L. Mytnik, J. Villa
The existence of self-intersection local time (SILT), when the time diagonal is intersected, of the $(α,d,β)$-superprocess is proved for $d/2<α$ and for a renormalized SILT when $d/(2+(1+β)^{-1})<α\leq d/2$. We also establish Tanaka-like formula for SILT.
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This paper examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions $d\leq3$, which through constructive methods, results in a Tanaka-like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin's proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler's method of calculating moments is extended to higher moments, from which existence follows.
On the exponential integrability of the derivative of intersection and self-intersection local time for fractional Brownian motion and a limit theorem related to the self-intersection local time for fractional Brownian motion
该论文修正了分数布朗运动交局部时导数存在的条件，证明了其指数可积性，并给出了存在条件违反时自交局部时导数的极限定理。
In the paper Dynkin construction for self-intersection local time of planar Wiener process is extended on Hilbert-valued weights.
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.
We prove joint Holder continuity and an occupation-time formula for the self-intersection local time of fractional Brownian motion. Motivated by an occupation-time formula, we also introduce a new version of the derivative of self-intersection local time for fractional Brownian motion and prove Holder conditions for this process. This process is related to a different version of the derivative of self-intersection local time studied by the authors in a previous work.
In present article we prove the existence of multiple self-intersection local times, describe its Ito-Wiener expansion and establish Clark representation for the class of Gaussian integrators generated by operators with a finite dimensional kernel.
In this article we study transformations of Gaussian field by stochastic flow on the plane. A stochastic flow is a solution to the equation with interaction whose coefficients depend on the occupation measure of the field. We consider nonsmooth Gaussian field, which has self-intersection local times of any multiplicity. In the article we prove the existence of self-intersection local times for the transformed field and study its asymptotics.
We obtain a large deviations principle for the self-intersection local times for a symmetric random walk in dimension d>4. As an application, we obtain moderate deviations for random walk in random sceneries in some region of parameters.
We study the mean and variance of the number of self-intersections of the equilateral isotropic random walk in the plane, as well as the corresponding quantities for isotropic equilateral random polygons (random walks conditioned to return to their starting point after a given number of steps). The expected number of self-intersections is $(2/π^2)n\log n + O(n)$ for both walks and polygons with $n$ steps. The variance is $O(n^2 \log n)$ for both walks and polygons, which shows that the number of self-intersections exhibits concentration around the mean.
In this paper, we prove two limit laws for functionals of self-intersection symmetric alpha-stable processes with alpha\in(1,2). The results are obtained based on the method of moments, the sample configuration and the chaining argument introduced in (Nualart and Xu 2013) are employed.
In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the $d$% -dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$. As a consequence, we derive the existence of some exponential moments for this random variable.
We present a Darboux-Wiener type lemma and apply it to obtain an exact asymptotic for the variance of the self-intersection of one and two-dimensional random walks. As a corollary, we obtain a central limit theorem for random walk in random scenery conjectured by Kesten and Spitzer in 1979.
In this paper we compute the $\frac 43$-variation of the derivative of the self-intersection Brownian local time $γ_t=\int_0^t \int_0^u δ'(B_u-B_s)dsdu\,, t\ge 0$, applying techniques from the theory of fractional martingales.
The existence condition $H<1/d$ for first-order derivative of self-intersection local time for $d\geq3$ dimensional fractional Brownian motion can be obtained in Yu (2021). In this paper, we show a limit theorem under the non-existence critical condition $H=1/d$.
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 $k=(k_1,k_2,\cdots, k_d)$. 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 (2014).
The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-i…
On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes
本文研究 alpha-稳定过程（含布朗运动）相交及自相交局部时间导数的指数可积性，并指出文献中分数布朗运动相关结论的一个反例。
In this article 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 $d\geqslant 4$ 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.
