Discrete-Time Langevin Motion in a Gibbs Potential

Affiliation(s)

Department of Mathematics, Iowa State University, Ames, USA.

Department of Electrical and Computer Engineering, University of California San Diego, San Diego, USA.

Department of Mathematics, Grand Valley State University, Allendale, USA.

Department of Mathematics, Iowa State University, Ames, USA.

Department of Electrical and Computer Engineering, University of California San Diego, San Diego, USA.

Department of Mathematics, Grand Valley State University, Allendale, USA.

ABSTRACT

We consider a multivariate Langevin equation in discrete time, driven by a force induced by certain Gibbs’states. The main goal of the paper is to study the asymptotic behavior of a random walk with stationary increments (which are interpreted as discrete-time speed terms) satisfying the Langevin equation. We observe that (stable) functional limit theorems and laws of iterated logarithm for regular random walks with i.i.d. heavy-tailed increments can be carried over to the motion of the Langevin particle.

Cite this paper

R. Rastegar, A. Roitershtein, V. Roytershteyn and J. Suh, "Discrete-Time Langevin Motion in a Gibbs Potential,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 2032-2037. doi: 10.4236/am.2012.312A280.

R. Rastegar, A. Roitershtein, V. Roytershteyn and J. Suh, "Discrete-Time Langevin Motion in a Gibbs Potential,"

References

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[5] D. Buraczewski, E. Damek and M. Mirek, “Asymptotics of Stationary Solutions of Multivariate Stochastic Recursions with Heavy Tailed Inputs and Related Limit Theorems,” Stochastic Processes and Their Applications, Vol. 122, No. 1, 2012, pp. 42-67. doi:10.1016/j.spa.2011.10.010

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[10] A. Novikov and N. Kordzakhia, “Martingales and First Passage Times of AR(1) Sequences,” Stochastics, Vol. 80, No. 2-3, 2008, pp. 197-210. doi:10.1080/17442500701840885

[11] P. Embrechts and C. M. Goldie, “Perpetuities and Random Equations,” Proceedings of 5th Prague Symposium, Physica, Heidelberg, 1993, pp. 75-86.

[12] R. F. Engle, “ARCH. Selected Readings,” Oxford University Press, Oxford, 1995.

[13] S. T. Rachev and G. Samorodnitsky, “Limit Laws for a Stochastic Process and Random Recursion Arising in Probabilistic Modelling,” Advances in Applied Probability, Vol. 27, No. 1, 1995, pp. 185-202.

[14] W. Vervaat, “On a Stochastic Difference Equations and a Representation of Non-Negative Infinitely Divisible Random Variables,” Advances in Applied Probability, Vol. 11, No. 4, 1979, pp. 750-783. doi:10.2307/1426858

[15] Y. Ephraim and N. Merhav, “Hidden Markov Processes,” IEEE Transactions on Information Theory, Vol. 48, No. 6, 2002, pp. 1518-1569. doi:10.1109/TIT.2002.1003838

[16] D. Hay, R. Rastegar and A. Roitershtein, “Multivariate Linear Recursions with Markov-Dependent Coefficients,” Journal of Multivariate Analysis, Vol. 102, No. 3, 2011, pp. 521-527. doi:10.1016/j.jmva.2010.10.011

[17] R. Stelzer, “Multivariate Markov-Switching ARMA Processes with Regularly Varying Noise,” Journal of Multivariate Analysis, Vol. 99, No. 6, 2008, pp. 1177-1190. doi:10.1016/j.jmva.2007.07.001

[18] P. Eloe, R. H. Liu, M. Yatsuki, G. Yin and Q. Zhang, “Optimal Selling Rules in a Regime-Switching Exponential Gaussian Diffusion Model,” SIAM Journal of Applied Mathematics, Vol. 69, No. 3, 2008, pp. 810-829. doi:10.1137/060652671

[19] E. F. Fama, “The Behavior of Stock Market Prices,” Journal of Business, Vol. 38, No. 1, 1965, pp. 34-105. doi:10.1086/294743

