Abstract:
We study large deviations for the renormalized self-intersection local time of d-dimensional stable processes of index \beta \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 \beta

Abstract:
We study $\gamma_{k}(x_2,...,x_k;t)$, the k-fold renormalized self-intersection local time for Brownian motion in $R^1$. Our main result says that $\gamma_{k}(x_2,...,x_k;t)$ is continuously differentiable in the spatial variables, with probability 1.

Abstract:
Let \beta_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+\delta moments. On a suitable probability space we can construct X_n and a planar Brownian motion W_t such that for each k\geq 2, |\beta_k(n)-\gamma_k(n)|=O(n^{-a}), a.s. for some a>0 where \gamma_k(n) is the renormalized self-intersection local time of order k at time 1 for the Brownian motion W_{nt}/\sqrt n.

Abstract:
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.

Abstract:
We define renormalized intersection local times for random interlacements of L\'evy processes in R^{d} and prove an isomorphism theorem relating renormalized intersection local times with associated Wick polynomials.

Abstract:
We study the problem of self-intersection local time of -dimensional subfractional Brownian motion based on the property of chaotic representation and the white noise analysis. 1. Introduction As an extension of Brownian motion, Bojdecki et al. [1] introduced and studied a rather special class of self-similar Gaussian process. This process arises from occupation time fluctuations of branching particles with Poisson initial condition. It is called the subfractional Brownian motion. The so-called subfractional Brownian motion with index is a mean zero Gaussian process with the covariance function for all . For , coincides with the standard Brownian motion. is neither a semimartingale nor a Markov process unless , so many of the powerful techniques from classical stochastic analysis are not available when dealing with . The subfractional Brownian motion has properties analogous to those of fractional Brownian motion, such as self-similarity, H？lder continuous paths, and so forth. But its increments are not stationary, because, for , we have the following estimates: Let be a -dimensional subfractional Brownian motion with multiparameters . Suppose that , we are interested in, when it exists, the self-intersection local time of subfractional Brownian motion which is formally defined as where is the Dirac delta function. It measures the amount of time that the processes spend intersecting itself on the time interval and has been an important topic of the theory of stochastic process. More precisely, we study the existence of the limit when tends to zero, of the following sequence of processes where For , the process is a classical Brownian motion. The self-intersection local time of the Brownian motion has been studied by many authors such as Albeverio et al. [2], Calais and Yor [3], He et al. [4], Hu [5], Varadhan [6], and so forth. In the case of planar Brownian motion, Varadhan [6] has proved that does not converge in but it can be renormalized so that converges in as tends to zero. The limit is called the renormalized self-intersection local time of the planar Brownian motion. This result has been extended by Rosen [7] to the (planar) fractional Brownian motion, where it is proved that for , converges in as tends to zero, where is a constant depending only on . Hu [8] showed that, under the condition , the (renormalized) self-intersection local time of fractional Brownian motion is in the Meyer-Watanabe test functional space, that is, the space of “differentiable” functionals. In 2005, Hu and Nualart [9] proved that the renormalized self-intersection

Abstract:
In this article we calculate the third and fourth moment of the renormalized intersection local time of a planar Brownian motion. The third moment is calculated anlaytically, the fourth moment numerically. For the closed planar random walk the third moment of the distribution of the multiple point range is also calculated in leading order.

Abstract:
We obtain direct, finite, descriptions of a renormalized quantum mechanical system with no reference to ultraviolet cutoffs and running coupling constants, in both the Hamiltonian and path integral pictures. The path integral description requires a modification to the Wiener measure on continuous paths that describes an unusual diffusion process wherein colliding particles occasionally stick together for a random interval of time before going their separate ways.

Abstract:
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an $L^p$-function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.

Abstract:
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^t\delta(B_t^H-B_s^H) ds dt -E\biggl(\int_0^T\int_0^t\delta (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(\epsilon)\ell_{\epsilon} converges in distribution to a normal law N(0,T\sigma^2), as \epsilon tends to zero, where \ell_{\epsilon} is an approximation of \ell, defined through (2), and r(\epsilon)=|\log\epsilon|^{-1} if H=3/(2d), and r(\epsilon)=\epsilon^{d-3/(2H)} if 3/(2d)