We obtain large deviations estimates for the self-intersection local times for a symmetric random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length $n$, comes from sites visited less than some power of $\log(n)$. This is opposite to the situation in dimensions larger or equal to 5. Finally, we present two applications of our estimates: (i) to moderate deviations estimates for the range of a random walk, and (ii) to moderate deviations for random walk in random sceneries.
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.
Let B_n be the number of self-intersections of a symmetric random walk with finite second moments in the integer planar lattice. We obtain moderate deviation estimates for B_n - E B_n and E B_n- B_n, which are given in terms of the best constant of a certain Gagliardo-Nirenberg inequality. We also prove the corresponding laws of the iterated logarithm.
The asymptotics of the probability that the self-intersection local time of a random walk on $\Z^d$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to large-deviation theory and variational analysis and because of the variety of the effects that can be observed. However, the proof of the upper bound is notoriously difficult and requires various sophisticated techniques. We survey some heuristics and some recently elaborated techniques and results. This is an extended summary of a talk held on the CIRM-confere…
We consider Random Walk in Random Scenery, denoted $X_n$, where the random walk is symmetric on $Z^d$, with $d>4$, and the random field is made up of i.i.d random variables with a stretched exponential tail decay, with exponent $α$ with $1<α$. We present asymptotics for the probability, over both randomness, that $\{X_n>n^β\}$ for $1/2<β<1$. To obtain such asymptotics, we establish large deviations estimates for the the self-intersection local times process.
We study large deviations for the renormalized self-intersection local time of d-dimensional stable processes of index β\in (2d/3,d]. We find a difference between the upper and lower tail. In addition, we find that the behavior of the lower tail depends critically on whether β<d or β=d.
Fourier-Wiener transform of the formal expression for multiple self-intersection local time is described in terms of the integral, which is divergent on the diagonals. The method of regularization we use in this work related to regularization of functions with non-integrable singularities. The strong local nondeterminism property, which is more restrictive than the property of local nondeterminism introduced by S.Berman is considered. Its geometrical meaning in the construction of the regularization is investigated. As the example the problem of regularization is solved for the compact perturb…
Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion
利用Wiener混沌展开，重新证明了分数布朗运动自相交局部时高阶导数的存在性，并引入Varadhan型重整化以扩展偶次导数的收敛范围。
If β_t is renormalized self-intersection local time for planar Brownian motion, we characterize when Ee^{γβ_1} is finite or infinite in terms of the best constant of a Gagliardo-Nirenberg inequality. We prove large deviation estimates for β_1 and -β_1. We establish lim sup and lim inf laws of the iterated logarithm for β_t as t\to\infty.
Let $(X_t, t \geq 0)$ be an $α$-stable random walk with values in $\Z^d$. Let $l_t(x) = \int_0^t δ_x(X_s) ds$ 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 $p(d -α) < d$, 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 König 2010, and Laurent 2012.
In the paper $k$-multiple self-intersection local time for planar Gaussian integrators generated by linear operator with nontrivial kernel is studied. In this case additional singularities arise in its formal Fourier--Wiener transform. In case $k=2$ the set of singularities is the finite number of points. In case $k>2$ it contains intervals and hyperplanes. In the first and the second cases using two different approaches related on structure of set of singularities we show that "new" singularities do not imply on the convergence of integral corresponding to the formal Fourier--Wiener transform…
