Search results for "Invariant measure"

On the enhancement of diffusion by chaos, escape rates and stochastic instability

1999

We consider stochastic perturbations of expanding maps of the interval where the noise can project the trajectory outside the interval. We estimate the escape rate as a function of the amplitude of the noise and compare it with the purely diffusive case. This is done under a technical hypothesis which corresponds to stability of the absolutely continuous invariant measure against small perturbations of the map. We also discuss in detail a case of instability and show how stability can be recovered by considering another invariant measure.

On the existence of conditionally invariant probability measures in dynamical systems

2000

Let T : X→X be a measurable map defined on a Polish space X and let Y be a non-trivial subset of X. We give conditions ensuring the existence of conditionally invariant probability measures to non-absorption in Y. For dynamics which are non-singular with respect to some fixed probability measure we supply sufficient conditions for the existence of absolutely continuous conditionally invariant measures. These conditions are satisfied for a wide class of dynamical systems including systems that are Φ-mixing and Gibbs.

Invariant measures for piecewise convex transformations of an interval

2002

Maximal regularity for Kolmogorov operators in L2 spaces with respect to invariant measures

2006

Abstract We prove an optimal embedding result for the domains of Kolmogorov (or degenerate hypoelliptic Ornstein–Uhlenbeck) operators in L 2 spaces with respect to invariant measures. We use an interpolation method together with optimal L 2 estimates for the space derivatives of T ( t ) f near t = 0 , where T ( t ) is the Ornstein–Uhlenbeck semigroup and f is any function in L 2 .

Cores for parabolic operators with unbounded coefficients

2009

Abstract Let A = ∑ i , j = 1 N q i j ( s , x ) D i j + ∑ i = 1 N b i ( s , x ) D i be a family of elliptic differential operators with unbounded coefficients defined in R N + 1 . In [M. Kunze, L. Lorenzi, A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., in press], under suitable assumptions, it has been proved that the operator G : = A − D s generates a semigroup of positive contractions ( T p ( t ) ) in L p ( R N + 1 , ν ) for every 1 ⩽ p + ∞ , where ν is an infinitesimally invariant measure of ( T p ( t ) ) . Here, under some additional conditions on the growth of the coefficients of A , which cover also some growths with an ex…

Lp-uniqueness for elliptic operators with unbounded coefficients in RN

AbstractLet A be an elliptic operator with unbounded and sufficiently smooth coefficients and let μ be a (sub)-invariant measure of the operator A. In this paper we give sufficient conditions guaranteeing that the closure of the operator (A,Cc∞(RN)) generates a sub-Markovian strongly continuous semigroup of contractions in Lp(RN,μ). Applications are given in the case when A is a generalized Schrödinger operator.

Almost sure rates of mixing for i.i.d. unimodal maps

2002

International audience; It has been known since the pioneering work of Jakobson and subsequent work by Benedicks and Carleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and Keller-Nowicki proved exponential decay of its correlation functions. Benedicks and Young, and Baladi and Viana studied stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations. The almost sure statistical properties of the sample stationary measures of i.i.d. itineraries are more difficult to estimate than the "averaged statistics". Adapting to random systems, on the one hand partitions associ…

Moduli spaces of discrete gravity

2003

Spectral triples describe and generalize Riemannian spin geometries by converting the geometrical information into algebraic data, which consist of an algebra $A$, a Hilbert space $H$ carrying a representation of $A$ and the Dirac operator $D$ (a selfadjoint operator acting on $H$). The gravitational action is described by the trace of a suitable function of $D$. In this paper we examine the (path-integral-) quantization of such a system given by a finite dimensional commutative algebra. It is then (in concrete examples) possible to construct the moduli space of the theory, i.e. to divide the space of all Dirac operators by the action of the diffeomorphism group, and to construct an invaria…

Non-Periodic Systems with Continuous Diffraction Measures

2015

The present state of mathematical diffraction theory for systems with continuous spectral components is reviewed and extended. We begin with a discussion of various characteristic examples with singular or absolutely continuous diffraction, and then continue with a more general exposition of a systematic approach via stationary stochastic point processes. Here, the intensity measure of the Palm measure takes the role of the autocorrelation measure in the traditional approach. We furthermore introduce a ‘Palm-type’ measure for general complex-valued random measures that are stationary and ergodic, and relate its intensity measure to the autocorrelation measure.

Applications de type Lasota–Yorke à trou : mesure de probabilité conditionellement invariante et mesure de probabilité invariante sur l'ensemble des …

2003

Abstract Let T :I→I be a Lasota–Yorke map on the interval I, let Y be a nontrivial sub-interval of I and g 0 :I→ R + , be a strictly positive potential which belongs to BV and admits a conformal measure m. We give constructive conditions on Y ensuring the existence of absolutely continuous (w.r.t. m) conditionally invariant probability measures to nonabsorption in Y. These conditions imply also existence of an invariant probability measure on the set X∞ of points which never fall into Y. Our conditions allow rather “large” holes.