# American Institute of Mathematical Sciences

November  2016, 21(9): 3175-3190. doi: 10.3934/dcdsb.2016092

## Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process

 1 Friedrich Schiller University Jena, Department of Mathematics and Computer Science, Institute for Mathematics, Ernst-Abbe-Platz 2, 07743 Jena, Germany, Germany

Received  November 2015 Revised  April 2016 Published  October 2016

We study a bistable gradient system perturbed by a stable-like additive process with a periodically varying stability index. Among a continuum of intrinsic time scales determined by the values of the stability index we single out the characteristic time scale on which the system exhibits the metastable behaviour, namely it behaves like a time discrete two-state Markov chain.
Citation: Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092
##### References:
 [1] R. B. Alley, S. Anandakrishnan and P. Jung, Stochastic resonance in the North Atlantic,, Paleoceanography, 16 (2001), 190.  doi: 10.1029/2000PA000518.  Google Scholar [2] N. Berglund and B. Gentz, Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations,, Stochastics and Dynamics, 2 (2002), 327.  doi: 10.1142/S0219493702000455.  Google Scholar [3] R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC, (2004).   Google Scholar [4] P. D. Ditlevsen, Anomalous jumping in a double-well potential,, Physical Review E, 60 (1999), 172.  doi: 10.1103/PhysRevE.60.172.  Google Scholar [5] P. D. Ditlevsen, Observation of $\alpha$-stable noise induced millenial climate changes from an ice record,, Geophysical Research Letters, 26 (1999), 1441.  doi: 10.1029/1999GL900252.  Google Scholar [6] P. D. Ditlevsen, M. S. Kristensen and K. K. Andersen, The reccurence time of Dansgaard-Oeschger events and limits of the possible periodic component,, Journal of Climate, 18 (2005), 2594.  doi: 10.1175/JCLI3437.1.  Google Scholar [7] B. Franke, The scaling limit behaviour of periodic stable-like processes,, Bernoulli, 12 (2006), 551.  doi: 10.3150/bj/1151525136.  Google Scholar [8] B. Franke, Correction to: The scaling limit behaviour of periodic stable-like processes,, Bernoulli, 13 (2007), 600.  doi: 10.3150/07-BEJ5127.  Google Scholar [9] M. Freidlin, Quasi-deterministic approximation, metastability and stochastic resonance,, Physica D, 137 (2000), 333.  doi: 10.1016/S0167-2789(99)00191-8.  Google Scholar [10] J. Gairing, M. Högele, T. Kosenkova and A. Kulik, On the calibration of Lévy driven time series with coupling distances. An application in paleoclimate,, in Mathematical Paradigms of Climate Science (eds. F. Ancona, (2016).   Google Scholar [11] A. Ganopolski and S. Rahmstorf, Abrupt glacial climate changes due to stochastic resonance,, Physical Review Letters, 88 (2002).  doi: 10.1103/PhysRevLett.88.038501.  Google Scholar [12] V. V. Godovanchuk, Asymptotic probabilities of large deviations due to large jumps of a Markov process,, Theory of Probability and its Applications, 26 (1981), 321.  doi: 10.1137/1126031.  Google Scholar [13] C. Hein, P. Imkeller and I. Pavlyukevich, Limit theorems for $p$-variations of solutions of SDEs driven by additive stable Lévy noise and model selection for paleo-climatic data,, in Recent Development in Stochastic Dynamics and Stochastic Analysis (eds. J. Duan, (2010), 161.  doi: 10.1142/9789814277266_0010.  