The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen and subsequently used by others to study the $L^2$ and $L^3$ moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) has also appeared in the literature. In the present work, we further its study by proving a Tanaka-style formula for it. In the course of this endeavor we provide 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 se…
Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $ l_T(x)= \int_0^T δ_x(X_s)ds$ the local time at the state $x$ and $ I_T= \sum\limits_{x\in\mathbb{Z}^d} l_T(x)^q $ the q-fold self-intersection local time (SILT). In \cite{Castell} Castell proves a large deviations principle for the SILT of the simple random walk in the critical case $q(d-2)=d$. In the supercritical case $q(d-2)>d$, Chen and Mörters obtain in \cite{ChenMorters} a large deviations principle for the intersection of $q$ independent random walks, and Asselah obtains in \cite{Asselah5} a large deviations principle for the…
Let $\{B_{t}\}_{t\geq0}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $0<H<1$, where $d\geq2$. Consider the approximation of the self-intersection local time of $B$, defined as \begin{align*} I_{T}^{\varepsilon} &=\int_{0}^{T}\int_{0}^{t}p_{\varepsilon}(B_{t}-B_{s})dsdt, \end{align*} where $p_\varepsilon(x)$ is the heat kernel. We prove that the process $\{I_{T}^{\varepsilon}-\mathbb{E}\left[I_{T}^{\varepsilon}\right]\}_{T\geq0}$, rescaled by a suitable normalization, converges in law to a constant multiple of a standard Brownian motion for $\frac{3}{2d}<H\leq\frac{3}{4…
Let $\{B_{t}\}_{t\geq0}$ be a fractional Brownian motion with Hurst parameter $\frac{2}{3}<H<1$. We prove that the approximation of the derivative of self-intersection local time, defined as \begin{align*} α_{\varepsilon} &= \int_{0}^{T}\int_{0}^{t}p'_{\varepsilon}(B_{t}-B_{s})\text{d}s\text{d}t, \end{align*} where $p_\varepsilon(x)$ is the heat kernel, satisfies a central limit theorem when renormalized by $\varepsilon^{\frac{3}{2}-\frac{1}{H}}$. We prove as well that for $q\geq2$, the $q$-th chaotic component of $α_{\varepsilon}$ converges in $L^{2}$ when $\frac{2}{3}<H<\frac{3}{4}$, and sat…
Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $ l_t(x)= \int_0^t δ_x(X_s)ds$ be the local time at site $x$ and $ I_t= \sum\limits_{x\in\mathbb{Z}^d} l_t(x)^p $ the p-fold self-intersection local time (SILT). Becker and König have recently proved a large deviations principle for $I_t$ for all $(p,d)\in\mathbb{R}^d\times\mathbb{Z}^d$ such that $p(d-2)<2$. We extend these results to a broader scale of deviations and to the whole subcritical domain $p(d-2)<d$. Moreover we unify the proofs of the large deviations principle using a method introduced by Castell for the critical case $p(…
Let β_k(n) be the number of self-intersections of order k, appropriately renormalized, for a mean zero random walk X_n in Z^2 with 2+δmoments. On a suitable probability space we can construct X_n and a planar Brownian motion W_t such that for each k\geq 2, |β_k(n)-γ_k(n)|=O(n^{-a}), a.s. for some a>0 where γ_k(n) is the renormalized self-intersection local time of order k at time 1 for the Brownian motion W_{nt}/\sqrt n.
Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $\Z^d$ up to time $t$. This is the $p$-norm of the vector of the walker's local times, $\ell_t$. We derive precise logarithmic asymptotics of the expectation of $\exp\{θ_t \|\ell_t\|_p\}$ for scales $θ_t>0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $θ_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the {\it Gagliardo-Nire…
Let B_t^H be a d-dimensional fractional Brownian motion with Hurst parameter H\in(0,1). Assume d\geq2. We prove that the renormalized self-intersection local time\ell=\int_0^T\int_0^tδ(B_t^H-B_s^H) ds dt -E\biggl(\int_0^T\int_0^tδ(B_t^H-B_s^H) ds dt\biggr) exists in L^2 if and only if H<3/(2d), which generalizes the Varadhan renormalization theorem to any dimension and with any Hurst parameter. Motivated by a result of Yor, we show that in the case 3/4>H\geq\frac{3}{2d}, r(ε)\ell_ε converges in distribution to a normal law N(0,Tσ^2), as εtends to zero, where \ell_ε is an approximation of \ell,…
We study $γ_{k}(x_2,...,x_k;t)$, the k-fold renormalized self-intersection local time for Brownian motion in $R^1$. Our main result says that $γ_{k}(x_2,...,x_k;t)$ is continuously differentiable in the spatial variables, with probability 1.
