The Local Times And The Self-intersection Local Times Of Additive Stable Process

This master thesis about local times and intersection local times is always concerned by scholars in physics and probability theory. It not only has profound theoretical background but also is widely applied in mathematics of finance, s-tatistical mechanics, quantum field theory, string theory and many other fields. In2003, Khoshnevisan, Xiao and Zhong [34]studied local times of additive stable processes with index α. The fractal properties of additive stable processes are im-protant for analysis other multi-parameter stochastic processes. In this paper, we investigate the local times and intersection local times of additive stable processes with different index. The main results are as follows:Let X=X1+…+XN be a type A additive stable process in Rd, where X1,…,XN with index α1,…,αN. Let L be its jointly continuous local time, and for I∈21, write L*(I)=supL(x,I).·If∑t=1Nα1>d, then for (?)r∈(0,1∧2/1(∑t=1Nα-d)) there are finite constant C1, c2, such that for every r∈(0,00)N, T∈21, and· Let Vr∈R+N and T∈21, there exist finite constants C3, C4such that, and·If∑t=1Nα>d, then for every r∈R+N and every T∈21there exist constants C5and C6such that andLet X=X1+…+XN be a type A additive stable process on Rd,where X1,…,XN with index α1…,α∈(0,2],Z(T)=(X(t2)-X(t1),…,X(tr)-X(tr-1))其中T=(t1,…,tr)∈R+Nr,tm∈R+N(1≤m≤r).write L*(I)=sup L(x,I),I∈R.·If Nrα>d(r-1),then for Vr∈(0,1∧(Nrα-d(r-1))/2∧Nrαα-dα(r-1)/2α(r-1)-αthere exist positive constants c7,c8,such that for every s∈(0,∞)Nr,s=(s1,…,sr), Sm=(sm1,…,smr)(1≤m≤r),Sm+1j>smj and every B∈R: and...