Kolokoltsov, V. N. (Vasiliĭ Nikitich), Schilling, René L. and Tyukov, Alexei E.. (2002) Transience and non-explosion of certain stochastic Newtonian systems. Electronic Journal of Probability, Vol.7 (No.19). ISSN 1083-6489

Preview

PDF WRAP_Kolokotsov_Transient_EJP-2002-1320.pdf - Published Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (251Kb)

Abstract

We give sufficient conditions for non-explosion and transience of the solution (xt,pt) (in dimensions >= 3) to a stochastic Newtonian system of the form { dxdt = ptdt dpt = -δV(xt)/δx dt - δc(xt)/δx dξt where {ξt}t>=0 is a d-dimensional Lévy process, dξt is an Itô differential and c ∈ C2(Rd,Rd), V ∈ C2(Rd,R) such that V >= 0.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Mechanics, Stochastic processes
Journal or Publication Title:	Electronic Journal of Probability
Publisher:	Institute of Mathematical Statistics
ISSN:	1083-6489
Date:	2002
Volume:	Vol.7
Number:	No.19
Number of Pages:	19
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	Nottingham Trent University (NTU), Nuffield Foundation (NF), Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	RF175 (NTU), NAL/00056/G (NF), GR/R200892/01 (EPSRC)
References:	[1] S. Albeverio, A.Klar, Longtime behaviour of stochastic Hamiltonian systems: The multidimensional case, Potential Anal. 12 (2000), 281{297. [2] J. Azema, M. Kaplan-Dulfo, D. Revuz, Recurrence fine des processus de Markov, Ann. Inst. Henri Poincare (Ser. B) 2 (1966), 185{220. [3] S. Albeverio, A. Hilbert, V. N. Kolokoltsov, Transience for stochastically perturbed Newton systems, Stochastics and Stochastics Reports 60 (1997), 41{55. [4] S. Albeverio, A. Hilbert, V. N. Kolokoltsov, Estimates Uniform in Time for the Transition Probability of Diffusions with Small Drift and for Stochastically Perturbed Newton Equations, J. Theor. Probab. 12 (1999), 293{300. [5] S. Albeverio, A. Hilbert, E. Zehnder, Hamiltonian systems with a stochastic force: nonlinear versus linear and a Girsanov formula, Stochastics and Stochastics Reports 39 (1992), 159{188. [6] S. Albeverio, V. N. Kolokoltsov, The rate of escape for some Gaussian processes and the scattering theory for their small perturbations, Stochastic Processes and their Applications 67 (1997), 139{159. [7] S. E. Ethier, T. Kurtz, Markov Processes: Characterization and Convergence, Wiley, Series in Probab. Math. Stat., New York 1986. [8] M. Freidlin, Functional Integration and Partial Differential Equations, Princeton Univ. Press, Princeton, NJ 1985. [9] I. Gradshteyn, I. Ryzhik, Tables of Integrals, Series, and Products. Corrected and Enlarged Edition, Academic Press, San Diego, CA 1992 (4th ed.). [10] N. Jacob, Pseudo-differential operators and Markov processes, Akademie-Verlag, Mathematical Research vol. 94, Berlin 1996. [11] N. Jacob, R. L. Schilling, Levy-type processes and pseudo-differential operators, in: Barndorff-Nielsen, O. E. et al. (eds.) Levy processes: Theory and Applications, Birkhauser, Boston (2001), 139{167. [12] D. Khoshnevisan, Z. Shi, Chung's law for integrated Brownian motion, Trans. Am. Math. Soc. 350 (1998), 4253{4264. [13] V. N. Kolokoltsov, Stochastic Hamilton-Jacobi-Bellman equation and stochastic Hamiltonian systems, J. Dyn. Control Syst. 2 (1996), 299-379. [14] V. N. Kolokoltsov, Application of quasi-classical method to the investigation of the Belavkin quantum filtering equation, Mat. Zametki, 50 (1991), 153{156. (English transl.: Math. Notes 50 (1991), 1204{1206.) [15] V. N. Kolokoltsov, A note on the long time asymptotics of the Brownian motion with applications to the theory of quantum measurement, Potential Anal. 7 (1997), 759{764. [16] V. N. Kolokoltsov, The stochastic HJB Equation and WKB Method. In: J. Gunawardena (ed.), Idempotency, Cambridge Univ. Press, Cambridge 1998, 285{302. [17] V. N. Kolokoltsov, Localisation and analytic properties of the simplest quantum filtering equation, Rev. Math. Phys. 10 (1998), 801{828. [18] V. N. Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, Springer, Lecture Notes Math. 1724, Berlin 2000. [19] V. N. Kolokoltsov, R. L. Schilling, A. E. Tyukov, Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations, submitted. [20] V. N. Kolokoltsov, A. E. Tyukov, The rate of escape of -stable Ornstein-Uhlenbeck processes, Markov Process. Relat. Fields 7 (2001), 603{625. [21] R. Z. Khasminski, Ergodic properties of recurrent diffusion processes and stabilization of the solutions to the Cauchy problem for parabolic equations, Theor. Probab. Appl. 5 (1960), 179{195. [22] T. Kurtz, F. Marchetti, Averaging stochastically perturbed Hamiltonian systems. In: M. Cranston (ed.), Stochastic analysis, Proc. Summer Research Institute on Stochastic Analysis, Am. Math. Soc., Proc. Symp. Pure Math. 57, Providence, RI 1995, 93{114. [23] L. Markus, A. Weerasinghe, Stochastic Oscillators, J. Differ. Equations 71 (1998), 288{314. [24] L. Mehta, Random matrices (2nd ed.), Academic Press, Boston, MA 1991. [25] E. Nelson, Dynamical theories of Brownian motion, Princton University Press, Mathematical Notes, Princeton, NJ 1967. [26] K. Norita, The Smoluchowski-Kramers approximation for the stochastic Lienard equation with mean-field, Adv. Appl. Prob. 23 (1991), 303{316. [27] K. Norita, Asymptotic behavior of velocity process in the Smoluchowski-Kramers approximation for stochastic differential equations, Adv. Appl. Prob. 23 (1991), 317{326. [28] S. Olla, S. Varadhan, Scaling limit for interacting Ornstein-Uhlenbeck processes, Commun. Math. Phys. 135 (1991), 355{378. [29] S. Olla, S. Varadhan, H. Yau, Hydrodynamical limit for a Hamiltonian system with weak noise, Commun. Math. Phys. 155 (1993), 523{560. [30] P. Protter, Stochastic Integration and Differential Equations, Springer, Appl. Math. vol. 21, Berlin 1990. [31] R. L. Schilling, Growth and Holder conditions for the sample paths of Feller processes, Probab. Theor. Relat. Fields 112 (1998), 565{611. [32] A. Truman, H. Zhao, Stochastic Hamilton-Jacobi equation end related topics. In: A.M. Etheridge (ed.), Stochastic Partial Differential Equations, Cambridge Univ. Press, LMS Lecture Notes 276, Cambridge 1995, 287{303. [33] A. Truman, H. Zhao, The stochastic Hamilton-Jacobi equation, stochastic heat equation and Schrodinger equations. In: A. Truman, I.M. Davis, K.D. Elworthy (eds.), Stochastic Analysis and Applications, World Scientific, Singapore 1996, 441{464.
URI:	http://wrap.warwick.ac.uk/id/eprint/39394

Request changes to a record

Actions (login required)

View Item