Google Scholar [14] S. Herrmann and P. Imkeller, Barrier crossings characterize stochastic resonance,, Stochastics and Dynamics, 2 (2002), 413.  doi: 10.1142/S0219493702000509.  Google Scholar [15] S. Herrmann and P. Imkeller, The exit problem for diffusions with time periodic drift and stochastic resonance,, The Annals of Applied Probability, 15 (2005), 39.  doi: 10.1214/105051604000000530.  Google Scholar [16] S. Herrmann, P. Imkeller, I. Pavlyukevich and D. Peithmann, Stochastic Resonance: A Mathematical Approach in the Small Noise Limit, vol. 194 of AMS Mathematical Surveys and Monographs,, American Mathematical Society, (2014).   Google Scholar [17] S. Herrmann, P. Imkeller and D. Peithmann, Transition times and stochastic resonance for multidimensional diffusions with time periodic drift: A large deviations approach,, Annals of Applied Probability, 16 (2006), 1851.  doi: 10.1214/105051606000000385.  Google Scholar [18] M. Högele and I. Pavlyukevich, Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Lévy type noise,, Stochastics and Dynamics, 15 (2015).  doi: 10.1142/S0219493715500197.  Google Scholar [19] P. Imkeller and I. Pavlyukevich, Model reduction and stochastic resonance,, Stochastics and Dynamics, 2 (2002), 463.  doi: 10.1142/S0219493702000583.  Google Scholar [20] P. Imkeller and I. Pavlyukevich, Lévy flights: transitions and meta-stability,, Journal of Physics A: Mathematical and General, 39 (2006).  doi: 10.1088/0305-4470/39/15/L01.  Google Scholar [21] P. Imkeller and I. Pavlyukevich, Metastable behaviour of small noise Lévy-driven diffusions,, ESAIM: Probaility and Statistics, 12 (2008), 412.  doi: 10.1051/ps:2007051.  Google Scholar [22] O. Kallenberg, Foundations of Modern Probability,, 2nd edition, (2002).  doi: 10.1007/978-1-4757-4015-8.  Google Scholar [23] V. N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, vol. 1724 of Lecture Notes in Mathematics,, Springer, (2000).  doi: 10.1007/BFb0112488.  Google Scholar [24] V. N. Kolokoltsov, Markov Processes, Semigroups, and Generators, vol. 38 of Studies in Mathematics,, Walter de Gruyter, (2011).   Google Scholar [25] I. Kuhwald, Small Noise Analysis of Time-Periodic Bistable Jump Diffusions,, PhD thesis, (2015).   Google Scholar [26] I. Kuhwald and I. Pavlyukevich, Stochastic resonance with multiplicative heavy-tailed Lévy noise: Optimal tuning on an algebraic time scale,, Stochastics and Dynamics, 17 (2017).  doi: 10.1142/S0219493717500277.  Google Scholar [27] I. Kuhwald and I. Pavlyukevich, Stochastic resonance in systems driven by $\alpha$-stable Lévy noise,, Procedia Engineering, 144 (2016), 1307.  doi: 10.1016/j.proeng.2016.05.129.  Google Scholar [28] F. W. J. Olver, Asymptotics and Special Functions,, Computer Science and Applied Mathematics, (1974).   Google Scholar [29] I. Pavlyukevich, First exit times of solutions of stochastic differential equations driven by multiplicative Lévy noise with heavy tails,, Stochastics and Dynamics, 11 (2011), 495.  doi: 10.1142/S0219493711003413.  Google Scholar [30] K. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in Advanced Mathematics,, Cambridge University Press, (2013).   Google Scholar [31] M. Schulz, On the 1470-year pacing of Dansgaard-Oeschger warm events,, Paleoceanography, 17 (2002), 4.  doi: 10.1029/2000PA000571.  Google Scholar

