Self-intersection Local Time Research Articles

The existence and smoothness of self-intersection local time for a class of Gaussian processes

In this paper sufficient conditions for the existence and smoothness of the self-intersection local time of a class of Gaussian processes are given in the sense of Meyer–Watanabe through L2 convergence and Wiener chaos expansion. Let X be a centered Gaussian process, whose canonical metric E[(X(t)−X(s)2)] is commensurate with σ2(|t−s|), where σ(⋅) is continuous, increasing and concave. If ∫0T1σ(γ)dγ<∞, then the self-intersection local time of the Gaussian process exists, and if ∫0T(σ(γ))−32dγ<∞, the self-intersection local time of the Gaussian process is smooth in the sense of Meyer–Watanabe.

Limit Theorem for Self-intersection Local Time Derivative of Multidimensional Fractional Brownian Motion

Limit Theorem for Self-intersection Local Time Derivative of Multidimensional Fractional Brownian Motion

Refinements of asymptotics at zero of Brownian self-intersection local times

In this paper, we establish some estimates related to the Gaussian densities and to Hermite polynomials in order to obtain an almost sure estimate for each term of the Itô-Wiener expansion of the self-intersection local times of the Brownian motion. In dimension [Formula: see text] the self-intersection local times of the Brownian motion can be considered as a family of measures on the classical Wiener space. We provide some asymptotics relative to these measures. Finally, we try to estimate the quadratic Wasserstein distance between these measures and the Wiener measure.

Derivative of Multiple Self-intersection Local Time for Fractional Brownian Motion

Derivative of Multiple Self-intersection Local Time for Fractional Brownian Motion

Stochastic analysis for vector-valued generalized grey Brownian motion

In this article, we show that the standard vector-valued generalization of a generalized grey Brownian motion (ggBm) has independent components if and only if it is a fractional Brownian motion. In order to extend ggBm with independent components, we introduce a vector-valued generalized grey Brownian motion (vggBm). The characteristic function of the corresponding measure is introduced as the product of the characteristic functions of the one-dimensional case. We show that for this measure, the Appell system and a calculus of generalized functions or distributions are accessible. We characterize these distributions with suitable transformations and give a d d -dimensional Donsker’s delta function as an example for such distributions. From there, we show the existence of local times and self-intersection local times of vggBm as distributions under some constraints, and compute their corresponding generalized expectations. At the end, we solve a system of linear SDEs driven by a vggBm noise in d d dimensions.

Self-intersection local time derivative for systems of non-linear stochastic heat equations

We consider the existence and Hölder continuity conditions for the higher-order derivative of self-intersection local time for u(t, x), where u(t,x)=u1(t,x),…,ud(t,x) is the solution to a system of non-linear stochastic heat equations driven by a d-dimensional space-time white noise. Moreover, we study the cases of intersection local time and collision local time.

Existence and Smoothness of Local Time and Self-intersection Local Time for Spherical Gaussian Random Fields

In this paper, we study properties of local time for spherical Gaussian random fields T on $${\mathbb{S}^2}$$ . For 2 < α < 4, we give a formally expression of local time for spherical Gaussian random fields, and obtain that the existence of local time in L2 if and only if (α − 2)d < 4 and the smoothness in the sense of Meyer–Watanabe if and only if (α − 2)(d + 2) < 4, respectively. Finally, the existence and smoothness of self-intersection local time of T are considered.

Exponential asymptotics for Brownian self-intersection local times under Dalang’s condition

Exponential asymptotics for Brownian self-intersection local times under Dalang’s condition

Higher-order derivatives of self-intersection local time for linear fractional stable processes

In this paper, we aim to consider the k = (k1, k2,..., kd)-th order derivatives ?(k)(T, x) of selfintersection local time ?(T, x) for the linear fractional stable process X?,H in Rd with indices ? ? (0, 2) and H = (H1,...,Hd) ? (0, 1)d. We first give sufficient condition for the existence and joint H?lder continuity of the derivatives ?(k)(T, x) using the local nondeterminism of linear fractional stable processes. As a related problem, we also study the power variation of ?(k)(T, x).

Derivative of self-intersection local time for the sub-bifractional Brownian motion

&lt;abstract&gt;&lt;p&gt;Let $ S^{H, K} = \{S^{H, K}_t, t\geq 0\} $ be the sub-bifractional Brownian motion (sbfBm) of dimension 1, with indices $ H\in (0, 1) $ and $ K\in (0, 1]. $ We mainly consider the existence of the self-intersection local time and its derivative for the sbfBm. Moreover, we prove its derivative is H$ \ddot{o} $lder continuous in space variable and time variable, respectively.&lt;/p&gt;&lt;/abstract&gt;

Renormalized self-intersection local time for sub-bifractional Brownian motion

Let SH,K = {SH,K(t), t ? 0} be a d?dimensional sub-bifractional Brownian motion with indices H ? (0, 1) and K ? (0,1]. Assuming d ? 2, as HKd &lt; 1, we mainly prove that the renormalized self-intersection local time ? t0 ? s0 ?(SH,K(s) ? SH,K(r))drds ? E [?t0 ?s0 ?(SH,K(s) ? SH,K(r))drds] exists in L2, where ?(x) is the Dirac delta function for x ? Rd.