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[21] S. Lalley, “Regenerative Representation for One-Dimensional Gibbs States,” Annals of Probability, Vol. 14, No. 4, 1986, pp. 1262-1271. doi:10.1214/aop/1176992367

[22] R. Fernández and G. Maillard, “Chains with Complete Connections and One-Dimensional Gibbs Measures,” Electronic Journal of Probability, Vol. 9, 2004, pp. 145-176. doi:10.1214/EJP.v9-149

[23] R. Fernández and G. Maillard, “Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties,” Journal of Statistical Physics, Vol. 118, No. 3-4, 2005, pp. 555-588. doi:10.1007/s10955-004-8821-5

[24] M. Iosifescu and S. Grigorescu, “Dependence with Complete Connections and Its Applications,” Cambridge Tracts in Mathematics, Vol. 96, Cambridge University Press, Cambridge, 2009.

[25] H. Berbee, “Chains with Infinite Connections: Uniqueness and Markov Representation,” Probability Theory and Related Fields, Vol. 76, No. 2, 1987, pp. 243-253. doi:10.1007/BF00319986

[26] R. Bowen, “Equillibrium States and the Ergodic Theory of Anosov Diffeomorthisms,” Lecture Notes in Mathematics, Vol. 470, Springer, Berlin, 1975.

[27] F. Comets, R. Fernández and P. A. Ferrari, “Processes with Long Memory: Regenerative Construction and Perfect Simulation,” Annals of Applied Probability, Vol. 12, No. 3, 2002, pp. 921-943. doi:10.1214/aoap/1031863175

[28] A. Galves and E. Locherbach, “Stochastic Chains with Memory of Variable Length,” TICSP Series, Vol. 38, 2008, pp. 117-133.

[29] J. Rissanen, “A Universal Data Compression System,” IEEE Transactional on Information Theory, Vol. 29, No. 5, 1983, pp. 656-664. doi:10.1109/TIT.1983.1056741

[30] R. Vilela Mendes, R. Lima and T. Araújo, “A Process-Reconstruction Analysis of Market Fluctuations,” International Journal of Theoretical Applied Finance, Vol. 5, No. 8, 2002, pp. 797-821. doi:10.1142/S0219024902001730

[31] S. I. Resnick, “On the Foundations of Multivariate Heavy Tail Analysis,” Journal of Applied Probability, Vol. 41, 2004, pp. 191-212. doi:10.1239/jap/1082552199

[32] A. Brandt, “The Stochastic Equation with Stationary Coefficients,” Advances in Applied Probability, Vol. 18, No. 1, 1986, pp. 211-220. doi:10.2307/1427243

[33] A. A. Borovkov and K. A. Borovkov, “Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions,” Encyclopedia of Mathematics and Its Applications, Vol. 118, Cambridge University Press, Cambridge, 2008.

[34] M. Kobus, “Generalized Poisson Distributions as Limits for Sums of Arrays of Dependent Random Vectors,” Journal of Multivariate Analysis, Vol. 52, No. 2, 1995, pp. 199-244. doi:10.1006/jmva.1995.1011

[35] P. Billingsley, “Convergence of Probability Measures,” John Wiley & Sons, New York, 1968.

[36] Z. S. Szewczak, “A Central Limit Theorem for Strictly Stationary Sequences in Terms of Slow Variation in the Limit,” Probability and Mathematical Statistics, Vol. 18, No. 2, 1998, pp. 359-368.

[37] H. Oodaria and K. Yoshihara, “The Law of the Iterated Logarithm for Stationary Processes Satisfying Mixing Conditions,” Kodai Mathematical Seminar Reports, Vol. 23, No. 3, 1971, pp. 311-334. doi:10.2996/kmj/1138846370

[1] W. T. Coffey, Yu. P. Kalmykov and J. T. Waldron, “The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering,” 2nd Edition, World Scientific Series in Contemporary Chemical Physics, Vol. 14, World Scientific Publishing Company, Singapore City, 2004.

[2] N. G. Van Kampen, “Stochastic Processes in Physics and Chemistry,” 3th Edition, North-Holland Personal Library, Amsterdam, 2003.

