Clark Representation for Local Times of Self-Intersection of Gaussian Integrators

Ukrainian Mathematical Journal - We prove the existence of multiple local times of self-intersection for a class of Gaussian integrators generated by operators with finite-dimensional kernels,,...     

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

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 in R ^d , i.e, the set of pairs (s, t), where sA, tB, and W _s=W _t, 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 (0pN). 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.     

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

Journal of Theoretical Probability - We consider the existence and Hölder continuity conditions for the k-th-order derivatives of self-intersection local time for d-dimensional fractional...     

Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

The fractional Brownian density process is a continuous centered Gaussian ( ^d )-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. ( ( ^d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on ^d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand¹². This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.     

Ornstein－Uhlenbeck型马氏过程局部时与占位时的存在性

本文考虑Ｏｒｎｓｔｅｉｎ－Ｕｈｌｅｎｂｅｃｋ型马氏过程的局部时，证明了在一定情形下局部时的存在性，并给出了不存在的反例，同时讨论了这类过程的占位时，指出了在某些限制性条件下，占位时密度的平方可积性．     

On the Local Times of Transient Random Walks

We study the properties of the local and occupation times of certain transient random walks. First, our recent results concerning simple symmetric random walk in higher dimension are surveyed, then we start to establish similar results for simple asymmetric random walk on the line.     

Local Times of Subdiffusive Biased Walks on Trees

Consider a class of null-recurrent randomly biased walks on a supercritical Galton–Watson tree. We obtain the scaling limits of the local times and the quenched local probability for the biased walk in the subdiffusive case. These results are a consequence of a sharp estimate on the return time, whose analysis is driven by a family of concave recursive equations on trees.     

Regularity of Intersection Local Times of Fractional Brownian Motions

Let \(B^{\alpha_{i}}\) be an (N _i ,d)-fractional Brownian motion with Hurst index α _i (i=1,2), and let \(B^{\alpha_{1}}\) and \(B^{\alpha_{2}}\) be independent. We prove that, if \(\frac{N_{1}}{\alpha_{1}}+\frac{N_{2}}{\alpha_{2}}>d\), then the intersection local times of \(B^{\alpha_{1}}\) and \(B^{\alpha_{2}}\) exist, and have a continuous version. We also establish Hölder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and intersection points.One of the main motivations of this paper is from the results of Nualart and Ortiz-Latorre (J. Theor. Probab. 20:759–767, 2007), where the existence of the intersection local times of two independent (1,d)-fractional Brownian motions with the same Hurst index was studied by using a different method. Our results show that anisotropy brings subtle differences into the analytic properties of the intersection local times as well as rich geometric structures into the sets of intersection times and intersection points.     

Decomposition of Order Statistics of Semimartingales Using Local Times

Given a finite collection of continuous semimartingales, a semimartingale decomposition of the corresponding ranked (order-statistics) processes was derived recently in [1 Banner , A.D. , and Ghomrasni , R. 2008 . Local times of ranked continuous semimartingales . Stochastic Processes and Applications 118 : 1244 – 1253 . [Google Scholar]]. In this paper, we obtain a more general result for semimartingales (not necessarily continuous) using a simpler approach. Furthermore, we also give a generalization of Ouknine [7 Ouknine , Y. 1988 . Généralisation d'un lemme de S. Nakao et applications . Stochastics 23 : 149 – 157 .[Taylor & Francis Online] , [Google Scholar], 8 Ouknine , Y. 1990 . Temps local du produit et du sup de deux semimartingales . Séminaire de Probabilités XXIV, 1988/89 . Lecture Notes in Mathematics , Vol. 1426 , pp. 477 – 479 . [Google Scholar]] and Yan's [11 Yan , J.A. 1985 . A formula for local times of semimartingales . Northeast. Math. J. 1 : 138 – 140 . [Google Scholar]] formula for local times of ranked processes.     

Towards Bounded Negativity of Self-Intersection on General Blown-up Projective Planes

We address the problem of bounding from below the self-intersection of integral curves on the projective plane blown-up at general points. In particular, by applying classical deformation theory, we obtain the expected bound in the case of either high ramification or low multiplicity.     

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.     

A Factorization Theorem for Curves with Vanishing Self-Intersection

Let C be a connected divisor in a compact Kähler manifold such that the self-intersection of C, computed with respect to a Kähler metric, vanishes. Assume that the normal closure of the image of \(\pi _{1}(C)\) in \(\pi _{1}(Y)\) has infinite index. Then there exists a holomorphic map f from Y to a curve B such that C is a fiber. The conclusion holds if one assumes that the image of \(\pi _{1}(C)\) is amenable but \(\pi _{1}(Y)\) is non-amenable.     

平面上Brown运动的重正化自交局部时的光滑性

本文运用实插值理论中的K-方法证明了平面上Brown运动自交局部时属于分数次Sob0lev空间Dpa,其中p＞1及α＜1/3.     

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Let, be two independent,
-dimensional sub-fractional Brownian motions with respective indices.
Assume. Our principal results are the necessary and sufficient condition for the
existence and smoothness of the collision local time and the intersection local time of
and through chaos expansion and elementary inequalities.     

Local Times and Related Properties of Multidimensional Iterated Brownian Motion

Let {W(t), tR} and {B(t), t0} be two independent Brownian motions in R with W(0) = B(0) = 0 and let

Two-parameter p, q-variation Paths and Integrations of Local Times

In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter -variation path integrals. Our condition of locally bounded -variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time pathwise and then give generalized It’s formula when is only of bounded -variation in . In the case that is of locally bounded variation in , the integral is the Lebesgue–Stieltjes integral and was used by Elworthy, Truman and Zhao. When is of only locally -variation, where , , and , the integral is a two-parameter Young integral of -variation rather than a Lebesgue–Stieltjes integral. In the special case that is independent of , we give a new condition for Meyer's formula and is defined pathwise as a Young integral. For this we prove the local time is of -variation in for each , for each almost surely (-variation in the sense of Lyons and Young, i.e. ).     

On Moduli of Continuity for Local Times of Fractional Stable Processes

We establish estimates for moduli of continuity of the local times of a class of self-similar stable processes. Our arguments yield also sharp lower bounds for the Hausdorff measure of the level sets of these processes. Finally, we prove the continuity in law of the local times with respect to the self-similarity index for a wide class of self-similar stable processes.     

Results on Local Times of a Class of Multiparameter Gaussian Processes

In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.     

Local Times of Markov Processes Approximated by a Generalized Iterated Brownian Motion

Csáki et al.⁽⁵⁾ have given strong approximations of continuous additive functional of Brownian motion. We establish here an extension of these results for a large class of Markov processes.     

Intersection Local Times of Independent Brownian Motions as Generalized White Noise Functionals

A 'chaos expansion' of the intersection local time functional of two independent Brownian motions in R ^d is given. The expansion is in terms of normal products of white noise (corresponding to multiple Wiener integrals). As a consequence of the local structure of the normal products, the kernel functions in the expansion are explicitly given and exhibit clearly the dimension dependent singularities of the local time functional. Their L ^p -properties are discussed. An important tool for deriving the chaos expansion is a computation of the 'S-transform' of the corresponding regularized intersection local times and a control about their singular limit.