The transition from ergodic to explosive behavior in a family of stochastic differential equations

Jeremiah Birrell; David P. Herzog; Jan Wehr

doi:10.1016/j.spa.2011.12.014

The transition from ergodic to explosive behavior in a family of stochastic differential equations

Jeremiah Birrell
, David P. Herzog
, Jan Wehr

Mathematics

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

14 Scopus citations

Abstract

We study a family of quadratic, possibly degenerate, stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, Hrmander's hypoellipticity theorem, and geometric control theory, we find a critical parameter value α ₁=α ₂ such that when α ₂>α ₁ the system is ergodic and when α ₂<α ₁ solutions are not defined for all times.

Original language	English (US)
Pages (from-to)	1519-1539
Number of pages	21
Journal	Stochastic Processes and their Applications
Volume	122
Issue number	4
DOIs	https://doi.org/10.1016/j.spa.2011.12.014
State	Published - Apr 2012

Keywords

Degenerate noise
Ergodic property
Geometric control theory
Invariant (probability) measures
Lyapunov functions
Stochastic differential equations

ASJC Scopus subject areas

Statistics and Probability
Modeling and Simulation
Applied Mathematics

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10.1016/j.spa.2011.12.014

Cite this

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title = "The transition from ergodic to explosive behavior in a family of stochastic differential equations",

abstract = "We study a family of quadratic, possibly degenerate, stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, Hrmander's hypoellipticity theorem, and geometric control theory, we find a critical parameter value α 1=α 2 such that when α 2>α 1 the system is ergodic and when α 2<α 1 solutions are not defined for all times.",

keywords = "Degenerate noise, Ergodic property, Geometric control theory, Invariant (probability) measures, Lyapunov functions, Stochastic differential equations",

author = "Jeremiah Birrell and Herzog, \{David P.\} and Jan Wehr",

note = "Funding Information: We were introduced to the general circle of problems that this paper addresses by K. Gawȩdzki. We thank M. Hairer for suggesting [18] and J. Zabczyk for a reference to [30] . We thank the referee for his/her comments and suggestions, and especially for pointing us to the works [11,2{\textendash}4] to help with studying the critical case α 1 = α 2 . J.W. was partially supported by the NSF grant DMS 1009508 . D.H. gratefully acknowledges support from an NSF VIGRE grant through the Mathematics Graduate Program at the University of Arizona. ",

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doi = "10.1016/j.spa.2011.12.014",

language = "English (US)",

volume = "122",

pages = "1519--1539",

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AU - Birrell, Jeremiah

AU - Herzog, David P.

AU - Wehr, Jan

N1 - Funding Information: We were introduced to the general circle of problems that this paper addresses by K. Gawȩdzki. We thank M. Hairer for suggesting [18] and J. Zabczyk for a reference to [30] . We thank the referee for his/her comments and suggestions, and especially for pointing us to the works [11,2–4] to help with studying the critical case α 1 = α 2 . J.W. was partially supported by the NSF grant DMS 1009508 . D.H. gratefully acknowledges support from an NSF VIGRE grant through the Mathematics Graduate Program at the University of Arizona.

PY - 2012/4

Y1 - 2012/4

N2 - We study a family of quadratic, possibly degenerate, stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, Hrmander's hypoellipticity theorem, and geometric control theory, we find a critical parameter value α 1=α 2 such that when α 2>α 1 the system is ergodic and when α 2<α 1 solutions are not defined for all times.

AB - We study a family of quadratic, possibly degenerate, stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, Hrmander's hypoellipticity theorem, and geometric control theory, we find a critical parameter value α 1=α 2 such that when α 2>α 1 the system is ergodic and when α 2<α 1 solutions are not defined for all times.

KW - Degenerate noise

KW - Ergodic property

KW - Geometric control theory

KW - Invariant (probability) measures

KW - Lyapunov functions

KW - Stochastic differential equations

UR - https://www.scopus.com/pages/publications/84857961858

UR - https://www.scopus.com/pages/publications/84857961858#tab=citedBy

U2 - 10.1016/j.spa.2011.12.014

DO - 10.1016/j.spa.2011.12.014

M3 - Article

AN - SCOPUS:84857961858

SN - 0304-4149

VL - 122

SP - 1519

EP - 1539

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 4

ER -

The transition from ergodic to explosive behavior in a family of stochastic differential equations

Abstract

Keywords

ASJC Scopus subject areas

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