Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum

Friesen M, Kondratiev Y (2018)
JOURNAL OF STATISTICAL PHYSICS 171(5): 842-877.

Zeitschriftenaufsatz | Veröffentlicht| Englisch
 
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Autor*in
Friesen, Martin; Kondratiev, YuriUniBi
Abstract / Bemerkung
We study a spatial birth-and-death process on the phase space of locally finite configurations Gamma(+) x Gamma(-) over R-d. Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator L+(gamma(-)) + 1/epsilon L-, epsilon > 0. Here L(-)describes the environment process on Gamma(-) and L+(gamma(-)) describes the system process on Gamma(+), where gamma(-)indicates that the corresponding birth-and-death rates depend on another locally finite configuration gamma(-) is an element of Gamma(-). We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states mu(epsilon)(t) on Gamma(+) x Gamma(-). Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let mu(inv) be the invariant measure for the environment process on Gamma(-). In the main part of this work we establish the stochastic averaging principle, i. e. we prove that the marginal of mu(epsilon)(t) onto Gamma(+) converges weakly to an evolution of states on Gamma(+) associated with the averaged Markov birth-and-death operator (L) over bar = integral(Gamma)-L+(gamma(-))D mu(inv) (gamma(-)).
Stichworte
Spatial birth-and-death processes; Fokker-Planck equation; Ergodicity; Averaging; Random evolution
Erscheinungsjahr
2018
Zeitschriftentitel
JOURNAL OF STATISTICAL PHYSICS
Band
171
Ausgabe
5
Seite(n)
842-877
ISSN
0022-4715
eISSN
1572-9613
Page URI
https://pub.uni-bielefeld.de/record/2920304

Zitieren

Friesen M, Kondratiev Y. Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum. JOURNAL OF STATISTICAL PHYSICS. 2018;171(5):842-877.
Friesen, M., & Kondratiev, Y. (2018). Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum. JOURNAL OF STATISTICAL PHYSICS, 171(5), 842-877. doi:10.1007/s10955-018-2042-9
Friesen, M., and Kondratiev, Y. (2018). Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum. JOURNAL OF STATISTICAL PHYSICS 171, 842-877.
Friesen, M., & Kondratiev, Y., 2018. Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum. JOURNAL OF STATISTICAL PHYSICS, 171(5), p 842-877.
M. Friesen and Y. Kondratiev, “Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum”, JOURNAL OF STATISTICAL PHYSICS, vol. 171, 2018, pp. 842-877.
Friesen, M., Kondratiev, Y.: Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum. JOURNAL OF STATISTICAL PHYSICS. 171, 842-877 (2018).
Friesen, Martin, and Kondratiev, Yuri. “Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum”. JOURNAL OF STATISTICAL PHYSICS 171.5 (2018): 842-877.