For a random walk $S_n, n\geq 0$ in $\mathbb{Z}^d$, let $l(n,x)$ be its local time at the site $x\in \mathbb{Z}^d$. Define the $α$-fold self intersection local time $L_n(α) := \sum_{x} l(n,x)^α$, and let $L_n(α|ε, d)$ the corresponding quantity for $d$-dimensional simple random walk. Without imposing any moment conditions, we show that the variances of the local times $\mathop{var}(L_n(α))$ of any genuinely $d$-dimensional random walk are bounded above by the corresponding characteristics of the simple symmetric random walk in $\mathbb{Z}^d$, i.e. $\mathop{var}(L_n(α)) \leq C \mathop{var}[L_n(…
We prove that the self-intersection local times for generalized grey Brownian motion $B^{β,α}$ in arbitrary dimension $d$ is a well defined object in a suitable distribution space for $dα<2$.
We study the fluctuations of self-intersection counts of random geodesic segments of length $t$ on a compact, negatively curved surface in the limit of large $t$. If the initial direction vector of the geodesic is chosen according to the \emph{Liouville measure}, then it is not difficult to show that the number $N (t)$ of self-intersections by time $t$ grows like $κt^{2}$, where $κ=κ_{M}$ is a positive constant depending on the surface $M$. We show that (for a smooth modification of $N (t)$) the fluctuations are of size $t$, and the limit distribution is a weak limit of Gaussian quadratic form…
We show a remarkable similarity between strategies to realize a large intersection or self-intersection local times in dimension five or more. This leads to the same rate functional for large deviation principles for the two objects obtained respectively by Chen and Morters, and by the present author. We also present a new estimate for the distribution of high level sets for a random walk, with application to the geometry of the intersection set of two high level sets of the local times of two independent random walks.
Strong transport noise prevents self-intersection in 2D fluid-structure interaction
针对二维流固耦合中弹性壳长期自相交的开放问题，本文通过在结构方程中加入足够强的输运噪声，证明在任意给定时间前自相交不会以任意小于1的概率发生。
In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is the number of letters of the corresponding word. The self-intersection of a free homotopy class (that is, the smallest number of self-crossings of a representative of a class) can be computed in terms of the word. For each of the free homotopy classes of length twenty on the punctured torus, we compute its self-intersection number and make a histogram of how…
We consider a periodic planar Lorentz process with strictly convex obstacles and finite horizon. This process describes the displacement of a particle moving in the plane with unit speed and with elastic reflection on the obstacles. We call number of self-intersections of this Lorentz process the number V(n) of couples of integers (k,m) smaller than n such that the particle hits a same obstacle both at the k-th and at the m-th collision times. The aim of this paper is to prove that the variance of V(n) is equivalent to cn^2 (such a result has recently been proved for simple planar random walks…
We reveal a shape transition for a transient simple random walk forced to realize an excess $q$-norm of the local times, as the parameter $q$ crosses the value $q_c(d)=\frac{d}{d-2}$. Also, as an application of our approach, we establish a central limit theorem for the $q$-norm of the local times in dimension 4 or more.
Self-intersection local times for Volterra Gaussian processes in stochastic flows with interaction
研究具有相互作用的随机流中Volterra高斯过程的自交局部时，通过建立变元公式将过程x的自交局部时表示为过程u的加权自交局部时，并证明其存在性及大时间渐近性。
On the self-intersection time of non-backtracking random walks
该研究分析了连通无向图上非回溯随机游走的自相交时间，得出不同条件下的上下界，并将其应用于改进Δ正则图上Ising模型Glauber动力学的混合时间界。
Local times of self-intersection and sample path properties of Volterra Gaussian processes
本文研究由连续核与 Wiener 过程定义的 Volterra 高斯过程，建立一维重对数律、局部时存在性及其对核的连续依赖，并证明平面情形 Rosen 重整化自相交局部时存在。
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