Advertisement

Some Results on Brownian Motion Perturbed by Alternating Jumps in Biological Modeling

  • Antonio Di Crescenzo
  • Antonella Iuliano
  • Barbara Martinucci
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8111)

Abstract

We consider the model of random evolution on the real line consisting in a Brownian motion perturbed by alternating jumps. We give the probability density of the process and pinpoint a connection with the limit density of a telegraph process subject to alternating jumps. We study the first-crossing-time probability in two special cases, in the presence of a constant upper boundary.

Keywords

Brownian motion alternating jumps first-crossing time 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berg, H.C., Brown, D.A.: Chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature 239, 500–504 (1972)CrossRefGoogle Scholar
  2. 2.
    Borodin, A.N., Salminen, P.: Handbook of Brownian motion – facts and formulae. Probability and its Applications. Birkhäuser Verlag, Basel (1996)CrossRefzbMATHGoogle Scholar
  3. 3.
    Buonocore, A., Di Crescenzo, A., Martinucci, B.: A stochastic model for the rising phase of myosin head displacements along actin filaments. In: Capasso, V. (ed.) 5th ESMTB Conference on Mathematical Modelling & Computing in Biology and Medicine, pp. 121–126. MIRIAM Proj., 1, Esculapio, Bologna (2003)Google Scholar
  4. 4.
    Buonocore, A., Di Crescenzo, A., Martinucci, B., Ricciardi, L.M.: A stochastic model for the stepwise motion in actomyosin dynamics. Sci. Math. Japon. 58, 245–254 (2003)zbMATHGoogle Scholar
  5. 5.
    Di Crescenzo, A., Martinucci, B.: On the generalized telegraph process with deterministic jumps. Methodol. Comput. Appl. Probab. 15, 2012–2235 (2013)Google Scholar
  6. 6.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of Integral Transforms, vol. I. McGraw-Hill Publisher, New York (1954)Google Scholar
  7. 7.
    Garcia, R., Moss, F., Nihongi, A., Strickler, J.R., Göller, S., Erdmann, U., Schimansky-Geier, L., Sokolov, I.M.: Optimal foraging by zooplankton within patches: the case of Daphnia. Math. Biosci. 207, 165–188 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lutscher, F., Pachepsky, E., Lewis, M.A.: The effect of dispersal patterns on stream populations. SIAM J. Appl. Math. 65, 1305–1327 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Orsingher, E.: Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff’s laws. Stoch. Proc. Appl. 34, 49–66 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Othmer, H.G., Dunbar, S.R., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ricciardi, L.M., Di Crescenzo, A., Giorno, V., Nobile, A.G.: An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling. Math. Japon. 50(2), 247–322 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Sacerdote, L., Sirovich, R.: Multimodality of the interspike interval distribution in a simple jump-diffusion model. Sci. Math. Japon. 58, 307–322 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Sacerdote, L., Sirovich, R.: A Wiener process with inverse Gaussian time distributed jumps as a model for neuronal activity. In: Capasso, V. (ed.) 5th ESMTB Conference on Mathematical Modelling & Computing in Biology and Medicine, pp. 134–140. MIRIAM Proj., 1, Esculapio, Bologna (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Antonio Di Crescenzo
    • 1
  • Antonella Iuliano
    • 2
  • Barbara Martinucci
    • 1
  1. 1.Dipartimento di MatematicaUniversità di SalernoFiscianoItaly
  2. 2.CNR, Istituto per le Applicazioni del Calcolo (IAC)NaplesItaly

Personalised recommendations