×

Smooth invariant densities for random switching on the torus. (English) Zbl 1390.93828

Summary: We consider a random dynamical system obtained by switching between the flows generated by two smooth vector fields on the 2d-torus, with the random switchings happening according to a Poisson process. Assuming that the driving vector fields are transversal to each other at all points of the torus and that each of them allows for a smooth invariant density and no periodic orbits, we prove that the switched system also has a smooth invariant density, for every switching rate. Our approach is based on an integration by parts formula inspired by techniques from Malliavin calculus.

MSC:

93E15 Stochastic stability in control theory
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
37A50 Dynamical systems and their relations with probability theory and stochastic processes
60J25 Continuous-time Markov processes on general state spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adams R A 1975 {\it Sobolev Spaces}{\it (Pure and Applied Mathematics vol 65)} (New York: Academic)
[2] Bally V, Bavouzet M-P and Messaoud M 2007 Integration by parts formula for locally smooth laws and applications to sensitivity computations {\it Ann. Appl. Probab.}17 33-66 · Zbl 1139.60025
[3] Bass R F and Cranston M 1986 The Malliavin calculus for pure jump processes and applications to local time {\it Ann. Probab.}14 490-532 · Zbl 0595.60044
[4] Benaïm M, Colonius F and Lettau R 2017 Supports of invariant measures for piecewise deterministic Markov processes {\it Nonlinearity}30 3400-18 · Zbl 1384.93025
[5] Bakhtin Y and Hurth T 2012 Invariant densities for dynamical systems with random switching {\it Nonlinearity}25 2937-52 · Zbl 1251.93132
[6] Bakhtin Y, Hurth T and Mattingly J C 2015 Regularity of invariant densities for 1d-systems with random switching {\it Nonlinearity}28 3755-87 · Zbl 1327.93396
[7] Benaïm M, Le Borgne S, Malrieu F and Zitt P-A 2012 Quantitative ergodicity for some switched dynamical systems {\it Electron. Commun. Probab.}17 14 · Zbl 1347.60118
[8] Cloez B and Hairer M 2015 Exponential ergodicity for Markov processes with random switching {\it Bernoulli}21 505-36 · Zbl 1330.60094
[9] Constantine G M and Savits T H 1996 A multivariate faà di bruno formula with applications {\it Trans. Am. Math. Soc.}348 503-20 · Zbl 0846.05003
[10] Davis M H A 1993 {\it Markov Models and Optimization}{\it (Monographs on Statistics and Applied Probability vol 49)} (London: Chapman and Hall)
[11] Faggionato A, Gabrielli D and Crivellari M R 2009 Non-equilibrium thermodynamics of piecewise deterministic Markov processes {\it J. Stat. Phys.}137 259-304 · Zbl 1179.82108
[12] Hersh R 2003 The birth of random evolutions {\it Math. Intelligencer}25 53-60 · Zbl 1483.60038
[13] Kac M 1974 A stochastic model related to the telegrapher’s equation {\it Rocky Mt. J. Math.}4 497-509 · Zbl 0314.60052
[14] Katok A and Hasselblatt B 1995 {\it Introduction to the Modern Theory of Dynamical Systems}{\it (Encyclopedia of Mathematics and its Applications vol 54)} (Cambridge: Cambridge University Press) · Zbl 0878.58020
[15] Lawley S D, Mattingly J C and Reed M C 2015 Stochastic switching in infinite dimensions with applications to random parabolic PDE {\it SIAM J. Math. Anal.}47 3035-63 · Zbl 1338.35515
[16] Löcherbach E 2017 Absolute continuity of the invariant measure in piecewise deterministic Markov processes having degenerate jumps {\it Stoch. Proc. Appl.} accepted (https://doi.org/10.1016/j.spa.2017.08.011)
[17] Malrieu F 2015 Some simple but challenging Markov processes {\it Ann. Fac. Sci. Toulouse Math.}24 857-83 · Zbl 1333.60185
[18] Yin G G and Zhu C 2010 {\it Hybrid Switching Diffusions (Properties and Applications)}{\it (Stochastic Modelling and Applied Probability vol 63)} (New York: Springer)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.