show all references

##### References:
 [1] R. B. Alley, S. Anandakrishnan and P. Jung, Stochastic resonance in the North Atlantic,, Paleoceanography, 16 (2001), 190.  doi: 10.1029/2000PA000518.  Google Scholar [2] N. Berglund and B. Gentz, Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations,, Stochastics and Dynamics, 2 (2002), 327.  doi: 10.1142/S0219493702000455.  Google Scholar [3] R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC, (2004).   Google Scholar [4] P. D. Ditlevsen, Anomalous jumping in a double-well potential,, Physical Review E, 60 (1999), 172.  doi: 10.1103/PhysRevE.60.172.  Google Scholar [5] P. D. Ditlevsen, Observation of $\alpha$-stable noise induced millenial climate changes from an ice record,, Geophysical Research Letters, 26 (1999), 1441.  doi: 10.1029/1999GL900252.  Google Scholar [6] P. D. Ditlevsen, M. S. Kristensen and K. K. Andersen, The reccurence time of Dansgaard-Oeschger events and limits of the possible periodic component,, Journal of Climate, 18 (2005), 2594.  doi: 10.1175/JCLI3437.1.  Google Scholar [7] B. Franke, The scaling limit behaviour of periodic stable-like processes,, Bernoulli, 12 (2006), 551.  doi: 10.3150/bj/1151525136.  Google Scholar [8] B. Franke, Correction to: The scaling limit behaviour of periodic stable-like processes,, Bernoulli, 13 (2007), 600.  doi: 10.3150/07-BEJ5127.  Google Scholar [9] M. Freidlin, Quasi-deterministic approximation, metastability and stochastic resonance,, Physica D, 137 (2000), 333.  doi: 10.1016/S0167-2789(99)00191-8.  Google Scholar [10] J. Gairing, M. Högele, T. Kosenkova and A. Kulik, On the calibration of Lévy driven time series with coupling distances. An application in paleoclimate,, in Mathematical Paradigms of Climate Science (eds. F. Ancona, (2016).   Google Scholar [11] A. Ganopolski and S. Rahmstorf, Abrupt glacial climate changes due to stochastic resonance,, Physical Review Letters, 88 (2002).  doi: 10.1103/PhysRevLett.88.038501.  Google Scholar [12] V. V. Godovanchuk, Asymptotic probabilities of large deviations due to large jumps of a Markov process,, Theory of Probability and its Applications, 26 (1981), 321.  doi: 10.1137/1126031.  Google Scholar [13] C. Hein, P. Imkeller and I. Pavlyukevich, Limit theorems for $p$-variations of solutions of SDEs driven by additive stable Lévy noise and model selection for paleo-climatic data,, in Recent Development in Stochastic Dynamics and Stochastic Analysis (eds. J. Duan, (2010), 161.  doi: 10.1142/9789814277266_0010.  Google Scholar [14] S. Herrmann and P. Imkeller, Barrier crossings characterize stochastic resonance,, Stochastics and Dynamics, 2 (2002), 413.  doi: 10.1142/S0219493702000509.  Google Scholar [15] S. Herrmann and P. Imkeller, The exit problem for diffusions with time periodic drift and stochastic resonance,, The Annals of Applied Probability, 15 (2005), 39.  doi: 10.1214/105051604000000530.  Google Scholar [16] S. Herrmann, P. Imkeller, I. Pavlyukevich and D. Peithmann, Stochastic Resonance: A Mathematical Approach in the Small Noise Limit, vol. 194 of AMS Mathematical Surveys and Monographs,, American Mathematical Society, (2014).   Google Scholar [17] S. Herrmann, P. Imkeller and D. Peithmann, Transition times and stochastic resonance for multidimensional diffusions with time periodic drift: A large deviations approach,, Annals of Applied Probability, 16 (2006), 1851.  doi: 10.1214/105051606000000385.  Google Scholar [18] M. Högele and I. Pavlyukevich, Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Lévy type noise,, Stochastics and Dynamics, 15 (2015).  doi: 10.1142/S0219493715500197.  Google Scholar [19] P. Imkeller and I. Pavlyukevich, Model reduction and stochastic resonance,, Stochastics and Dynamics, 2 (2002), 463.  doi: 10.1142/S0219493702000583.  Google Scholar [20] P. Imkeller and I. Pavlyukevich, Lévy flights: transitions and meta-stability,, Journal of Physics A: Mathematical and General, 39 (2006).  doi: 10.1088/0305-4470/39/15/L01.  Google Scholar [21] P. Imkeller and I. Pavlyukevich, Metastable behaviour of small noise Lévy-driven diffusions,, ESAIM: Probaility and Statistics, 12 (2008), 412.  doi: 10.1051/ps:2007051.  Google Scholar [22] O. Kallenberg, Foundations of Modern Probability,, 2nd edition, (2002).  