scholarly journals On mix-norms and the rate of decay of correlations

Nonlinearity ◽  
2021 ◽  
Vol 34 (6) ◽  
pp. 3762-3782
Author(s):  
Bryan W Oakley ◽  
Jean-Luc Thiffeault ◽  
Charles R Doering
Keyword(s):  
Decay Of Correlations ◽  
Rate Of Decay

scholarly journals A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems

2012 ◽  
Vol 33 (2) ◽  
pp. 475-498 ◽  
Author(s):  
NICOLAI HAYDN ◽  
MATTHEW NICOL ◽  
TOMAS PERSSON ◽  
SANDRO VAIENTI
Keyword(s):  
Dynamical Systems ◽  
Return Time ◽  
Decay Of Correlations ◽  
Cantelli Lemma ◽  
Target Problem ◽  
Exponential Decay Of Correlations ◽  
Hyperbolic Dynamical Systems ◽  
Rate Of Decay ◽  
Dispersing Billiards ◽  
Lozi Maps

AbstractLet (Bi) be a sequence of measurable sets in a probability space (X,ℬ,μ) such that ∑ ∞n=1μ(Bi)=∞. The classical Borel–Cantelli lemma states that if the sets Bi are independent, then μ({x∈X:x∈Bi infinitely often})=1. Suppose (T,X,μ) is a dynamical system and (Bi) is a sequence of sets in X. We consider whether Tix∈Bi infinitely often for μ almost every x∈X and, if so, is there an asymptotic estimate on the rate of entry? If Tix∈Bi infinitely often for μ almost every x, we call the sequence (Bi) a Borel–Cantelli sequence. If the sets Bi :=B(p,ri) are nested balls of radius ri about a point p, then the question of whether Tix∈Bi infinitely often for μ almost every x is often called the shrinking target problem. We show, under certain assumptions on the measure μ, that for balls Bi if μ(Bi)≥i−γ, 0<γ<1, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observables implies that the sequence is Borel–Cantelli. If μ(Bi)≥C1 /i, then exponential decay of correlations implies that the sequence is Borel–Cantelli. We give conditions in terms of return time statistics which quantify Borel–Cantelli results for sequences of balls such that μ(Bi)≥C/i. Corollaries of our results are that for planar dispersing billiards and Lozi maps, sequences of nested balls B(p,1/i) are Borel–Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.

UV Stimulation of Chromosomal Marker Exchange in Sulfolobus acidocaldarius: Implications for DNA Repair, Conjugation and Homologous Recombination at Extremely High Temperatures

Genetics ◽  
1999 ◽  
Vol 152 (4) ◽  
pp. 1407-1415 ◽  
Author(s):  
Katherine J Schmidt ◽  
Kristen E Beck ◽  
Dennis W Grogan
Keyword(s):  
Homologous Recombination ◽  
Incubation Temperature ◽  
Uv Light ◽  
Dna Lesions ◽  
Sulfolobus Acidocaldarius ◽  
Chromosomal Marker ◽  
Marker Exchange ◽  
Chromosomal Markers ◽  
Rate Of Decay ◽  
Uv Photoproducts

Abstract The hyperthermophilic archaeon Sulfolobus acidocaldarius exchanges and recombines chromosomal markers by a conjugational mechanism, and the overall yield of recombinants is greatly increased by previous exposure to UV light. This stimulation was studied in an effort to clarify its mechanism and that of marker exchange itself. A variety of experiments failed to identify a significant effect of UV irradiation on the frequency of cell pairing, indicating that subsequent steps are primarily affected, i.e., transfer of DNA between cells or homologous recombination. The UV-induced stimulation decayed rather quickly in parental cells during preincubation at 75°, and the rate of decay depended on the incubation temperature. Preincubation at 75° decreased the yield of recombinants neither from unirradiated parental cells nor from parental suspensions subsequently irradiated. We interpret these results as evidence that marker exchange is stimulated by recombinogenic DNA lesions formed as intermediates in the process of repairing UV photoproducts in the S. acidocaldarius chromosome.

scholarly journals Mixing rates for potentials of non-summable variations

2021 ◽  
pp. 1-18
Author(s):  
CHRISTOPHE GALLESCO ◽  
DANIEL Y. TAKAHASHI
Keyword(s):  
Invariance Principle ◽  
Type Inequality ◽  
Upper Bounds ◽  
Decay Of Correlations ◽  
Relaxation Rates ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
Coupling Inequality ◽  
Mixing Rates

Abstract Mixing rates, relaxation rates, and decay of correlations for dynamics defined by potentials with summable variations are well understood, but little is known for non-summable variations. This paper exhibits upper bounds for these quantities for dynamics defined by potentials with square-summable variations. We obtain these bounds as corollaries of a new block coupling inequality between pairs of dynamics starting with different histories. As applications of our results, we prove a new weak invariance principle and a Hoeffding-type inequality.

scholarly journals ASYMPTOTIC BEHAVIOR OF THE PRINCIPAL EIGENVALUE FOR A CLASS OF NON-LOCAL ELLIPTIC OPERATORS RELATED TO BROWNIAN MOTION WITH SPATIALLY DEPENDENT RANDOM JUMPS

2011 ◽  
Vol 13 (06) ◽  
pp. 1077-1093
Author(s):  
NITAY ARCUSIN ◽  
ROSS G. PINSKY
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Large Time ◽  
Dirichlet Boundary ◽  
Principal Eigenvalue ◽  
Probability Measures ◽  
Rate Of Decay ◽  
Non Local ◽  
Random Jumps ◽  
Explicit Constant

Let D ⊂ Rd be a bounded domain and let [Formula: see text] denote the space of probability measures on D. Consider a Brownian motion in D which is killed at the boundary and which, while alive, jumps instantaneously according to a spatially dependent exponential clock with intensity γV to a new point, according to a distribution [Formula: see text]. From its new position after the jump, the process repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator -Lγ,μ, defined by [Formula: see text] with the Dirichlet boundary condition, where Cμ is the "μ-centering" operator defined by [Formula: see text] The principal eigenvalue, λ0(γ, μ), of Lγ, μ governs the exponential rate of decay of the probability of not exiting D for large time. We study the asymptotic behavior of λ0(γ, μ) as γ → ∞. In particular, if μ possesses a density in a neighborhood of the boundary, which we call μ, then [Formula: see text] If μ and all its derivatives up to order k - 1 vanish on the boundary, but the kth derivative does not vanish identically on the boundary, then λ0(γ, μ) behaves asymptotically like [Formula: see text], for an explicit constant ck.

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