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State-dependent jump processes: Itô-Stratonovich interpretations, potential, and transient solutions

Publication ,  Journal Article
Bartlett, MS; Porporato, A
Published in: Physical Review E
November 26, 2018

The abrupt changes that are ubiquitous in physical systems often are well characterized by shot noise with a state-dependent recurrence frequency and jump amplitude. For such state-dependent behavior, we derive the transition probability for both the Itô and Stratonovich jump interpretations and subsequently use the transition probability to pose a master equation for the jump process. For exponentially distributed inputs, we present a class of transient solutions, as well as a generic steady-state solution in terms of a potential function and the Pope-Ching formula. These results allow us to describe state-dependent jumps in a double-well potential for steady-state particle dynamics, as well as transient salinity dynamics forced by state-dependent jumps. Both examples showcase a stochastic description that is more general than the limiting case of Brownian motion to which the jump process defaults in the limit of infinitely frequent and small jumps. Accordingly, our analysis may be used to explore a continuum of stochastic behavior from infrequent, large jumps to frequent, small jumps approaching a diffusion process.

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Published In

Physical Review E

DOI

EISSN

2470-0053

ISSN

2470-0045

Publication Date

November 26, 2018

Volume

98

Issue

5

Related Subject Headings

  • 51 Physical sciences
  • 49 Mathematical sciences
  • 40 Engineering
 

Citation

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Bartlett, M. S., & Porporato, A. (2018). State-dependent jump processes: Itô-Stratonovich interpretations, potential, and transient solutions. Physical Review E, 98(5). https://doi.org/10.1103/PhysRevE.98.052132
Bartlett, M. S., and A. Porporato. “State-dependent jump processes: Itô-Stratonovich interpretations, potential, and transient solutions.” Physical Review E 98, no. 5 (November 26, 2018). https://doi.org/10.1103/PhysRevE.98.052132.
Bartlett, M. S., and A. Porporato. “State-dependent jump processes: Itô-Stratonovich interpretations, potential, and transient solutions.” Physical Review E, vol. 98, no. 5, Nov. 2018. Scopus, doi:10.1103/PhysRevE.98.052132.

Published In

Physical Review E

DOI

EISSN

2470-0053

ISSN

2470-0045

Publication Date

November 26, 2018

Volume

98

Issue

5

Related Subject Headings

  • 51 Physical sciences
  • 49 Mathematical sciences
  • 40 Engineering