Self-intersection Local Time Research Articles

Probabilistic models for vortex filaments based on fractional Brownian motion

We consider a vortex structure based on a three-dimensional fractional Brownian motion with Hurst parameter $H>\frac{1}{2}.$ We show that the energy $\mathbb{H}$\vspace*{-1pt} has moments of any order under suitable conditions. When $H\in (\frac{1}{2},\frac{1}{3})$ we prove that the intersection energy $\mathbb{H}_{xy}$ can be decomposed into four terms, one of them being a weighted self-intersection local time of the fractional Brownian motion in $\mathbb{R}^{3}$.

Self-Intersection Local Time for ′(ℝd)-Ornstein-Uhlenbeck Processes Arising from Immigration Systems

Fluctuation limits of an immigration branching particle system and an immigration branching measure-valued process yield different types of 𝒮′(ℝd)-valued Ornstein-Uhlenbeck processes whose covariances are given in terms of an excessive measure for the underlying motion in Rd, which is taken to be a symmetric α-stable process. In this paper we prove existence and path continuity results for the self-intersection local time of these Ornstein-Uhlenbeck processes. The results depend on relationships between the dimension d and the parameter α.

Self-intersection local time of Additive levy process

This article discusses the problem of existence of jointly continuous self-intersection local time for an additive levy process. Here, “local time” is understood in the sense of occupation density, and by an additive Levy process the authors mean a process X={X(t),t∈R+N)} which has the decomposition X=X1⊕X2⊕⋯⊕XN, each X· has the lower index αℓ,α=min {α1,⋯,αN}. Let Z=(Xt2−Xt1,⋯,Xtr−Xtr−1). They prove that if Nrα > d(r – 1), then a jointly continuous local time of Z, i.e. the self-intersection local time of X, can be obtained.

Bivariate occupation measure dimension of multidimensional processes

Bivariate occupation measure dimension is a new dimension for multidimensional random processes. This dimension is given by the asymptotic behavior of its bivariate occupation measure. Firstly, we compare this dimension with the Hausdorff dimension. Secondly, we study relations between these dimensions and the existence of local time or self-intersection local time of the process. Finally, we compute the local correlation dimension of multidimensional Gaussian and stable processes with local Hölder properties and show it has the same value that the Hausdorff dimension of its image have. By the way, we give a new a.s. convergence of the bivariate occupation measure of a multidimensional fractional Brownian or particular stable motion (and thus of a spatial Brownian or Lévy stable motion).

DIVERGENCE RESULTS FOR SELF-INTERSECTION LOCAL TIMES OF GAUSSIAN ${\mathscr S}' ({\mathbb R}^d)$-PROCESSES

For various types of Gaussian [Formula: see text]-processes we consider the case when the self-intersection local time (SILT) does not exist. We study the rate of divergence of the corresponding approximating processes obtaining, after suitable normalizations convergence in law to some [Formula: see text]-valued processes (not necessarily Gaussian). We also obtain some new necessary conditions for the existence of SILT. We give examples associated with fluctuation limits of α-stable particle systems.

Self-intersection local time of order k for Gaussian processes in [formula omitted

We study existence and continuity of self-intersection local time of any order of Gaussian S′( R d) processes. In particular, we give results for Wiener and Ornstein–Uhlenbeck processes. We study processes associated with several classes of covariances, which arise in examples mainly as fluctuation limits of particle systems.

Self-Intersection Local Time for SomeS′(ℝd)-Ornstein-Uhlenbeck Processes Related to Inhomogeneous Fields

We study existence and path continuity of the self-intersection local time (SILT) for some S′(ℝd)-valued Ornstein-Uhlenbeck processes. Examples of such processes arise in particular as fluctuation limits ofparticle systems. We analyze the effect that different types ofmeasures involved in the covariances of the processes have on existence and continuity of SILT. These measures do not necessarily have the regularity and homogeneity properties of those that have been considered before; this precludes some key ingredients of the previous techniques. We develop new technical tools and prove sufficient conditions and some necessary conditions for existence of SILT, and sufficient conditions for path continuity of SILT. The questions of existence and continuity involve problems of existence of integrals and singular integrals. We give examples which illustrate how different types of measures e.g., atomic or with L2-density) may produce different critical dimensions for existence of SILT. One of our motivations is the desire for a better understanding ofwhat SILT represents for S′(ℝd)-valued processes.

