Kolmogorov's Example (A Survey of Actions of Infinite-Dimensional Groups with an Invariant Probability Measure)

2004 ◽  
Vol 48 (2) ◽  
pp. 373-378 ◽  
Author(s):  
A. M. Vershik
Keyword(s):  
Probability Measure ◽  
Invariant Probability Measure ◽  
Infinite Dimensional

scholarly journals Closability of quadratic forms associated to invariant probability measures of SPDEs

2017 ◽  
Vol 20 (04) ◽  
pp. 1750023
Author(s):  
Michael Röckner ◽  
Feng-Yu Wang
Keyword(s):  
Probability Measure ◽  
Hamiltonian Systems ◽  
General Result ◽  
Quadratic Forms ◽  
Invariant Probability Measure ◽  
Markov Operator ◽  
Probability Measures ◽  
Infinite Dimensional ◽  
Stochastic Hamiltonian Systems ◽  
Integration By Parts Formula

By using the integration by parts formula of a Markov operator, the closability of quadratic forms associated to the corresponding invariant probability measure is proved. The general result is applied to the study of semilinear SPDEs, infinite-dimensional stochastic Hamiltonian systems, and semilinear SPDEs with delay.

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

scholarly journals Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

2019 ◽  
Vol 23 ◽  
pp. 797-802
Author(s):  
Raphaël Cerf ◽  
Joseba Dalmau
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Branching Process ◽  
Invariant Probability Measure ◽  
Classical Formula ◽  
Multitype Branching Process ◽  
Primitive Matrix ◽  
Watson Process

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

Invariant measures for interval maps with different one-sided critical orders

2013 ◽  
Vol 35 (3) ◽  
pp. 835-853 ◽  
Author(s):  
HONGFEI CUI ◽  
YIMING DING
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Mild Condition ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Jump Discontinuities ◽  
Point Set ◽  
Invent Math

AbstractFor an interval map whose critical point set may contain critical points with different one-sided critical orders and jump discontinuities, under a mild condition on critical orbits, we prove that it has an invariant probability measure which is absolutely continuous with respect to Lebesgue measure by using the methods of Bruin et al [Invent. Math. 172(3) (2008), 509–533], together with ideas from Nowicki and van Strien [Invent. Math. 105(1) (1991), 123–136]. We also show that it admits no wandering intervals.

Continuity of Lyapunov exponents for random two-dimensional matrices

2016 ◽  
Vol 37 (5) ◽  
pp. 1413-1442 ◽  
Author(s):  
CARLOS BOCKER-NETO ◽  
MARCELO VIANA
Keyword(s):  
Probability Measure ◽  
Lyapunov Exponents ◽  
Invariant Probability Measure ◽  
Two Dimensional ◽  
Bernoulli Shifts

The Lyapunov exponents of locally constant$\text{GL}(2,\mathbb{C})$-cocycles over Bernoulli shifts vary continuously with the cocycle and the invariant probability measure.

Peculiarities of the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors in the presence of noise

2002 ◽  
Vol 65 (3) ◽  
Author(s):  
Vadim S. Anishchenko ◽  
Tatjana E. Vadivasova ◽  
Andrey S. Kopeikin ◽  
Jürgen Kurths ◽  
Galina I. Strelkova
Keyword(s):  
Probability Measure ◽  
Invariant Probability Measure ◽  
Chaotic Attractors

scholarly journals Ergodic optimization in dynamical systems

2018 ◽  
Vol 39 (10) ◽  
pp. 2593-2618 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Probability Measure ◽  
Invariant Measures ◽  
Model Problem ◽  
Thermodynamic Formalism ◽  
Invariant Probability Measure ◽  
Temperature Limit ◽  
Zero Temperature ◽  
Ergodic Optimization ◽  
Maximizing Measure ◽  
Time Average

Ergodic optimization is the study of problems relating to maximizing orbits and invariant measures, and maximum ergodic averages. An orbit of a dynamical system is called$f$-maximizing if the time average of the real-valued function$f$along the orbit is larger than along all other orbits, and an invariant probability measure is called$f$-maximizing if it gives$f$a larger space average than any other invariant probability measure. In this paper, we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.

scholarly journals A FRAMEWORK FOR INTERNEURAL DYNAMICS IN A CEREBRAL CORTEX CENTER

2013 ◽  
Vol 23 (09) ◽  
pp. 1350151
Author(s):  
ABRAHAM BOYARSKY ◽  
ZHENYANG LI ◽  
PAWEŁ GÓRA
Keyword(s):  
Cerebral Cortex ◽  
Steady State ◽  
Probability Measure ◽  
Discrete Time ◽  
Sensory Input ◽  
Invariant Probability Measure ◽  
Chaotic State ◽  
Time Map

We model the innervation dynamics of interneurons in a cerebral cortex center A between the time of initial sensory input and acquisition of a sustained steady state. The model assumes that interneurons in A are heavily interconnected allowing synchronization. This invites modeling the dynamics by means of a discrete time map. The model takes into account the influence of excitatory and inhibitory cells and reflects the architecture of synapses along the axons. The acquisition of a sustained chaotic state is characterized by means of a natural invariant probability measure. The time to attain this probability measure can be estimated.

Sign in / Sign up

Export Citation Format

Share Document