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Stationary points in coalescing stochastic flows on R

Author

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  • Dorogovtsev, Andrey A.
  • Riabov, Georgii V.
  • Schmalfuß, Björn

Abstract

This work is devoted to long-time properties of the Arratia flow with drift – a stochastic flow on R whose one-point motions are weak solutions to a stochastic differential equation dX(t)=a(X(t))dt+dw(t) that move independently before the meeting time and coalesce at the meeting time. We study special modification of such flow that gives rise to a random dynamical system and thus allows to discuss stationary points. Existence of a unique stationary point is proved in the case of a strictly monotone Lipschitz drift by developing a variant of a pullback procedure. Connections between the existence of a stationary point and properties of a dual flow are discussed.

Suggested Citation

  • Dorogovtsev, Andrey A. & Riabov, Georgii V. & Schmalfuß, Björn, 2020. "Stationary points in coalescing stochastic flows on R," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4910-4926.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:8:p:4910-4926
    DOI: 10.1016/j.spa.2020.02.005
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    References listed on IDEAS

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    1. Harris, Theodore E., 1984. "Coalescing and noncoalescing stochastic flows in R1," Stochastic Processes and their Applications, Elsevier, vol. 17(2), pages 187-210, July.
    2. Goncharuk, Nataliya Yu. & Kotelenez, Peter, 1998. "Fractional step method for stochastic evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 73(1), pages 1-45, January.
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