Smoothness of higher order derivative of self-intersection local time for fractional Brownian motion

In a recent paper, Yu (2021) define a -th order derivative of self-intersection local times with respect to d-dimensional fractional Brownian motion. By the existence and Hölder continuity proved in Yu (2021), we study the smoothness of derivative of self-intersection local times with respect to fractional Brownian motion in this paper. We also consider the cases of intersection local times and collision local times, respectively.

Existence and Hölder continuity conditions for self-intersection local time of Rosenblatt process

We consider the existence and Hölder continuity conditions for the self-intersection local time of Rosenblatt process. Moreover, we study the cases of intersection local time and collision local time, respectively.

Higher-Order Derivative of Self-Intersection Local Time for Fractional Brownian Motion

We consider the existence and Hölder continuity conditions for the k-th-order derivatives of self-intersection local time for d-dimensional fractional Brownian motion, where $$k=(k_1,k_2,\ldots , k_d)$$ . Moreover, we show a limit theorem for the critical case with $$H=\frac{2}{3}$$ and $$d=1$$ , which was conjectured by Jung and Markowsky [7].

Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion

In a recent paper by Yu (arXiv:2008.05633, 2020), higher order derivatives of self-intersection local time of fractional Brownian motion were defined, and existence over certain regions of the Hurst parameter H was proved. Utilizing the Wiener chaos expansion, we provide new proofs of Yu’s results and show how a Varadhan-type renormalization can be used to extend the range of convergence for the even derivatives.

Asymptotic properties for q-th chaotic component of derivative of self-intersection local time of fractional Brownian motion

Let {BtH}t≥0 be a fractional Brownian motion with Hurst parameter H∈(0,1). Consider the approximation of derivative of self-intersection local time of BH, defined asαε=∫0T∫0tpε′(BtH−BsH)dsdt, where pε(x) is the heat kernel. For q≥2, renormalized by different factors, we prove two different central limit theorems for q-th chaotic component of αε when H=34 and H=4q−34q−2, respectively. The results are obtained based on the fourth moment theorem introduced in [8].

Spectral Rigidity of Random Schrödinger Operators via Feynman–Kac Formulas

We develop a technique for proving number rigidity (in the sense of Ghosh and Peres in Duke Math J 166(10):1789–1858, 2017) of the spectrum of general random Schrodinger operators (RSOs). Our method makes use of Feynman–Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman–Kac formulas for RSOs with multiplicative noise (Gaudreau Lamarre in Semigroups for one-dimensional Schrodinger operators with multiplicative Gaussian noise, Preprint arXiv:1902.05047v3 , 2019; Gaudreau Lamarre and Shkolnikov in Ann Inst Henri Poincare Probab Stat 55(3):1402–1438, 2019; Gorin and Shkolnikov in Ann Probab 46(4):2287–2344, 2018) by Gorin, Shkolnikov, and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form $$-\frac{1}{2}\Delta +V+\xi $$ , where V is a deterministic potential and $$\xi $$ is a stationary Gaussian noise. Our results require only very mild assumptions on the domain on which the operator is defined, the boundary conditions on that domain, the regularity of the potential V, and the singularity of the noise $$\xi $$ .

Fractional smoothness of derivative of self-intersection local times with respect to bi-fractional Brownian motion

Let BHi,Ki={BtHi,Ki,t≥0},i=1,2 be two independent bifractional Brownian motions with respective indices Hi∈(0,1) and Ki∈(0,1]. In this paper, we study the fractional smoothness of derivative of the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motion.

Self-intersection local times for multifractional Brownian motion in higher dimensions: A white noise approach

In this paper, we study the self-intersection local times of multifractional Brownian motion (mBm) in higher dimensions in the framework of white noise analysis. We show that when a suitable number of kernel functions of self-intersection local times of mBm are truncated then we obtain a Hida distribution. In addition, we present the expansion of the self-intersection local times in terms of Wick powers of white noises. Moreover, we obtain the convergence of the regularized truncated self-intersection local times in the sense of Hida distributions.

Functional limit theorem for the self-intersection local time of the fractional Brownian motion

Soit $\{B_{t}\}_{t\geq0}$ un mouvement brownien fractionnaire $d$-dimensionel avec parametre de Hurst $0<H<1$, ou $d\geq2$. On considere l’approximation du temps local d’auto-intersection du processus $B$, defini comme \[I_{T}^{\varepsilon}=\int_{0}^{T}\int_{0}^{t}p_{\varepsilon}(B_{t}-B_{s})\,ds\,dt,\] ou $p_{\varepsilon}(x)$ est le noyau de la chaleur. Nous demontrons que le processus $\{I_{T}^{\varepsilon}-\mathbb{E}[I_{T}^{\varepsilon}]\}_{T\geq0}$, reechelonne avec une normalisation convenable, converge en loi vers un mouvement brownien multiplie par une constante si $\frac{3}{2d}<H\leq\frac{3}{4}$ et vers une somme de processus de Hermite independants multipliee par une constante si $\frac{3}{4}<H<1$, dans l’espace $C[0,\infty)$, muni de la topologie de la convergence uniforme sur les compacts.