A finite difference method for piecewise deterministic processes with memory

References

• M. Annunziato A finite difference method for piecewise deterministic Markov processes. http://arxiv.org/abs/math.NA/0606588.
   Google Scholar
• Annunziato , M. 2002 . Non‐gaussian equilibrium distribution arising from the Langevin equation . Phys. Rev. E , 65 21113–1–6
   Google Scholar
• Annunziato , M. , Grigolini , P. and Riccardi , J. 2000 . Fluctuation‐dissapation process without a time scale . Phys. Rev. E , 61 : 4801 – 4808 .
   Web of Science ®Google Scholar
• Annunziato , M. , Grigolini , P. and West , B. J. 2001 . Canonical and noncanonical equilibrium distribution . Phys. Rev. E , 64 011107–1–13
   Google Scholar
• Bena , I. , Van den Broeck , C. , Kawai , R. and Lindenberg , K. 2003 . Drift by dichotomous Markov noise . Phys. Rev. E , 68 041111–1–12
   Google Scholar
• Bologna , M. , Grigolini , P. and West , B. J. 2002 . Strange kinetics: conflict between density and trajectory description . Chem. Phys. , 284 : 115 – 128 .
   Google Scholar
• Brunner , H. and Va der Houwen , P.J. 1986 . The numerical solution of Volterra equation North‐Holland
   Google Scholar
• Budini , A. and Cáceres , O. M. 2004 . Functional characterization of generalized Langevin equations . J. Phys. A: Math. Gen. , 37 : 5959 – 5981 .
   Google Scholar
• Cáceres , O. M. 2003 . Computing a non‐Maxwellian velocity distribution from first principles . Phys. Rev E , 67 : 016102 – 016104 .
   Web of Science ®Google Scholar
• Costa , O. L V. and Dufour , F. 2003 . On the Poisson equation for piecewise‐deterministic Markov processes . SIAM J. Control Optim. , 42 : 985 – 1001 .
   Web of Science ®Google Scholar
• Cox , D.R. 1955 . The analysis of non‐Markovian stochastic processes by the inclusion of supplementary variables . Proc. Camb. Phil. Soc. , 51 : 433 – 441 .
   Google Scholar
• Davis , M.H.A. 1984 . Piecewise‐deterministic Markov processes: A general class of non‐diffusion stochastic models . J. of the Royal Stat. Soc. Series B , 46 : 353 – 388 .
   Google Scholar
• Davis , M.H.A. 1993 . Markov Models and Optimizations , London : Chapman & Hall/CRC .
   Google Scholar
• Filliger , R. and Hongler , M. O. 2004 . Supersymmetry in random two‐velocity processes . Physica A , 332 : 141 – 150 .
   Web of Science ®Google Scholar
• Gardiner , C. W. 1985 . Handbook of Stochastic Methods , 2nd Ed. , Berlin Heidelberg New York : Springer‐Verlag .
   Google Scholar
• Hillen , T. and Othmer , H. G. 2000 . The diffusion limit of transport equations derived from velocity‐jump processes . SIAM J. Appl Math. , 61 : 751 – 775 .
   Web of Science ®Google Scholar
• Horsthemke , W. 1999 . Spatial instabilities in reaction random walks with direction‐independent kinetics . Phys. Rev E , 60 : 2651 – 2663 .
   PubMed Web of Science ®Google Scholar
• Jakeman , E. and Renshaw , E. 1987 . Correlated random‐walk model for scattering . J. Opt. Soc. Am. A , 4 : 1206 – 1212 .
   Google Scholar
• Jakeman , E. and Ridley , K. D. 1999 . The statistics of a filtered telegraph signal . J. Phys. A , 32 : 8803 – 8821 .
   Google Scholar
• Kitahara , K. 1980 . Phase diagrams of noise induced transitions . Prog. Theor. Phys. , 64 : 1233 – 1247 .
   Web of Science ®Google Scholar
• Veque , Randall J. Le . 1992 . Numerical Methods for conservation Laws , Birkhäuser .
   Google Scholar
• Mankin , R. , Ainsaar , A. , Haljas , A. and Reiter , E. 2002 . Trichotomous‐noise‐induced catastrophic shifts in symbiotic ecosystems . Phys. Rev. E , 65 051108–1–9
   Google Scholar
• Masoliver , J. and Porrà , J. M. 1993 . Harmonic oscillators driven by colored noise: Crossovers, resonances, and spectra . Phys. Rev. E , 48 : 4309 – 4319 .
   Web of Science ®Google Scholar
• Milstein , G.N. and Repin , Yu.M. 1972 . Action of Markov process on a system of differential equations . Differential Equations , 5 : 1010 – 1019 . Translated from: Differentsial Uravneniya 5, 1969 pp. 1371–1384
   Google Scholar
• Mori , H. 1958 . Statistical‐mechanical theory of transport in fluids . Phys. Rev. , 112 : 1829 – 1842 .
   Web of Science ®Google Scholar
• Morita , A. 1990 . Free Brownian motion of a particle driven by a dichotomous random force . Phys. Rev. A , 41 : 754 – 760 .
   PubMed Web of Science ®Google Scholar
• Morton , K. W. and Mayers , D. F. 1994 . Numerical Solution of Partial Differential Equations , Cambridge University Press .
   Google Scholar
• Othmer , H. G. , Dunba , S. R. and Alt , W. 1988 . Models of dispersal in biological systems . J. Math. Biol. , 26 : 263 – 298 .
   PubMed Web of Science ®Google Scholar
• Pawula , R. F. 1987 . Approximating distributions from moments . Phys.Rev.A , 36 : 4996 – 5007 .
   PubMed Web of Science ®Google Scholar
• Pawula , R. F. and Rice , O. 1986 . On filtered binary processes . IEEE Trans. Inf. Th. , IT‐32 : 63 – 72 .
   Google Scholar
• Perger , W. F. , Bhalla , A. and Nardin , M. 1993 . A numerical evaluator for the generalized hypergeometric 1 series . Comp. Phys. Commun. , 77 : 249 – 254 .
   Google Scholar
• B. E. Barrowes Central File Exchange). This routine uses a gamma function for complex arguments, see P. Godfrey http://my.fit.edu/∼gabdo/gamma.m.
   Google Scholar
• Van den Broeck , C. and P&H;auml;nggi , P. 1984 . Activation rates for nonlinear stochastic flows driven by non‐Gaussian noise . Phys. Rev. A , 30 : 2730 – 2736 .
   Google Scholar
• Wonham , W. M. 1959 . J. Electron. Control , 6 : 376 – 383 .
   Google Scholar
• Zwanzig , R. 1961 . “ (Boulder) ” . In Lectures in Theoretical Physics , Edited by: Broton , W.E. volume 3 , New York : Interscience .
   Google Scholar