Abstract

We consider a planar dynamical system generated by two stable linear vector fields with distinct fixed points and random switching between them. We characterize singularities of the invariant density in terms of the switching rates and contraction rates. We prove boundedness away from those singularities. We also discuss some motivating biological examples.

Keywords

  1. randomly switched ODEs
  2. invariant densities
  3. singularities
  4. integral equations

MSC codes

  1. 93E15
  2. 93C30
  3. 37A50
  4. 60J25

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
Y. Bakhtin and T. Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity, 25 (2012), pp. 2937--2952.
2.
Y. Bakhtin, T. Hurth, S. D. Lawley, and J. C. Mattingly, Smooth invariant densities for random switching on the torus, Nonlinearity, 31 (2018), pp. 1331--1350.
3.
Y. Bakhtin, T. Hurth, and J. C. Mattingly, Regularity of invariant densities for 1d-systems with random switching, Nonlinearity, 28 (2015), pp. 3755--3787.
4.
M. Benaïm, S. Le Borgne, F. Malrieu, and P.-A. Zitt, Quantitative ergodicity for some switched dynamical systems, Electron. Commun. Probab., 17 (2012), 56, https://doi.org/10.1214/ECP.v17-1932.
5.
M. Benaïm, S. Le Borgne, F. Malrieu, and P.-A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), pp. 1040--1075, https://doi.org/10.1214/14-AIHP619.
6.
P. C. Bressloff, Stochastic switching in biology: from genotype to phenotype, J. Phys. A, 50 (2017), 133001.
7.
P. C. Bressloff and S. D. Lawley, Dynamically active compartments coupled by a stochastically gated gap junction, J. Nonlinear Sci, 27 (2017), pp. 1487--1512.
8.
M. H. A. Davis, Markov models and optimization, Monogr. Statist. Appl. Probab. 49, Chapman & Hall, London, 1993.
9.
A. Faggionato, D. Gabrielli, and M. Ribezzi Crivellari, Non-equilibrium thermodynamics of piecewise deterministic Markov processes, J. Stat. Phys., 137 (2009), pp. 259--304, https://doi.org/10.1007/s10955-009-9850-x.
10.
R. Hersh, The birth of random evolutions, Math. Intelligencer, 25 (2003), pp. 53--60, https://doi.org/10.1007/BF02985641.
11.
C. Jia, M. Q. Zhang, and H. Qian, Emergent Lévy behavior in single-cell stochastic gene expression, Phys. Rev. E (3), 96 (2017), 040402.
12.
T. B. Kepler and T. C. Elston, Stochasticity in transcriptional regulation: Origins, consequences, and mathematical representations, Biophys. J., 81 (2001), pp. 3116--3136.
13.
S. D. Lawley, Boundary value problems for statistics of diffusion in a randomly switching environment: PDE and SDE perspectives, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1410--1433, https://doi.org/10.1137/15M1038426.
14.
S. D. Lawley, A probabilistic analysis of volume transmission in the brain, SIAM J. Appl. Math., 78 (2018), pp. 942--962, https://doi.org/10.1137/16M1102252.
15.
S. D. Lawley, J. C. Mattingly, and M. C. Reed, Stochastic switching in infinite dimensions with applications to random parabolic PDE, SIAM J. Math. Anal., 47 (2015), pp. 3035--3063, https://doi.org/10.1137/140976716.
16.
F. Malrieu, Some simple but challenging Markov processes, Ann. Fac. Sci. Toulouse Math. (6), 24 (2015), pp. 857--883, https://doi.org/10.5802/afst.1468.
17.
J. Paulsson, Models of stochastic gene expression, Phys. Life Rev., 2 (2005), pp. 157--175.
18.
M. W. Smiley and S. R. Proulx, Gene expression dynamics in randomly varying environments, J. Math. Biol., 61 (2010), pp. 231--251.
19.
G. G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Stoch. Model. Appl. Probab. 63, Springer, New York, 2010, https://doi.org/10.1007/978-1-4419-1105-6.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 1917 - 1958
ISSN (online): 1536-0040

History

Submitted: 4 September 2020
Accepted: 8 July 2021
Published online: 11 October 2021

Keywords

  1. randomly switched ODEs
  2. invariant densities
  3. singularities
  4. integral equations

MSC codes

  1. 93E15
  2. 93C30
  3. 37A50
  4. 60J25

Authors

Affiliations

Funding Information

Division of Mathematical Sciences https://doi.org/10.13039/100000121 : DMS-1460595, DMS-1613337, DMS-1811444, DMS-1814832, DMS-1944574
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung https://doi.org/10.13039/501100001711 : 200021-175728/1

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.