Subexponential Decay Research Articles

Band-limited wavelets beyond Gevrey regularity

It is known that a smooth function of exponential decay at infinity cannot be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey-type subexponential decay. We weaken the Gevrey-type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that precise asymptotics of functions from such classes can be given in terms of the Lambert [Formula: see text]-function.

Decay rates of convergence for Fokker-Planck equations with confining drift

We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or sub-exponential decay rates, as time goes to infinity, of zero average solutions, under some diffusivity condition on the Levy process, which includes the fractional Laplace operator as a model example. Our approach relies on the long time oscillation estimates of the adjoint problem and applies to (the possible superposition of) both local and nonlocal diffusions, as well as to strongly or weakly confining drifts. Our results extend, with a unifying perspective, many previous works based on different analytic or probabilistic methods, with several interesting connections. On one hand, we make a link between the (nonlinear) PDE methods used for the long time behavior of Hamilton-Jacobi equations and the decay estimates of Fokker-Planck equations; on another hand, we give a purely analytical approach towards some oscillation decay estimates which were obtained so far only with probabilistic coupling methods.

Deviation and concentration inequalities for dynamical systems with subexponential decay of correlations

We obtain large and moderate deviation estimates, as well as concentration inequalities, for a class of nonuniformly expanding maps with stretched exponential decay of correlations. In the large deviation regime, we also exhibit examples showing that the obtained upper bounds are essentially optimal.

Spectral Gap Formation to Kinetic Equations with Soft Potentials in Bounded Domain

It has been unknown in kinetic theory whether the linearized Boltzmann or Landau equation with soft potentials admits a spectral gap in the spatially inhomogeneous setting. Most of existing works indicate a negative answer because the spectrum of two linearized self-adjoint collision operators is accumulated to the origin in case of soft interactions. In the paper we rather prove it in an affirmative way when the space domain is bounded with an inflow boundary condition. The key strategy is to introduce a new Hilbert space with an exponential weight function that involves the inner product of space and velocity variables and also has the strictly positive upper and lower bounds. The action of the transport operator on such space-velocity dependent weight function induces an extra non-degenerate relaxation dissipation in large velocity that can be employed to compensate the degenerate spectral gap and hence give the exponential decay for solutions in contrast with the sub-exponential decay in either the spatially homogeneous case or the case of torus domain. The result reveals a new insight of hypocoercivity for kinetic equations with soft potentials in the specified situation.

Well-Separating Common Complements for Sequences of Subspaces of the Same Codimension are Generic in Hilbert Spaces

Given a family of subspaces of a Banach or Hilbert space, we investigate existence, quantity and quality of its common complements. In particular, we are interested in common complements for countable families of closed subspaces of finite codimension. For those families, we show that common complements with subexponential decay of quality are generic in Hilbert spaces. Moreover, we prove that the existence of one such complement in a Banach space already implies that they are generic.

Renewal Model for Dependent Binary Sequences

We suggest to construct infinite stochastic binary sequences by associating one of the two symbols of the sequence with the renewal times of an underlying renewal process. Focusing on stationary binary sequences corresponding to delayed renewal processes, we investigate correlations and the ability of the model to implement a prescribed autocovariance structure, showing that a large variety of subexponential decay of correlations can be accounted for. In particular, robustness and efficiency of the method are tested by generating binary sequences with polynomial and stretched-exponential decay of correlations. Moreover, to justify the maximum entropy principle for model selection, an asymptotic equipartition property for typical sequences that naturally leads to the Shannon entropy of the waiting time distribution is demonstrated. To support the comparison of the theory with data, a law of large numbers and a central limit theorem are established for the time average of general observables.

The large amplitude solution of the Boltzmann equation with soft potential

In this paper, we deal with the (angular cut-off) Boltzmann equation with soft potential (−3<γ<0). In particular, we construct a unique global solution in Lx,v∞ which converges to global equilibrium asymptotically provided that initial data has a large amplitude but with sufficiently small relative entropy. Because frequency multiplier is not uniformly positive anymore, unlike hard potential case, time-involved velocity weight will be used to derive sub-exponential decay of the solution. Motivated by recent development of L2-L∞ approach also, we introduce some modified estimates of quadratic nonlinear terms. Linearized collision kernel will be treated in a subtle manner to control singularity of soft potential kernel.

Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators A_a^{varphi _1,varphi _2} with symbols a in the wide weighted modulation space M^infty _{w}({mathbb {R}^{2d}}), and windows varphi _1, varphi _2 in the Gelfand–Shilov space mathcal {S}^{left( 1right) }(mathbb {R}^d). If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of A_a^{varphi _1,varphi _2} have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space mathcal {S}^{(gamma )} (mathbb {R^{d}}) , where the parameter gamma ge 1 is related to the growth of the considered weight. An important role is played by tau -pseudodifferential operators Op_{tau } (sigma ). In that direction we show convenient continuity properties of Op_{tau } (sigma )when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Op_{tau } (sigma ) when the symbol sigma belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.

