Self-intersection Local Time Research Articles

Renormalized self-intersection local time of bifractional Brownian motion

Let B^{H,K}={B^{H,K}(t), t geq 0} be a d-dimensional bifractional Brownian motion with Hurst parameters Hin (0,1) and Kin (0,1]. Assuming dgeq 2, we prove that the renormalized self-intersection local time \t\t\t∫0T∫0tδ(BH,K(t)−BH,K(s))dsdt−E(∫0T∫0tδ(BH,K(t)−BH,K(s))dsdt)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\int^{T}_{0} \\int^{t}_{0}\\delta \\bigl(B^{H,K}(t)-B^{H,K}(s) \\bigr)\\,ds\\,dt-\\mathbb{E} \\biggl( \\int^{T}_{0} \\int^{t}_{0}\\delta \\bigl(B^{H,K}(t)-B^{H,K}(s) \\bigr)\\,ds\\,dt \\biggr) \\end{aligned}$$ \\end{document} exists in L^{2} if and only if HKd< 3/2, where δ denotes the Dirac delta function. Our work generalizes the result of the renormalized self-intersection local time for fractional Brownian motion.

Smoothness of self-intersection local time of multidimensional fractional Brownian motion

In this paper, we study the existence, Hölder continuity and fractional smoothness of the self-intersection local time with respect to fractional Brownian motion:where BH is a d-dimensional fractional Brownian motion with Hurst index . We also consider the case of intersection local time for two independent multidimensional fractional Brownian motions.

Hilbert-valued self-intersection local times for planar Brownian motion

ABSTRACTDynkin's construction for self-intersection local time of a planar Wiener process is extended to Hilbert-valued weights. In Dynkin's construction, the weight is bounded and measurable. Since the weight function describes the properties of the medium in which the Brownian motion moves, relative to the external medium's properties, the weight function can be random and unbounded. In this article, we discuss the possibility to consider Hilbert-space-valued weights. It appears that the existence of Hilbert-valued Dynkin-renormalized self-intersection local time is equivalent to the embedding of the values of Hilbert-valued weight into a Hilbert–Schmidt brick. Using Dorogovtsev's sufficient condition for the embedding of compact sets into a Hilbert–Schmidt brick in terms of an isonormal process, we prove the existence of Hilbert-valued Dynkin-renormalized self-intersection local time. Also using Dynkin's construction we construct the self-intersection local time for the deterministic image of the planar Wiener process.

The Schrödinger equation with spatial white noise: The average wave function

We prove a representation for the average wave function of the Schrödinger equation with a white noise potential in d=1,2, in terms of the renormalized self-intersection local time of a Brownian motion.

Annealed scaling for a charged polymer in dimensions two and higher**The research in this paper was supported through ERC Advanced Grant 267356-VARIS.

This paper considers an undirected polymer chain on , , with i.i.d. random charges attached to its constituent monomers. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. The object of interest is the annealed free energy per monomer in the limit as the length n of the polymer chain tends to infinity.We show that there is a critical curve in the parameter plane spanned by the charge bias and the inverse temperature separating an extended phase from a collapsed phase. We derive the scaling of the critical curve for small and for large charge bias and the scaling of the annealed free energy for small inverse temperature. We argue that in the collapsed phase the polymer chain is subdiffusive, namely, on scale it moves like a Brownian motion conditioned to stay inside a ball with a deterministic radius and a randomly shifted center. We further expect that in the extended phase the polymer chain scales like a weakly self-avoiding walk.The scaling of the critical curve for small charge bias and the scaling of the annealed free energy for small inverse temperature are both anomalous. Proofs are based on a detailed analysis for simple random walk of the downward large deviations of the self-intersection local time and the upward large deviations of the range. Part of our scaling results are rough. We formulate conjectures under which they can be sharpened. The existence of the free energy remains an open problem, which we are able to settle in a subset of the collapsed phase for a subclass of charge distributions.

Strong approximations for the p-fold integrated empirical process with applications to statistical tests

The main purpose of this paper is to investigate the strong approximation of the $p$-fold integrated empirical process, $p$ being a fixed positive integer. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on results of Koml\'os, Major and Tusn\'ady (1975). We also obtain an exponential bound for the tail probability of the weighted approximation to the $p$-fold integrated empirical process. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of integrated empirical processes when the parameters are estimated. We study the behavior of the self-intersection local time of the partial sum process representation of integrated empirical processes. Finally, simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the integrated empirical processes.

Fractional smoothness of derivative of self-intersection local times

In this paper, we study the fractional smoothness of derivative of self-intersection local times with respect to Brownian motion and fractional Brownian motion, and also consider the case of intersection local times.

Self-Intersection Local Times of Generalized Mixed Fractional Brownian Motion as White Noise Distributions

The generalized mixed fractional Brownian motion is defined by taking linear combinations of a finite number of independent fractional Brownian motions with different Hurst parameters. It is a Gaussian process with stationary increments, posseses self-similarity property, and, in general, is neither a Markov process nor a martingale. In this paper we study the generalized mixed fractional Brownian motion within white noise analysis framework. As a main result, we prove that for any spatial dimension and for arbitrary Hurst parameter the self-intersection local times of the generalized mixed fractional Brownian motions, after a suitable renormalization, are well-defined as Hida white noise distributions. The chaos expansions of the self-intersection local times in the terms of Wick powers of white noises are also presented.

