Cone paths for the planar Brownian snake

Summary. For the Brownian path-valued process of Le Gall (or Brownian snake) in , the times at which the process is a cone path are considered as a function of the size of the cone and the terminal position of the path. The results show that the paths for the path-valued process have local properties unlike those of a standard Brownian motion. Received: 29 January 1996 / In revised form: 21 June 1996     

Numerical bounds for critical exponents of crossing Brownian motion

We consider -dimensional crossing Brownian motion in a truncated Poissonian potential conditioned to reach a fixed hyperplane at distance from the starting point. The transverse fluctuation of the path is expected to be of order . We prove that for : . As a second critical exponent we introduce , which describes the fluctuations of naturally defined distance functions for crossing Brownian motion. The numerical bound we obtain is an improvement of Corollary 3.1 in Scaling identity for crossing Brownian motion in a Poissonian potential (Probab. Theory Related Fields 112 (1998), 299-319), resulting in if and if the killing rate is strictly positive (0$">).

     

Polygons on two-dimensional Brownian paths

Journal d'Analyse Mathématique -     

An iterated logarithm law for families of Brownian paths

Summary If n is large, a plot of log n independent Brownian paths over [0, n] is nearly certain to give the appearance of a shaded region having square root boundaries.Research of this author was partially supported by the National Science Foundation under grant MCS 82-00786This work was undertaken while B. Schreiber, of Wayne State University, was a Visiting Professor of Mathematics at Michigan State University, whose hospitality he wishes to acknowledge with gratitude.     

Geometric properties of 2-dimensional Brownian paths

Summary Let A be the set of all points of the plane , visited by 2-dimensional Brownian motion before time 1. With probability 1, all points of A are twist points except a set of harmonic measure zero. Twist points may be continuously approached in \A only along a special spiral. Although negligible in the sense of harmonic measure, various classes of cone points are dense in A, with probability 1. Cone points may be approached in \A within suitable wedges.Research supported in part by NSF Grant DMS 8419377     

Fast orthogonal transforms and generation of Brownian paths

We present a number of fast constructions of discrete Brownian paths that can be used as alternatives to principal component analysis and Brownian bridge for stratified Monte Carlo and quasi-Monte Carlo. By fast we mean that a path of length n$n$ can be generated in O(nlog(n))$O(nlog\left(n\right))$ floating point operations. We highlight some of the connections between the different constructions and we provide some numerical examples.     

Marcinkiewicz exponents and integrals over non‐rectifiable paths

We introduce and study certain distributions generalizing the operation of curvilinear integration for the case where the path of integration is not rectifiable. Then we apply that distributions for solving of boundary value problems of Riemann—Hilbert type in domains with non‐rectifiable boundaries. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths

In this paper, we establish asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths under the condition that the phase function has finitely many non-degenerate minima. Our main tool is the Banach space-valued rough path theory of T. Lyons. We use a large deviation principle and the stochastic Taylor expansion with respect to the topology of the space of geometric rough paths. This is a continuation of a series of papers by Inahama [Y. Inahama, Laplace's method for the laws of heat processes on loop spaces, J. Funct. Anal. 232 (2006) 148-194] and by Inahama and Kawabi [Y. Inahama, H. Kawabi, Large deviations for heat kernel measures on loop spaces via rough paths, J. London Math. Soc. 73 (3) (2006) 797-816], [Y. Inahama, H. Kawabi, On asymptotics of certain Banach space-valued Itô functionals of Brownian rough paths, in: Proceedings of the Abel Symposium 2005, Stochastic Analysis and Applications, A Symposium in Honor of Kiyosi Itô, Springer, Berlin, in press. Available at: http://www.abelprisen.no/no/abelprisen/deltagere_2005.html].     

Packing measure of the sample paths of fractional Brownian motion

Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure of .

     

Convergence rates for the full Brownian rough paths with applications to limit theorems for stochastic flows

A quantitative version of a well-known limit theorem for stochastic flows is establishing; our main tool is rough path analysis.     

Estimation of system reliability in Brownian stress-strength models based on sample paths

Reliability of many stochastic systems depends on uncertain stress and strength patterns that are time dependent. In this paper, we consider the problem of estimating the reliability of a system when bothX(t) andY(t) are assumed to be independent Brownian motion processes, whereX(t) is the system stress, andY(t) is the system strength, at timet.This research was partially supported by the Air-Force Office of Scientific Research Grants AFOSR-89-0402 and AFOSR-90-0402.     

Almost sure asymptotic behaviour of the r-neighbourhood surface area of Brownian paths

We show that whenever the q-dimensional Minkowski content of a subset A ⊂ ℝ ^d exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in ℝ ^d , d ⩾ 3.     

Small ball estimates for Brownian motion and the Brownian sheet

Small ball estimates are obtained for Brownian motion and the Brownian sheet when balls are given by certain Hölder norms. As an application of these results we include a functional form of Chung's LIL in this setting.Both authors were supported in part by NSF Grant Number DMS-9024961.     

Lyapunov exponents for continuous random transformations

In this paper, the concept of Lyapunov exponent is generalized to random transformations that are not necessarily differentiable. For a class of random repellers and of random hyperbolic sets obtained via small perturbations of deterministic ones respectively, the new exponents are shown to coincide with the classical ones.