doi: 10.1007/978-1-4757-4015-8.  Google Scholar [23] V. N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, vol. 1724 of Lecture Notes in Mathematics,, Springer, (2000).  doi: 10.1007/BFb0112488.  Google Scholar [24] V. N. Kolokoltsov, Markov Processes, Semigroups, and Generators, vol. 38 of Studies in Mathematics,, Walter de Gruyter, (2011).   Google Scholar [25] I. Kuhwald, Small Noise Analysis of Time-Periodic Bistable Jump Diffusions,, PhD thesis, (2015).   Google Scholar [26] I. Kuhwald and I. Pavlyukevich, Stochastic resonance with multiplicative heavy-tailed Lévy noise: Optimal tuning on an algebraic time scale,, Stochastics and Dynamics, 17 (2017).  doi: 10.1142/S0219493717500277.  Google Scholar [27] I. Kuhwald and I. Pavlyukevich, Stochastic resonance in systems driven by $\alpha$-stable Lévy noise,, Procedia Engineering, 144 (2016), 1307.  doi: 10.1016/j.proeng.2016.05.129.  Google Scholar [28] F. W. J. Olver, Asymptotics and Special Functions,, Computer Science and Applied Mathematics, (1974).   Google Scholar [29] I. Pavlyukevich, First exit times of solutions of stochastic differential equations driven by multiplicative Lévy noise with heavy tails,, Stochastics and Dynamics, 11 (2011), 495.  doi: 10.1142/S0219493711003413.  Google Scholar [30] K. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in Advanced Mathematics,, Cambridge University Press, (2013).   Google Scholar [31] M. Schulz, On the 1470-year pacing of Dansgaard-Oeschger warm events,, Paleoceanography, 17 (2002), 4.  doi: 10.1029/2000PA000571.  Google Scholar
 [1] Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028 [2] Massimiliano Berti, Philippe Bolle. Fast Arnold diffusion in systems with three time scales. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 795-811. doi: 10.3934/dcds.2002.8.795 [3] Sung Kyu Choi, Namjip Koo. Stability of linear dynamic equations on time scales. Conference Publications, 2009, 2009 (Special) : 161-170. doi: 10.3934/proc.2009.2009.161 [4] Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69 [5] Donny Citra Lesmana, Song Wang. A numerical scheme for pricing American options with transaction costs under a jump diffusion process. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1793-1813. doi: 10.3934/jimo.2017019 [6] Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298 [7] Christian Pötzsche, Stefan Siegmund, Fabian Wirth. A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1223-1241. doi: 10.3934/dcds.2003.9.1223 [8] Ishak Alia. A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion. Mathematical Control & Related Fields, 2019, 9 (3) : 541-570. doi: 10.3934/mcrf.2019025 [9] Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 671-691. doi: 10.3934/dcds.2011.29.671 [10] Fatih Bayazit, Britta Dorn, Marjeta Kramar Fijavž. Asymptotic periodicity of flows in time-depending networks. Networks & Heterogeneous Media, 2013, 8 (4) : 843-855. doi: 10.3934/nhm.2013.8.843 [11] Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235 [12] Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257 [13] Christophe Cheverry, Thierry Paul. On some geometry of propagation in diffractive time scales. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 499-538. doi: 10.3934/dcds.2012.32.499 [14] Yunfei Peng, X. Xiang, W. Wei. Backward problems of nonlinear dynamical equations on time scales. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1553-1564. doi: 10.3934/dcdss.2011.4.1553 [15] Petr Hasil, Petr Zemánek. Critical second order operators on time scales. Conference Publications, 2011, 2011 (Special) : 653-659. doi: 10.3934/proc.2011.2011.653 [16] B. Kaymakcalan, R. Mert, A. Zafer. Asymptotic equivalence of dynamic systems on time scales. Conference Publications, 2007, 2007 (Special) : 558-567. doi: 10.3934/proc.2007.2007.558 [17] Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823 [18] Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526 [19] Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315 [20] Amadeu Delshams, Rodrigo G. Schaefer. Arnold diffusion for a complete family of perturbations with two independent harmonics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6047-6072. doi: 10.3934/dcds.2018261

2018 Impact Factor: 1.008