Analysis On Occupation Measure Of Spatial Brownian Sheet Which Come From Local Propagation On Line

Long after the time when Ciesielski and Taylor obtained the law of iterated logarithm of occupation measure of spatial Brownian motion, Dembo,Peres,Rosen and Zeitouni made a multifractal analysis of occupation measure of spatial Brownian motion, and got the multifractal spectrum of the occupation measure.On the other side,there are less people who focus on the occupation measure of Brownain sheet,which contributes to less results.In this paper,we mainly discuss the occupation measure of spatial Brownian sheet which come from the local propagation of thick points on line.We divide this paper into three parts as follows:In the first part,we introduce some preliminary notations and basic lemma,which are convenient for later disscussion.In the second part,we give another proof of the existence of Brownian sheet which all sample paths are continuous.This proof uses two-parameter Borel-Cantelli lemma and maximum inequality of Brownian sheet, which shows more characteristic in probability way.And the process of the proof is similar to that of single-parameter Brownian motion.In the last part,as the core of this paper,we analyze the occupation measure of spatial Brownian sheet which come from the local propagation of thick points on line,and obtain the following result: Suppose that {WS)t} is a two-parameter Brownian sheet in Rd(d > 3), and that (B(Ws,t, e)) denotes the occupation measure of Brownian sheet which come from the local propagation of thick points on line in the interval A and in the ball B(Ws,t, ε), here we let A denotes the interval [po,Po + 2] x [q0, q0 + 2] for any p0,q0 > 0. Let k0 = mm{p0,q0},and qd denote the first positive zero of the Bessel functions Jd/2-2(x).For any a (0, 4/ ), we haveMoreover,except for a common null set in Ω respectively,for L-a.e.s,we have,and...