Thermodynamic Formalism and Singular Invariant Measures for Critical Circle Maps

As is well known, the renormalization group transformation in the space of analytic circle homeomorphisms with one cubic critical point and rotation number equal to the golden section has a single fixed point T ₀. We construct the thermodynamic formalism for the critical map T ₀ and use it to calculate the Hölder indices for the singular invariant measure of T ₀.     

Large Deviations of Solutions of Hyperbolic SPDE's in the Hölder Norm

In this paper, we prove the large deviations principle for solutions of a hyperbolic stochastic partial differential equation, in the Hölder topology of index for all 0 < . This result generalizes those in [5] and [10] to the Hölder norm, and the result in [3] for solutions of a class fo stochastic differential equations involving a two-parameter Wiener process. These solutions are obtained by small perturbations of the noise.     

Justifying the convergence of the rectangular method for complete singular integral equations with continuous coefficients on the circle

For an integral equation on the unit circle of the form (aI + bS + K)f = g, where a and b are Hölder functions, S is a singular integration operator, and K is an integral operator with Hölder kernel, we consider a method of solution based on the discretization of integral operators using the rectangle rule. This method is justified under the assumption that the equation is solvable in L ₂() and the coefficients a and b satisfy the strong ellipticity condition.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 163–175.Original Russian Text Copyright © 2005 by M. É. Abramyan.This revised version was published online in April 2005 with a corrected issue number.     

Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

The fractional Brownian density process is a continuous centered Gaussian ( ^d )-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. ( ( ^d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on ^d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand¹². This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.     

Smoothness of Nonlinear Median-Interpolation Subdivision

We give a refined analysis of the Hölder regularity for the limit functions arising from a nonlinear pyramid algorithm for robust removal of non-Gaussian noise proposed by Donoho and Yu [6,7,17]. The synthesis part of this algorithm can be interpreted as a nonlinear triadic subdivision scheme where new points are inserted based on local quadratic polynomial median interpolation and imputation. We introduce the analogon of the Donoho–Yu scheme for dyadic refinement, and show that its limit functions are in C for >log₄(128/31)=1.0229.... In the triadic case, we improve the lower bound of >log₂(135/121)=0.0997... previously obtained in [6] to >log₃(135/53)=0.8510.... These lower bounds are relatively close to the anticipated upper bounds of log₂(16/7)=1.1982... in the dyadic, respectivly 1 in the triadic cases, and have been obtained by deriving recursive inequalities for the norm of second rather than first order differences of the sequences arising in the subdivision process.     

Hölder continuity of local minimizers of vectorial integral functionals

We study the regularity of vector-valued local minimizers in $ W^{1,p}, p > 1 $, of the integral functional where is an open set in $ \mathbb{R}^N $ and f is a continuous function, convex with respect to the last variable, such that $ 0 \leq f(x,u,t)\leq C(1+t^p) $.We prove that if f = f(x, t), or f = f(x, u, t) and $ p \leq N $, then local minimizers are locally Hölder continuous for any exponent less than 1. If f = f(x, u, t) and p < N then local minimizers are Höolder continuous for every exponent less than 1 in an open set $ \Omega_0 $ such that the Hausdorff dimension of $ \Omega \backslash \Omega_0 $ is less than N–p.AMS Subject Classification: 49N60.     

On the two-parameter fractional Brownian motion and Stieltjes integrals for Hölder functions

We extend the Stieltjes integral to Hölder functions of two variables and prove an existence and uniqueness result for the corresponding deterministic ordinary differential equations and also for stochastic equations driven by a two-parameter fractional Brownian motion.     

Normalized Solutions of Schrödinger Equations with Potentially Bounded Measures

Given a potentially bounded signed measure on a Brelot space (X,) with Green function G, it is well known that -harmonic functions (i.e., in the classical case, finely continuous versions of solutions to u–u=0) may be very discontinuous. In this paper it is shown that under very general assumptions on G (satisfied for large classes of elliptic second-order linear differential operators) normalized perturbation, however, leads to a Brelot space (X, ) admitting a Green function T (G) which is locally (or even globally) comparable with G and has all properties required of G before. In particular, iterated perturbation is possible. Moreover, intrinsic Hölder continuity of quotients of harmonic functions with respect to the local quasimetric :=(G ^–1+^* G ^–1)/2 yields -Hölder continuity for quotients of -harmonic functions as well.     

Small ball estimates for Brownian motion and the Brownian sheet

Small ball estimates are obtained for Brownian motion and the Brownian sheet when balls are given by certain Hölder norms. As an application of these results we include a functional form of Chung's LIL in this setting.Both authors were supported in part by NSF Grant Number DMS-9024961.     

Estimation of Local Smoothness Coefficients for Continuous Time Processes

We consider processes that satisfied a local Hölder condition with coefficient ₀. According to the sampling times of observations given by i _n with i=0,...,n–1, we study two general classes of estimators for ₀. Their almost sure rates of convergence depend on asymptotic independence of the observed processes, on _n and eventually on an extra parameter ₀. Since this last parameter is in general unknown, we construct a family of preliminary estimators for ₀ with their rates of almost sure convergence. Finally we present some numerical simulations in order to compare the behaviour of our various estimators.     