Intersection Local Times, Loop Soups and Permanental Wick Powers

Several stochastic processes related to transient Levy processes with potential densities u(x,y) = u(y x), that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures V endowed with a metric d. Su- cient conditions are obtained for the continuity of these processes on (V,d). The processes include n-fold self-intersection local times of tran- sient Levy processes and permanental chaoses, which are 'loop soup n- fold self-intersection local times' constructed from the loop soup of the Levy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of n-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional proper- ties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.

Optimal Bounds for the Variance of Self-Intersection Local Times

For a Zd-valued random walk (Sn)n∈N0, let l(n,x) be its local time at the site x∈Zd. For α∈N, define the α-fold self-intersection local time as Ln(α)≔∑xl(n,x)α. Also let LnSRW(α) be the corresponding quantities for the simple random walk in Zd. Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely d-dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, var(Ln(α))=O(var⁡(LnSRW(α))). In particular, for any genuinely d-dimensional random walk, with d≥4, we have var⁡(Ln(α))=O(n). On the other hand, in dimensions d≤3 we show that if the behaviour resembles that of simple random walk, in the sense that lim infn→∞var⁡Lnα/var⁡(LnSRW(α))&gt;0, then the increments of the random walk must have zero mean and finite second moment.

Asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion

Let {Bt}t≥0 be a fractional Brownian motion with Hurst parameter 23<H<1. We prove that the approximation of the derivative of self-intersection local time, defined as αε=∫0T∫0tpε′(Bt−Bs)dsdt, where pε(x) is the heat kernel, satisfies a central limit theorem when renormalized by ε32−1H. We prove as well that for q≥2, the qth chaotic component of αε converges in L2 when 23<H<34, and satisfies a central limit theorem when renormalized by a multiplicative factor ε1−34H in the case 34<H<4q−34q−2.

Some asymptotic results for the integrated empirical process with applications to statistical tests

ABSTRACTThe main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (1975)'s results. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial-sum process representation of the integrated empirical process.

Chaos Decomposition and Gap Renormalization of Brownian Self-Intersection Local Times

We study the chaos decomposition of self-intersection local times and their regularization, with a particular view towards Varadhan's renormalization for the planar Edwards model.

Properties of Gaussian local times

We investigate properties of local time for one class of Gaussian processes. These processes are called integrators since every function from L2([0; 1]) can be integrated over it. Using the white noise representation, we can associate integrators with continuous linear operators in L2([0; 1]). In terms of these operators, we discuss the existence and properties of local time for integrators. Also, we study the asymptotic behavior of Brownian self-intersection local time as its end-point tends to infinity.

Smoothness of local times and self-intersection local times of Gaussian random fields

This paper is concerned with the smoothness (in the sense of Meyer-Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.

Derivative for self-intersection local time of multidimensional fractional Brownian motion

Let be a fractional Brownian motion taking values in with Hurst index . In this paper, we consider the self-intersection local time and its derivative in the spatial variable . In particular, we introduce the so-called integrated quadratic covariation and show that the Bouleau-Yor type identityholds for some suitable .

Polymer measure: Varadhan's renormalization revisited

Through chaos decomposition, we improve the Varadhan estimate for the rate of convergence of the centered approximate self-intersection local time of planar Brownian motion.

A WHITE NOISE APPROACH TO THE SELF-INTERSECTION LOCAL TIMES OF A GAUSSIAN PROCESS

In this paper we show that for any spatial dimension the renormalized self-intersection local times of a certain Gaussian process defined by indefinite Wiener integral exist as Hida distributions. An explicit expression for the chaos decomposition in terms of Wick tensor powers of white noise is also obtained. We also study a regularization of the self-intersection local times and prove a convergence result in the space of Hida distributions.DOI : http://dx.doi.org/10.22342/jims.20.2.136.111-124

On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion

The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen (2005) and subsequently used by others to study the L2 and L3 moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) was introduced in Yan et al. (2008); however, the definition given there presents difficulties, since it is motivated by an incorrect application of the fractional Itô formula. To rectify this, we introduce a modified DSLT for fBm and prove existence using an explicit Wiener chaos expansion. We will then argue that our modification is the natural version of the DSLT by rigorously proving the corresponding Tanaka formula. This formula corrects a formal identity given in both Rosen (2005) and Yan et al. (2008). In the course of this endeavor we prove a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously seen in the literature. As a further byproduct of our investigation, we also provide a small correction to an important technical second-moment bound for fBm which has appeared in the literature many times.

Exponential moments of self-intersection local times of stable random walks in subcritical dimensions

Let $(X_t, t \geq 0)$ be an $\alpha$-stable random walk with values in $\Z^d$. Let $l_t(x) = \int_0^t \delta_x(X_s) ds$ be its local time. For $p>1$, not necessarily integer, $I_t = \sum_x l_t^p(x)$ is the so-called $p$-fold self- intersection local time of the random walk. When $p(d -\alpha) < d$, we derive precise logarithmic asymptotics of the probability $P(I_t \geq r_t)$ for all scales $r_t \gg \E(I_t)$. Our result extends previous works by Chen, Li and Rosen 2005, Becker and Konig 2010, and Laurent 2012.