Concentration inequalities for polynomials in α-sub-exponential random variables

In this work we derive multi-level concentration inequalities for polynomial functions in independent random variables with a $\alpha$-sub-exponential tail decay. A particularly interesting case is given by quadratic forms $f(X_1, \ldots, X_n) = \langle X,A X \rangle$, for which we prove Hanson-Wright-type inequalities with explicit dependence on various norms of the matrix $A$. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in $\alpha$-sub-exponential random variables, such as quadratic Poisson chaos. We provide various applications of these inequalities. Among these are generalizations the results given by Rudelson-Vershynin from sub-Gaussian to $\alpha$-sub-exponential random variables, i. e. concentration of the Euclidean norm of the linear image of a random vector, small ball probability estimates and concentration inequalities for the distance between a random vector and a fixed subspace. Moreover, we obtain concentration inequalities for the excess loss in a fixed design linear regression and the norm of a randomly projected random vector.

Hypocoercivity and sub-exponential local equilibria

Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the heat equation obtained in the diffusion limit. The method applies to Fokker-Planck and scattering collision operators. The main tools are a weighted Poincar\'e inequality (in the Fokker-Planck case) and norms with various weights. The advantage of weighted Poincar\'e inequalities compared to the more classical weak Poincar\'e inequalities is that the description of the convergence rates to the local equilibrium does not require extra regularity assumptions to cover the transition from super-exponential and exponential local equilibria to sub-exponential local equilibria.

The existence of full dimensional invariant tori for 1-dimensional nonlinear wave equation

In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed by Bourgain [9].

Beyond universal behavior in the one-dimensional chain with random nearest-neighbor hopping

We study the one-dimensional nearest neighbor tight binding model of electrons with independently distributed random hopping and no on-site potential (i.e. off-diagonal disorder with particle-hole symmetry, leading to sub-lattice symmetry, for each realization). For non-singular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states $\rho(E) \sim 1/|E \ln^3|E||$ and of the localization length $\xi(E) \sim |\ln|E||$, near the band center $E = 0$. (This singular behavior is also applicable to random XY and Heisenberg spin chains; it was first obtained by Dyson for a specific random harmonic oscillator chain). Simultaneously, the state at $E = 0$ shows a universal, sub-exponential decay at large distances $\sim \exp [ -\sqrt{r/r_0} ]$. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small $t$ is of the form $\sim 1/ [t \ln^{\lambda+1}(1/t) ]$, characterized by a single, continuously tunable parameter $\lambda > 0$. We find, using a combination of analytic and numerical methods, that while the universal result applies for $\lambda > 2$, it no longer holds in the interval $0 < \lambda < 2$. In particular, we find that the form of the density of states singularity is enhanced (relative to the Dyson result) in a continuous manner depending on the non-universal parameter $\lambda$; simultaneously, the localization length shows a less divergent form at low energies, and ceases to diverge below $\lambda = 1$. For $\lambda < 2$, the fall-off of the $E = 0$ state at large distances also deviates from the universal result, and is of the form $\sim \exp [-(r/r_0)^{1/\lambda}]$, which decays faster than an exponential for $\lambda < 1$.

Multiresolution expansions and wavelets in Gelfand–Shilov spaces

We study approximation properties generated by highly regular scaling functions and orthonormal wavelets. These properties are conveniently described in the framework of Gelfand-Shilov spaces. Important examples of multiresolution analyses for which our results apply arise in particular from Dziuba\'{n}ski-Hern\'{a}ndez construction of band-limited wavelets with subexponential decay. Our results are twofold. Firstly, we obtain approximation properties of multiresolution expansions of Gelfand-Shilov functions and (ultra)distributions. Secondly, we establish convergence of wavelet series expansions in the same regularity framework.

Weak and strong type estimates for the multilinear pseudo-differential operators

In this paper, we investigate the boundedness of the multilinear pseudo-differential operator Tσ. First, we establish the local exponential decay estimates for Tσ. In terms of the corresponding commutators Tσ,Σb, we obtain the local subexponential decay estimates. Secondly, we derive the weighted mixed weak type inequality for Tσ, which parallels Sawyer's conjecture for Calderón-Zygmund operators and covers the endpoint weighted inequalities. Last but not least, we present the sharp weighted estimates for Tσ and Tσ,Σb. It is worth mentioning that our results are totally new even in the linear case.

