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 and 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), 190-198. doi: 10.1029/2000PA000518.

[2]

N. Berglund and B. Gentz, Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations, Stochastics and Dynamics, 2 (2002), 327-356. doi: 10.1142/S0219493702000455.

[3]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004.

[4]

P. D. Ditlevsen, Anomalous jumping in a double-well potential, Physical Review E, 60 (1999), 172-179. doi: 10.1103/PhysRevE.60.172.

[5]

P. D. Ditlevsen, Observation of $\alpha$-stable noise induced millenial climate changes from an ice record, Geophysical Research Letters, 26 (1999), 1441-1444. doi: 10.1029/1999GL900252.

[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), 2594-2603. doi: 10.1175/JCLI3437.1.

[7]

B. Franke, The scaling limit behaviour of periodic stable-like processes, Bernoulli, 12 (2006), 551-570. doi: 10.3150/bj/1151525136.

[8]

B. Franke, Correction to: The scaling limit behaviour of periodic stable-like processes, Bernoulli, 13 (2007), 600-600. doi: 10.3150/07-BEJ5127.

[9]

M. Freidlin, Quasi-deterministic approximation, metastability and stochastic resonance, Physica D, 137 (2000), 333-352. doi: 10.1016/S0167-2789(99)00191-8.

[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, P. Cannarsa, C. Jones and A. Portaluri), vol. 15 of Springer INdAM Series, Springer, 2016.

[11]

A. Ganopolski and S. Rahmstorf, Abrupt glacial climate changes due to stochastic resonance, Physical Review Letters, 88 (2002), 038501. doi: 10.1103/PhysRevLett.88.038501.

[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), 321-334. doi: 10.1137/1126031.

[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, S. Luo and C. Wang), vol. 8 of Interdisciplinary Mathematical Sciences, 2010, 161-175. doi: 10.1142/9789814277266_0010.

[14]

S. Herrmann and P. Imkeller, Barrier crossings characterize stochastic resonance, Stochastics and Dynamics, 2 (2002), 413-436. doi: 10.1142/S0219493702000509.

[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), 39-68. doi: 10.1214/105051604000000530.

[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, Providence, RI, 2014.

[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), 1851-1892. doi: 10.1214/105051606000000385.

[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), 1550019. doi: 10.1142/S0219493715500197.

[19]

P. Imkeller and I. Pavlyukevich, Model reduction and stochastic resonance, Stochastics and Dynamics, 2 (2002), 463-506. doi: 10.1142/S0219493702000583.

[20]

P. Imkeller and I. Pavlyukevich, Lévy flights: transitions and meta-stability, Journal of Physics A: Mathematical and General, 39 (2006), L237-L246. doi: 10.1088/0305-4470/39/15/L01.

[21]

P. Imkeller and I. Pavlyukevich, Metastable behaviour of small noise Lévy-driven diffusions, ESAIM: Probaility and Statistics, 12 (2008), 412-437. doi: 10.1051/ps:2007051.

[22]

O. Kallenberg, Foundations of Modern Probability, 2nd edition, Probability and Its Applications, Springer, 2002. doi: 10.1007/978-1-4757-4015-8.

[23]

V. N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, vol. 1724 of Lecture Notes in Mathematics, Springer, Berlin, 2000. doi: 10.1007/BFb0112488.

[24]

V. N. Kolokoltsov, Markov Processes, Semigroups, and Generators, vol. 38 of Studies in Mathematics, Walter de Gruyter, Berlin, 2011.

[25]

I. Kuhwald, Small Noise Analysis of Time-Periodic Bistable Jump Diffusions, PhD thesis, Friedrich Schiller University, Jena, 2015.

[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), 1750027. doi: 10.1142/S0219493717500277.

[27]

I. Kuhwald and I. Pavlyukevich, Stochastic resonance in systems driven by $\alpha$-stable Lévy noise, Procedia Engineering, 144 (2016), 1307-1314. doi: 10.1016/j.proeng.2016.05.129.

[28]

F. W. J. Olver, Asymptotics and Special Functions, Computer Science and Applied Mathematics, Academic Press, New York, 1974.

[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), 495-519. doi: 10.1142/S0219493711003413.

[30]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2013.

[31]

M. Schulz, On the 1470-year pacing of Dansgaard-Oeschger warm events, Paleoceanography, 17 (2002), 4-1-4-9. doi: 10.1029/2000PA000571.

show all references

References:
[1]

R. B. Alley, S. Anandakrishnan and P. Jung, Stochastic resonance in the North Atlantic, Paleoceanography, 16 (2001), 190-198. doi: 10.1029/2000PA000518.