[3] O. E. Barndorff-Nielsen and N. Shephard, “Non-Gaussian Ornstein-Uhlenbeck-Based Models and Some of Their Uses in Financial Economics (with Discussion),” Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 63, No. 2, 2001, pp. 167-241. doi:10.1111/1467-9868.00282

[4] S. C. Kou and X. S. Xie, “Generalized Langevin Equation with Fractional Gaussian Noise: Sub-Diffusion within a Single Protein Molecule,” Physical Review Letters, Vol. 93, 2004, Article ID: 18.

[5] D. Buraczewski, E. Damek and M. Mirek, “Asymptotics of Stationary Solutions of Multivariate Stochastic Recursions with Heavy Tailed Inputs and Related Limit Theorems,” Stochastic Processes and Their Applications, Vol. 122, No. 1, 2012, pp. 42-67. doi:10.1016/j.spa.2011.10.010

[6] H. Larralde, “A First Passage Time Distribution for a Discrete Version of the Ornstein-Uhlenbeck Process,” Journal of Physics A, Vol. 37, No. 12, 2004, pp. 3759-3767. doi:10.1088/0305-4470/37/12/003

[7] H. Larralde, “Statistical Properties of a Discrete Version of the Ornstein-Uhlenbeck Process,” Physical Review E, Vol. 69, 2004, Article ID: 027102.

[8] E. Renshaw, “The Discrete Uhlenbeck-Ornstein Process,” Journal of Applied Probability, Vol. 24, No. 4, 1987, pp. 908-917. doi:10.2307/3214215

[9] M. Lefebvre and J.-L. Guilbault, “First Hitting Place Probabilities for a Discrete Version of the Ornstein-Uhlenbeck Process,” International Journal of Mathematics and Mathematical Sciences, Vol. 2009, 2009, Article ID: 909835.

[10] A. Novikov and N. Kordzakhia, “Martingales and First Passage Times of AR(1) Sequences,” Stochastics, Vol. 80, No. 2-3, 2008, pp. 197-210. doi:10.1080/17442500701840885

[11] P. Embrechts and C. M. Goldie, “Perpetuities and Random Equations,” Proceedings of 5th Prague Symposium, Physica, Heidelberg, 1993, pp. 75-86.

[12] R. F. Engle, “ARCH. Selected Readings,” Oxford University Press, Oxford, 1995.

[13] S. T. Rachev and G. Samorodnitsky, “Limit Laws for a Stochastic Process and Random Recursion Arising in Probabilistic Modelling,” Advances in Applied Probability, Vol. 27, No. 1, 1995, pp. 185-202.

[14] W. Vervaat, “On a Stochastic Difference Equations and a Representation of Non-Negative Infinitely Divisible Random Variables,” Advances in Applied Probability, Vol. 11, No. 4, 1979, pp. 750-783. doi:10.2307/1426858

[15] Y. Ephraim and N. Merhav, “Hidden Markov Processes,” IEEE Transactions on Information Theory, Vol. 48, No. 6, 2002, pp. 1518-1569. doi:10.1109/TIT.2002.1003838

[16] D. Hay, R. Rastegar and A. Roitershtein, “Multivariate Linear Recursions with Markov-Dependent Coefficients,” Journal of Multivariate Analysis, Vol. 102, No. 3, 2011, pp. 521-527. doi:10.1016/j.jmva.2010.10.011

[17] R. Stelzer, “Multivariate Markov-Switching ARMA Processes with Regularly Varying Noise,” Journal of Multivariate Analysis, Vol. 99, No. 6, 2008, pp. 1177-1190. doi:10.1016/j.jmva.2007.07.001

[18] P. Eloe, R. H. Liu, M. Yatsuki, G. Yin and Q. Zhang, “Optimal Selling Rules in a Regime-Switching Exponential Gaussian Diffusion Model,” SIAM Journal of Applied Mathematics, Vol. 69, No. 3, 2008, pp. 810-829. doi:10.1137/060652671