The Chebyshev method for the implied volatility

The implied volatility is a crucial element in any financial toolbox, since it is used to both quote and hedge options as well as for model calibration. In contrast to the Black–Scholes formula, its inverse, the implied volatility is not explicitly avail- able, and numerical approximation is required. In this paper, we propose a bivariate interpolation of the implied volatility surface based on Chebyshev polynomials. This yields a closed-form approximation of the implied volatility, which is easy to implement and to maintain. We prove a subexponential error decay. This allows us to obtain an accuracy close to machine precision with polynomials of a low degree. We compare the performance of our chosen method in terms of runtime and accuracy with the most common reference methods. In contrast to existing interpolation methods, our method is able to compute the implied volatility for all relevant option data. We use numerical experiments to confirm this results in a considerable increase in efficiency, especially for large data sets.

Fast decay of eigenfunction correlators in long-range continuous random alloys

Abstract We study random Anderson Hamiltonians in Euclidean spaces with a long-range particle-media interaction potential {\mathfrak{u}(r)=r^{-A}} . Improving earlier results, for any {A&gt;2d} , we establish spectral and strong dynamical localization with sub-exponential decay of eigenfunction correlators, both in the strong disorder regime and at low energies.

Sub-exponential decay of eigenfunctions for some discrete Schrödinger operators

Following the method of Froese and Herbst, we show for a class of potentials V that an embedded eigenfunction \psi with eigenvalue E of the multi-dimensional discrete Schrödinger operator H = \Delta + V on \mathbb Z^d decays sub-exponentially whenever the Mourre estimate holds at E . In the one-dimensional case we further show that this eigenfunction decays exponentially with a rate at least of cosh ^{-1}((E-2)/(\theta_E-2)) , where \theta_E is the nearest threshold of H located between E and 2 . A consequence of the latter result is the absence of eigenvalues between 2 and the nearest thresholds above and below this value. The method of Combes–Thomas is also reviewed for the discrete Schrödinger operators.

Localization in Asymmetric Random Displacements Models with Infinite Range of Interaction

We consider a random displacements model in a Euclidean space with an infinite-range, polynomially decaying interaction potential, without assuming any symmetry properties of the latter, and give an elementary proof of eigenvalue concentration estimates without resorting to a more traditional Wegner-type analysis or explicit ground state energy minimization. In the strong disorder/semi-classical regime, we prove exponential spectral and dynamical localization, with sub-exponential decay of the eigenfunction correlators. Prior works focused on compactly supported interactions.

Large-deviation probabilities for correlated Gaussian processes and intermittent dynamical systems.

In its classical version, the theory of large deviations makes quantitative statements about the probability of outliers when estimating time averages, if time series data are identically independently distributed. We study large-deviation probabilities (LDPs) for time averages in short- and long-range correlated Gaussian processes and show that long-range correlations lead to subexponential decay of LDPs. A particular deterministic intermittent map can, depending on a control parameter, also generate long-range correlated time series. We illustrate numerically, in agreement with the mathematical literature, that this type of intermittency leads to a power law decay of LDPs. The power law decay holds irrespective of whether the correlation time is finite or infinite, and hence irrespective of whether the central limit theorem applies or not.

Achievable Moderate Deviations Asymptotics for Streaming Compression of Correlated Sources

Motivated by streaming multi-view video coding and wireless sensor networks, we consider the problem of blockwise streaming compression of a pair of correlated sources, which we term streaming Slepian-Wolf coding. We study the moderate deviations regime in which the rate pairs of a sequence of codes converge, along a straight line, to various points on the boundary of the Slepian-Wolf region at a speed slower than the inverse square root of the blocklength $n$, while the error probability decays subexponentially fast in $n$. Our main result focuses on directions of approaches to corner points of the Slepian-Wolf region. It states that for each correlated source and all corner points, there exists a non-empty subset of directions of approaches such that the moderate deviations constant (the constant of proportionality for the subexponential decay of the error probability) is enhanced (over the non-streaming case) by at least a factor of $T$, the block delay of decoding source block pairs. We specialize our main result to the setting of streaming lossless source coding and generalize this result to the setting where we have different delay requirements for each of the two source blocks. The proof of our main result involves the use of various analytical tools and amalgamates several ideas from the recent information-theoretic streaming literature. We adapt the so-called truncated memory encoding idea from Draper and Khisti (2011) and Lee, Tan, and Khisti (2016) to ensure that the effect of error accumulation is nullified in the limit of large blocklengths. We also adapt the use of the so-called minimum weighted empirical suffix entropy decoder which was used by Draper, Chang, and Sahai (2014) to derive achievable error exponents for symbolwise streaming Slepian-Wolf coding.