TY - JOUR
T1 - The transition from ergodic to explosive behavior in a family of stochastic differential equations
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
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U2 - 10.1016/j.spa.2011.12.014
DO - 10.1016/j.spa.2011.12.014
M3 - Article
AN - SCOPUS:84857961858
VL - 122
SP - 1519
EP - 1539
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
SN - 0304-4149
IS - 4
ER -