[19] E. F. Fama, “The Behavior of Stock Market Prices,” Journal of Business, Vol. 38, No. 1, 1965, pp. 34-105. doi:10.1086/294743

[20] J. D. Hamilton, “A New Approach to the Economic Analysis of Non-Stationary Time Series and the Business Cycle,” Econometrica, Vol. 57, No. 2, 1989, pp. 357-384. doi:10.2307/1912559

[21] S. Lalley, “Regenerative Representation for One-Dimensional Gibbs States,” Annals of Probability, Vol. 14, No. 4, 1986, pp. 1262-1271. doi:10.1214/aop/1176992367

[22] R. Fernández and G. Maillard, “Chains with Complete Connections and One-Dimensional Gibbs Measures,” Electronic Journal of Probability, Vol. 9, 2004, pp. 145-176. doi:10.1214/EJP.v9-149

[23] R. Fernández and G. Maillard, “Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties,” Journal of Statistical Physics, Vol. 118, No. 3-4, 2005, pp. 555-588. doi:10.1007/s10955-004-8821-5

[24] M. Iosifescu and S. Grigorescu, “Dependence with Complete Connections and Its Applications,” Cambridge Tracts in Mathematics, Vol. 96, Cambridge University Press, Cambridge, 2009.

[25] H. Berbee, “Chains with Infinite Connections: Uniqueness and Markov Representation,” Probability Theory and Related Fields, Vol. 76, No. 2, 1987, pp. 243-253. doi:10.1007/BF00319986

[26] R. Bowen, “Equillibrium States and the Ergodic Theory of Anosov Diffeomorthisms,” Lecture Notes in Mathematics, Vol. 470, Springer, Berlin, 1975.

[27] F. Comets, R. Fernández and P. A. Ferrari, “Processes with Long Memory: Regenerative Construction and Perfect Simulation,” Annals of Applied Probability, Vol. 12, No. 3, 2002, pp. 921-943. doi:10.1214/aoap/1031863175

[28] A. Galves and E. Locherbach, “Stochastic Chains with Memory of Variable Length,” TICSP Series, Vol. 38, 2008, pp. 117-133.

[29] J. Rissanen, “A Universal Data Compression System,” IEEE Transactional on Information Theory, Vol. 29, No. 5, 1983, pp. 656-664. doi:10.1109/TIT.1983.1056741

[30] R. Vilela Mendes, R. Lima and T. Araújo, “A Process-Reconstruction Analysis of Market Fluctuations,” International Journal of Theoretical Applied Finance, Vol. 5, No. 8, 2002, pp. 797-821. doi:10.1142/S0219024902001730

[31] S. I. Resnick, “On the Foundations of Multivariate Heavy Tail Analysis,” Journal of Applied Probability, Vol. 41, 2004, pp. 191-212. doi:10.1239/jap/1082552199

[32] A. Brandt, “The Stochastic Equation with Stationary Coefficients,” Advances in Applied Probability, Vol. 18, No. 1, 1986, pp. 211-220. doi:10.2307/1427243

[33] A. A. Borovkov and K. A. Borovkov, “Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions,” Encyclopedia of Mathematics and Its Applications, Vol. 118, Cambridge University Press, Cambridge, 2008.

[34] M. Kobus, “Generalized Poisson Distributions as Limits for Sums of Arrays of Dependent Random Vectors,” Journal of Multivariate Analysis, Vol. 52, No. 2, 1995, pp. 199-244. doi:10.1006/jmva.1995.1011

[35] P. Billingsley, “Convergence of Probability Measures,” John Wiley & Sons, New York, 1968.

[36] Z. S. Szewczak, “A Central Limit Theorem for Strictly Stationary Sequences in Terms of Slow Variation in the Limit,” Probability and Mathematical Statistics, Vol. 18, No. 2, 1998, pp. 359-368.

[37] H. Oodaria and K. Yoshihara, “The Law of the Iterated Logarithm for Stationary Processes Satisfying Mixing Conditions,” Kodai Mathematical Seminar Reports, Vol. 23, No. 3, 1971, pp. 311-334. doi:10.2996/kmj/1138846370