Dirichlet processes and an intrinsic characterization of renormalized intersection local times

We show that the nth order renormalized self-intersection local time γ n ( μ; t) for the symmetric stable process in R 2, where the n-fold multiple points are weighted by an arbitrary measure μ, can be characterized as the continuous process of zero quadratic variation in the decomposition of a natural Dirichlet process. This Dirichlet process is the potential of a random measure associated with γ n−1 ( μ; t).

Self-intersection local time of fractional Brownian motions-via chaos expansion

Let $B_{1,t}^{H},\ldots ,B_{d,t}^{H}$ be $d$ independent fractional Brownian motions with Hurst parameter $H\in (0, 1)$. Denote $X_{t}=(B_{1,t}^{H},\ldots ,B_{d,t}^{H})$ and let $\delta$ be the Dirac delta function. It is shown that when $H< \mathrm{min}(3/(2d), 2/(d+2))$, the (renormalized) self-intersection local time of fractional Brownian motion, $\int _{0}^{T}\int _{0}^{t}\delta (X_{t}-X_{s})dsdt-\mathbb{E}\int _{0}^{T}\int _{0}^{t}\delta (X_{t}-X_{s})dsdt$, is in $D_{1,2}$, where $D_{1,2}$ is the Meyer-Watanabe test functional space, i.e. the $L^{2}$ space of “differentiable” functionals, whose precise meaning is given in Section 2.

SELF-INTERSECTION LOCAL TIME FOR ${\mathcal S}' ({\mathbb R}^d)$-WIENER PROCESSES AND RELATED ORNSTEIN–UHLENBECK PROCESSES

Existence and continuity results are obtained for self-intersection local time of [Formula: see text]-valued Ornstein–Uhlenbeck processes [Formula: see text], where X0 is Gaussian, Wt is an [Formula: see text]-Wiener process (independent of X0), and T't is the adjoint of a semigroup Tt on [Formula: see text]. Two types of covariance kernels for X0 and for W are considered: square tempered kernels and homogeneous random field kernels. The case where Tt corresponds to the spherically symmetric α-stable process in ℝd, α∈(0,2], is treated in detail. The method consists in proving first results for self-intersection local times of the ingredient processes: Wt, T't X0 and [Formula: see text], from which the results for Xt are derived. As a by-product, a class of non-finite tempered measures on ℝd whose Fourier transforms are functions is identified. The tools are mostly analytical.

Self-intersection local time of an S′(Rd)-valued process involving motions of two types

We study existence and continuity of self-intersection local time (SILT) for a Gaussian S ′( R d ) -valued process which arises as a high-density fluctuation limit of a particle system in R d , where the particle motion switches back and forth between symmetric stable processes of indices α 1 and α 2 at exponential time intervals. We prove that SILT exists if and only if d<2 min {α 1 ,α 2 }. This means that existence of SILT is determined by the “most mobile” of the two types, and we interpret this result in terms of the particle picture. In contrast with the single-type case, there are technical difficulties due to the lack of self-similarity of the particle paths.

Joint continuity and a Doob-Meyer type decomposition for renormalized intersection local times

We study renormalized self-intersection local times γ n ( μ x ; t) for Lévy processes, where the n-fold multiple points are weighted by the translate μ x of an arbitrary measure μ. General sufficient conditions are provided which insure that γ n ( μ x ; t) is a.s. jointly continuous in x and t. Our proof involves a new Doob-Meyer type decomposition of γ n ( μ x ; t) as the difference of a martingale and a lower order renormalized self-intersection local time.

