Matter Phys. 127, 549–557.Google Scholar
Wilkinson, M. (2010). Perturbation theory for a stochastic process with Ornstein–Uhlenbeck noise. J. Statist. Phys. 139, 345–353. DOI: 10.1007/s10955-010-9944-5.10.1007/s10955-010-9944-5CrossRefGoogle Scholar
Wissel, C. (1989). Metastability, a consequence of stochastics in multiple stable population dynamics. Theoret. Pop. Biol. 36, 296–310.10.1016/0040-5809(89)90036-1CrossRefGoogle Scholar
Yuan, Ch and Mao, X. (2003). Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stoch. Process. Appl. 103, 277–291.10.1016/S0304-4149(02)00230-2CrossRefGoogle Scholar
Zhang, G., Liu, Y. and He, L. (2022). Stochastic resonance of an asymmetric tristable system driven by cross-correlated Ornstein–Uhlenbeck noise. Chinese J. Phys. 77, 1405–1418.10.1016/j.cjph.2021.10.004CrossRefGoogle Scholar