Abstract
This paper addresses the long–term behaviour –in a suitable probabilistic sense– of map iteration in subsets of Banach spaces that are randomly perturbed. The law of the latter change in state is allowed to depend on state. We provide quite general conditions under which a stable fixed point of the deterministic map iteration induces an asymptotically stable ergodic measure of the Markov chain defined by the perturbed system, which is regarded as ‘persistence of stability’. The support of this invariant measure is characterized. The applicability of the framework is illustrated for deterministic dynamical systems that are subject to random interventions at fixed equidistant time points. In particular, we consider systems motivated by population dynamics: a model in ordinary differential equations, a model derived from a reaction–diffusion system and a class of delay equations.
Similar content being viewed by others
References
Alkurdi, T., Hille, S.C., van Gaans, O.: Ergodicity and stability of a dynamical system perturbed by impulsive random interventions. J. Math. Anal. Appl. 63, 480–494 (2013). doi:10.1016/j.jmaa.2013.05.047
Amann, H.: Dual semigroups and second order linear elliptic boundary value problems. Israel J. Math. 45 (2–3), 225–254 (1983)
Bàtkai, A., Piazzera, S.: Semigroups for Delay Equations, Research Notes in Mathematics vol. 10, Wellesley MA (2005)
Conway, E., Hoff, D., Smoller, J.: Large time behaviour of solutions of systems of nonlinear reaction–diffsion equations. SIAM, J. Appl. Math. 35 (1), 1–16 (1978)
Costa, O.L.V., Dufour, F.: Stability and ergodicity of piecewise deterministic Markov processses. SIAM, J. Control Optim. 47 (2), 1053–1077 (2008)
Da Prato, G., Gatarekb, D., Zabczykc, J.: Invariant measures for semilinear stochastic equations. Stochastic Anal. Appl. 10 (4), 387–408 (1992)
Davis, M.H.A.: Piecewise–deterministic Markov processes: a general class of non–diffusion stochastic models. J. R. Statist. Soc., Series B 46 (3), 353–388 (1984)
Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.-O.: Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Springer (1995)
Engel, K., Nagel, R.: One-parameter Semigroups for Linear Evolution Equations. Springer (2000)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley Interscience (2005)
Freedman, H.: Deterministic Mathematical Models in Population Ecology.M. Dekker. (1980)
van Gaans, O., Verduyn Lunel, S.M.: Long term behaviour of dichotomous stochastic differential equations in Hilbert spaces. Commun. Contemp. Math. 6 (3), 349–376 (2004)
Hendriks, A. J., Mulder, C.: Delayed logistic and Rosenzweig-MacArthur models with allometric parameter setting estimate population cycles at lower trophic levels well. Ecol. Complexity 9, 43–54 (2012)
Hille, S.C., Horbacz, K., Szarek, T.: Unique steady–state molecular distribution for a regulatory network with random bursting. submitted.
Hoppensteadt, F., Saleki, H., Skorokhod, A.: Discrete time semigroup transformations with random perturbations. J. Dyn. Diff. Eq. 9 (3), 463–505 (1997)
Ichikawa, A.: Semilinear stochastic evolution equations: Boundedness, stability and invariant measures. Stochast. 12, 1–39 (1984)
Istratescu, V.I.: Fixed Point Theory; An Introduction, Mathematics and Its Applications, Vol. 7. Reidel Publishing Company, Dordrecht (1981)
Jacobsen, M.: Point Process Theory and Applications; Marked Point and Piecewise Deterministic Processes. Birkhuser, Basel (2006)
Kot, M.: Discrete time travelling waves: ecological examples. J. Math. Biol. 30, 413–436 (1992)
Lasota, A., Mackey, M.C.: Cell division and the stability of cellular populations. J. Math. Biol. 38, 241–261 (1999)
Lasota, A., Myjak, J.: Fractals, Semifractals and Markov operators. Int. J. Bifurcation Chaos 9 (2), 307–325 (1999)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhauser, Basel (1995)
Meyn, S. P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Cambridge University Press, Cambridge (2009)
Myjak, J.: Andrzej Lasota’s selected results. Opuscula Math. 28 (4), 363–394 (2008)
Mőnch, H.: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear An. Theor. Methods Appl. 4 (5), 985–999 (1980)
Neubert, M.G., Kot, M., Lewis, M.A.: Dispersal and pattern formation in a discrete time predator–prey model. Theor. Pop. Biol. 48, 7–43 (1995)
Okubo, A.: Diffusion and Ecological Problems: Mathematical Models. Springer, Berlin (1980)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, Berlin, Heidelberg, Tokyo (1983)
Riedler, M. G., Thieullen, M., Wainrib, G.: Limit theorems for infinite dimensional piecewise–deterministic Markov processes. Applications to stochastic excitable membrane models. Electron. J. Probab. 17 (55), 1–48 (2012)
Rosenzweig, M.L., MacArthur, R.H.: Graphical representations and stability conditions of predator-prey interactions. Am. Nat. 97, 209–223 (1963)
Szarek, T.: Markov operators acting on Polish spaces. Ann. Polon. Math. 67, 247–257 (1997)
Szarek, T.: The stability of Markov operators on Polish spaces. Stud. Math. 134 (2), 145–152 (2000)
Szarek, T., Worm, D.: Ergodic measures of Markov semigroups with the e–property. Ergod. Th. Dynam. Sys. 32, 1117–1135 (2012)
Walter, W.: Ordinary Differential Equations, Graduate Texts in Mathematics. Springer (1998)
Worm, D.: Semigroups on Spaces of Measures, PhD thesis, Leiden University. http://www.math.leidenuniv.nl/scripties/WormThesis.pdf (2010)
Young, L.S., Masmoudi, N.: Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs. Commun. Math. Phys. 227, 461–481 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alkurdi, T., Hille, S.C. & van Gaans, O. Persistence of Stability for Equilibria of Map Iterations in Banach Spaces Under Small Random Perturbations. Potential Anal 42, 175–201 (2015). https://doi.org/10.1007/s11118-014-9429-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-014-9429-2