Equivalence of physical and SRB measures in random dynamical systems

Alex Blumenthal; Lai Sang Young

doi:10.1088/1361-6544/aafaa8

Equivalence of physical and SRB measures in random dynamical systems

Alex Blumenthal, Lai Sang Young

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We give a geometric proof, offering a new and quite different perspective on an earlier result of Ledrappier and Young on random transformations (Ledrappier and Young 1988 Probab. Theory Relat. Fields 80 217-40). We show that under mild conditions, sample measures of random diffeomorphisms are SRB measures. As sample measures are the limits of forward images of stationary measures, they can be thought of as the analog of physical measures for deterministic systems. Our results thus show the equivalence of physical and SRB measures in the random setting, a hoped-for scenario that is not always true for deterministic maps.

Original language	English (US)
Pages (from-to)	1494-1524
Number of pages	31
Journal	Nonlinearity
Volume	32
Issue number	4
DOIs	https://doi.org/10.1088/1361-6544/aafaa8
State	Published - Mar 21 2019

Keywords

SRB measures
physical measures
random dynamics

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
General Physics and Astronomy
Applied Mathematics

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10.1088/1361-6544/aafaa8

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abstract = "We give a geometric proof, offering a new and quite different perspective on an earlier result of Ledrappier and Young on random transformations (Ledrappier and Young 1988 Probab. Theory Relat. Fields 80 217-40). We show that under mild conditions, sample measures of random diffeomorphisms are SRB measures. As sample measures are the limits of forward images of stationary measures, they can be thought of as the analog of physical measures for deterministic systems. Our results thus show the equivalence of physical and SRB measures in the random setting, a hoped-for scenario that is not always true for deterministic maps.",

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AU - Young, Lai Sang

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AB - We give a geometric proof, offering a new and quite different perspective on an earlier result of Ledrappier and Young on random transformations (Ledrappier and Young 1988 Probab. Theory Relat. Fields 80 217-40). We show that under mild conditions, sample measures of random diffeomorphisms are SRB measures. As sample measures are the limits of forward images of stationary measures, they can be thought of as the analog of physical measures for deterministic systems. Our results thus show the equivalence of physical and SRB measures in the random setting, a hoped-for scenario that is not always true for deterministic maps.

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KW - random dynamics

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Equivalence of physical and SRB measures in random dynamical systems

Abstract

Keywords

ASJC Scopus subject areas

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