Critical Dimensions for the Existence of Self-Intersection Local Times of the N-Parameter Brownian Motion in Rd

Fix two rectangles A, B in [0, 1]N. Then the size of the random set of double points of the N-parameter Brownian motion \((W_t )_{t \in } [0,1]^N \) in Rd, i.e, the set of pairs (s, t), where s∈A, t∈B, and Ws=Wt, can be measured as usual by a self-intersection local time. If A=B, we show that the critical dimension below which self-intersection local time does not explode, is given by d=2N. If A ∩ B is a p-dimensional rectangle, it is 4N–2p (0≤p≤N). If A ∩ B = ∅, it is infinite. In all cases, we derive the rate of explosion of canonical approximations of self-intersection local time for dimensions above the critical one, and determine its smoothness in terms of the canonical Dirichlet structure on Wiener space.

Renormalized self-intersection local times and Wick power chaos processes

Introduction Wick products Wick power chaos processes Isomorphism theorems Equivalence of two versions of renormalized self-intersection local times Continuity Stable mixtures Examples A large deviation result Appendix A. Necessary conditions Appendix B. The case $n=3$ Bibliography.

Polylinear Additive Functionals of Superprocesses

LetXbe a superdiffusion in a domainEof Rd. A polylinear additive functional ofXcorresponding to a positive Borel functionρis given by the formulaA(B)=∫Bdt1, …, dtk∫Ekρ(t1, z1; …; tk, zk)Xt1(dz1)…Xtk(dzk).By a passage to the limit, we extend this definition to a certain class of generalized functionsρ. More precisely, we associate an additive functionalAνof (Xt, Pμ) with every measureνsubject to the condition∫ν(dt1, dz1; …; dtk, dzk)ν(dt′1, dz′1; …; dt′k, dz′k) emsp;×qμ(t1, z1; …; tk, dzk; t′1, z′1; …; t′k, dz′k)<∞for a certain kernelqμ. We prove that, ifν{t:ti=s}=0 fori=1, …, kand for alls, then measureAνhas, a.s., the same property. This result is applicable, in particular, to self-intersection local times ofX.

The fluctuation result for the multiple point range of two-dimensional recurrent random walks

We study the fluctuation problem for the multiple point range of random walks in the two dimensional integer lattice with mean 0 and finite variance. The $p$-multiple point range means the number of distinct sites with multiplicity $p$ of random walk paths before time $n$. The suitably normalized multiple point range is proved to converge to a constant, which is independent of the multiplicity, multiple of the renormalized self-intersection local time of a planar Brownian motion.

A remark on non-smoothness of the self-intersection local time of planar Brownian motion

Using the Itô-Wiener chaos expansion we prove that the normalized self-intersection local time of planar Brownian motion is not differentiable in the sense of Meyer-Watanabe.

Intersection local times as generalized white noise functionals

For any dimension we present the expansions of Brownian motion self-intersection local times in terms of multiple Wiener integrals. Suitably subtracted, they exist in the sense of generalized white noise functionals; their kernel functions are given in closed (and remarkably simple) form.

On the self intersection local time of Brownian motion-via chaos expansion

We discuss the weak compactness problem related to the self-intersection local time of Brownian motion. We also propose a regular renormalization for self-intersection local time of higher dimensional Brownian motion.

Self-intersection local time for Gaussian [formula omitted]′( [formula omitted]d)-processes: Existence, path continuity and examples

In this paper we develop a criterion for existence or non-existence of self-intersection local time (SILT) for a wide class of Gaussian L ′( R d )-valued processes, we show that quite generally the SILT process has continuous paths, and we give several examples which illustrate existence of SILT for different ranges of dimensions (e.g., d ≤ 3, d ≤ 7 and 5 ≤ d ≤ 11 in the Brownian case). Some of the examples involve branching and exhibit “dimension gaps”. Our results generalize the work of Adler and coauthors, who studied the special case of “density processes” and proved that SILT paths are cadlag in the Brownian case making use of a “particle picture” approximation (this technique is not available for our general formulation).