Local Times and Related Properties of Multidimensional Iterated Brownian Motion

Let {W(t), tR} and {B(t), t0} be two independent Brownian motions in R with W(0) = B(0) = 0 and let

Brown运动在Hlder范数下的拟必然Strassen重对数律的收敛速率

本文研究了Brown运动的泛函极限问题.利用Brown运动在Hlder范数下关于容度的大偏差与小偏差,获得了Brown运动在Hlder范数下的Strassen型重对数律的拟必然收敛速率,推广了文[2]中的结果.     

Estimates of the Hölder constant for functions satisfying a uniformly elliptic or a uniformly parabolic quasilinear inequality with unbounded coefficients

One obtains inner and boundary estimates of the Hölder constants for functions u(·) satisfying a uniformly elliptic or uniformly parabolic quasilinear inequality of nondivergence form with unbounded coefficients. It is shown that the Hölder exponents in them depend only on the dimension W and on the constants and occurring in the ellipticity conditions. In the boundary estimates they depend also on the constant ₀, occurring in the condition (A) on the boundary and on the Hölder exponent for the boundary values of u(·).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 72–94, 1985.     

Large Deviations of a Storage Process with Fractional Brownian Motion as Input

We study probabilities of large extremes of the storage process Y(t) = sup _t (X() - X(t) - c( - t)), where X(t) is the fractional Brownian motion. We derive asymptotic behavior of the maximum tail distribution for the process on fixed or slowly increased intervals by a reduction the problem to a large extremes problem for a Gaussian field.     

Hlder范数下关于Brown运动增量的泛函局部收敛速度

本文研究了Brown运动在H?lder范数与容度下的泛函极限问题.利用大偏差小偏差方法,获得了Brown运动增量局部泛函极限的收敛速度,推广了文[4]中的结果.     

A Note on Transient Gaussian Fluid Models

In this paper we study the distribution of the supremum over interval [0,T] of a centered Gaussian process with stationary increments with a general negative drift function. This problem is related to the distribution of the buffer content in a transient Gaussian fluid queue Q(T) at time T, provided that at time 0 the buffer is empty. The general theory is illustrated by detailed considerations of different cases for the integrated Gaussian process and the fractional Brownian motion. We give asymptotic results for P(Q(T)>x) and P(sup _0tT Q(t)>x) as x.     

Necessary and Sufficient Condition for the Functional Central Limit Theorem in Hölder Spaces

Let (X _i ) _i1 be an i.i.d. sequence of random elements in the Banach space B, S _n X ₁++X _n and _n be the random polygonal line with vertices (k/n,S _k ), k=0,1,...,n. Put (h)=h L(1/h), 0h1 with 0<1/2 and L slowly varying at infinity. Let H ⁰ (B) be the Hölder space of functions x:[0,1]B, such that x(t+h)–x(t)=o((h)), uniformly in t. We characterize the weak convergence in H ⁰ (B) of n ^–1/2 _n to a Brownian motion. In the special case where B= and (h)=h , our necessary and sufficient conditions for such convergence are E X ₁=0 and P(|X ₁|>t)=o(t ^–p()) where p()=1/(1/2–). This completes Lamperti (1962) invariance principle.     

Minimal Paths between Maximal Chains in Finite Rank Semimodular Lattices

We study paths between maximal chains, or flags, in finite rank semimodular lattices. Two flags are adjacent if they differ on at most one rank. A path is a sequence of flags in which consecutive flags are adjacent. We study the union of all flags on at least one minimum length path connecting two flags in the lattice. This is a subposet of the original lattice. If the lattice is modular, the subposet is equal to the sublattice generated by the flags. It is a distributive lattice which is determined by the Jordan-Hölder permutation between the flags. The minimal paths correspond to all reduced decompositions of this permutation. In a semimodular lattice, the subposet is not uniquely determined by the Jordan-Hölder permutation for the flags. However, it is a join sublattice of the distributive lattice corresponding to this permutation. It is semimodular, unlike the lattice generated by the two flags, which may not be ranked. The minimal paths correspond to some reduced decompositions of the permutation, though not necessarily all. We classify the possible lattices which can arise in this way, and characterize all possibilities for the set of shortest paths between two flags in a semimodular lattice.     

Structure of Free Interpolation Sets for Analytic Function Spaces Determined by a Continuity Modulus

We describe the evolution of boundary interpolation sets between the disk-algebra and Hölder spaces of analytic functions. For the disk-algebra, interpolation sets are sets of zero measure, while for Hölder spaces of order , interpretation sets are porous. For the Hölder-type classes corresponding to a continuity modulus , a necessary condition for free interpolation turns into a certain condition of -porosity. Any set of zero measure is -porous for some . We establish a Muckenhoupt-type estimate; such estimates may be useful in the proof of sufficiency of -porosity conditions. Bibliography: 7 titles.     

Some small ball probabilities for Gaussian processes under nonuniform norms

We establish lower and upper bounds for the small ball probability of a centered Gaussian process(X(t)) _t[0,1] ^N under Hölder-type norms as well as upper bounds for some more general functionals. This extends recently established results for the uniform norm. In addition, our proof of the lower bound is considerably simpler. In the special caseN=1 we establish precise estimates under a wider class of norms including in particular the Besov norms.