Linear Stochastic Dyadic Model

AbstractWe discuss a stochastic interacting particles’ system connected to dyadic models of turbulence, defining suitable classes of solutions and proving their existence and uniqueness. We investigate the regularity of a particular family of solutions, called moderate, and we conclude with existence and uniqueness of invariant measures associated with such moderate solutions.

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On invariant measures of nonlinear Markov processes

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953393000310 ◽

1993 ◽

Vol 6 (4) ◽

pp. 385-406 ◽

Cited By ~ 4

Author(s):

N. U. Ahmed ◽

Xinhong Ding

Keyword(s):

Markov Process ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Markov Processes ◽

Existence And Uniqueness ◽

Invariant Measures ◽

Nonlinear Markov Processes

We consider a nonlinear (in the sense of McKean) Markov process described by a stochastic differential equations in Rd. We prove the existence and uniqueness of invariant measures of such process.

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Conditionally invariant measures for Anosov maps with small holes

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798117492 ◽

1998 ◽

Vol 18 (5) ◽

pp. 1049-1073 ◽

Cited By ~ 30

Author(s):

N. CHERNOV ◽

R. MARKARIAN ◽

S. TROUBETZKOY

Keyword(s):

Invariant Measure ◽

Existence And Uniqueness ◽

Invariant Measures ◽

Smooth Boundary ◽

Piecewise Smooth ◽

Piecewise Smooth Boundary ◽

Conditional Distributions ◽

The Past ◽

Set Of Points ◽

Small Holes

We study Anosov diffeomorphisms on surfaces in which some small ‘holes’ are cut. The points that are mapped into those holes disappear and never return. We assume that the holes are arbitrary open domains with piecewise smooth boundary, and their sizes are small enough. The set of points whose trajectories never enter holes under the past iterations of the map is a Cantor-like union of unstable fibers. We establish the existence and uniqueness of a conditionally invariant measure on this set, whose conditional distributions on unstable fibers are smooth. This generalizes previous works by Pianigiani, Yorke, and others.

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Existence and Uniqueness of Invariant Measures for Stochastic Reaction–Diffusion Equations in Unbounded Domains

Journal of Theoretical Probability ◽

10.1007/s10959-015-0606-z ◽

2015 ◽

Vol 29 (3) ◽

pp. 996-1026 ◽

Cited By ~ 12

Author(s):

Oleksandr Misiats ◽

Oleksandr Stanzhytskyi ◽

Nung Kwan Yip

Keyword(s):

Existence And Uniqueness ◽

Invariant Measures ◽

Unbounded Domains ◽

Reaction Diffusion ◽

Reaction Diffusion Equations ◽

Diffusion Equations ◽

Stochastic Reaction

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Existence and uniqueness of invariant measures for a class of transition semigroups on Hilbert spaces

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2008.12.031 ◽

2009 ◽

Vol 353 (2) ◽

pp. 497-507 ◽

Cited By ~ 3

Author(s):

Abdelhadi Es-Sarhir

Keyword(s):

Hilbert Spaces ◽

Existence And Uniqueness ◽

Invariant Measures ◽

Transition Semigroups

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Existence and Uniqueness of Invariant Measures for Stochastic Evolution Equations with Weakly Dissipative Drifts

Electronic Communications in Probability ◽

10.1214/ecp.v16-1643 ◽

2011 ◽

Vol 16 (0) ◽

pp. 447-457 ◽

Cited By ~ 10

Author(s):

Wei Liu ◽

Jonas Toelle

Keyword(s):

Existence And Uniqueness ◽

Invariant Measures ◽

Evolution Equations ◽

Stochastic Evolution ◽

Stochastic Evolution Equations

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On invariant measures for simple branching processes

Journal of Applied Probability ◽

10.2307/3211836 ◽

1971 ◽

Vol 8 (1) ◽

pp. 43-51 ◽

Cited By ~ 17

Author(s):

E. Seneta

Keyword(s):

Invariant Measure ◽

Existence And Uniqueness ◽

Invariant Measures ◽

Branching Processes ◽

Watson Process

This paper was initially motivated by a problem raised earlier by the author (Seneta (1969), Section 5.3) viz. that of the existence and uniqueness of an invariant measure for a supercritical Galton-Watson process with immigration; and, indeed, in the sequel we show that such a measure always exists, but is not in general unique.

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