The uniform modulus of continuity of iterated Brownian motion

Abstract

Summary. Let X be a Brownian motion defined on the line (with X(0)=0) and let Y be an independent Brownian motion defined on the nonnegative real numbers. For all t ≥ 0, we define the iterated Brownian motion (IBM), Z, by setting Z t ∆ = X(Y t ). In this paper we determine the exact uniform modulus of continuity of the process Z.

Diffusion in multiscale spacetimes

“…An example of particular importance for us is iterated Brownian motion (IBM) [86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102][103][104]. To illustrate iterated Brownian motion, we set α = 1 and consider the ordinary D-dimensional higher-order operator ∇ n , Eq.…”

The large deviation principle for stochastic processes. I

“…Wlth «°>*) = Mx)-Deheuvels and Mason [11], Burdzy ([6], [7]), Arcones [1], Hu and Shi [20], Shi [29], Hu, Pierre-Loti-Viaud, and Shi [21], Csaki; Csorgo, Foldes, and Revesz [9], Khoshnevisan and Lewis ( [22], [23]) and Csaki, Foldes and Revesz [10] have studied different properties of the iterated Brownian mo tion. We will prove that for each 0 < M < oo,…”

The first hitting time and place of a half-line by a biharmonic pseudo process

“…In 1956, Gel'fand-Yaglom [7] proposed to study a pseudo Markov process whose 'transition probability density' is taken to be a fundamental solution of It should be noted that BPP is also constructed as iterated Brownian motion which was introduced by Funaki [4], and Khoshnevisan and Lewis [9] and the oth ers studied several path-properties of iterated Brownian motion. Further to them, Burdzy and Madrecki [5] treated BPP as an extension of the usaul stable process and defined stochastic integrals and Ito's formula.…”