[2]

N. Berglund and B. Gentz, Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations, Stochastics and Dynamics, 2 (2002), 327-356. doi: 10.1142/S0219493702000455.

[3]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004.

[4]

P. D. Ditlevsen, Anomalous jumping in a double-well potential, Physical Review E, 60 (1999), 172-179. doi: 10.1103/PhysRevE.60.172.

[5]

P. D. Ditlevsen, Observation of $\alpha$-stable noise induced millenial climate changes from an ice record, Geophysical Research Letters, 26 (1999), 1441-1444. doi: 10.1029/1999GL900252.

[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), 2594-2603. doi: 10.1175/JCLI3437.1.

[7]

B. Franke, The scaling limit behaviour of periodic stable-like processes, Bernoulli, 12 (2006), 551-570. doi: 10.3150/bj/1151525136.

[8]

B. Franke, Correction to: The scaling limit behaviour of periodic stable-like processes, Bernoulli, 13 (2007), 600-600. doi: 10.3150/07-BEJ5127.

[9]

M. Freidlin, Quasi-deterministic approximation, metastability and stochastic resonance, Physica D, 137 (2000), 333-352. doi: 10.1016/S0167-2789(99)00191-8.

[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, P. Cannarsa, C. Jones and A. Portaluri), vol. 15 of Springer INdAM Series, Springer, 2016.

[11]

A. Ganopolski and S. Rahmstorf, Abrupt glacial climate changes due to stochastic resonance, Physical Review Letters, 88 (2002), 038501. doi: 10.1103/PhysRevLett.88.038501.

[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), 321-334. doi: 10.1137/1126031.

[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, S. Luo and C. Wang), vol. 8 of Interdisciplinary Mathematical Sciences, 2010, 161-175. doi: 10.1142/9789814277266_0010.

[14]

S. Herrmann and P. Imkeller, Barrier crossings characterize stochastic resonance, Stochastics and Dynamics, 2 (2002), 413-436. doi: 10.1142/S0219493702000509.

[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), 39-68. doi: 10.1214/105051604000000530.

[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, Providence, RI, 2014.

[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), 1851-1892. doi: 10.1214/105051606000000385.

[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), 1550019. doi: 10.1142/S0219493715500197.

[19]

P. Imkeller and I. Pavlyukevich, Model reduction and stochastic resonance, Stochastics and Dynamics, 2 (2002), 463-506. doi: 10.1142/S0219493702000583.

[20]

P. Imkeller and I. Pavlyukevich, Lévy flights: transitions and meta-stability, Journal of Physics A: Mathematical and General, 39 (2006), L237-L246. doi: 10.1088/0305-4470/39/15/L01.

[21]

P. Imkeller and I. Pavlyukevich, Metastable behaviour of small noise Lévy-driven diffusions, ESAIM: Probaility and Statistics, 12 (2008), 412-437. doi: 10.1051/ps:2007051.

[22]

O. Kallenberg, Foundations of Modern Probability, 2nd edition, Probability and Its Applications, Springer, 2002. doi: 10.1007/978-1-4757-4015-8.

[23]

V. N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, vol. 1724 of Lecture Notes in Mathematics, Springer, Berlin, 2000. doi: 10.1007/BFb0112488.

[24]

V. N. Kolokoltsov, Markov Processes, Semigroups, and Generators, vol. 38 of Studies in Mathematics, Walter de Gruyter, Berlin, 2011.

[25]

I. Kuhwald, Small Noise Analysis of Time-Periodic Bistable Jump Diffusions, PhD thesis, Friedrich Schiller University, Jena, 2015.

[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), 1750027. doi: 10.1142/S0219493717500277.

[27]

I. Kuhwald and I. Pavlyukevich, Stochastic resonance in systems driven by $\alpha$-stable Lévy noise, Procedia Engineering, 144 (2016), 1307-1314. doi: 10.1016/j.proeng.2016.05.129.

[28]

F. W. J. Olver, Asymptotics and Special Functions, Computer Science and Applied Mathematics, Academic Press, New York, 1974.

[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), 495-519. doi: 10.1142/S0219493711003413.

[30]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2013.

[31]

M. Schulz, On the 1470-year pacing of Dansgaard-Oeschger warm events, Paleoceanography, 17 (2002), 4-1-4-9. doi: 10.1029/2000PA000571.

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