The Local Times Of Additive Brownian Sheets And The Intersection Local Times Of Some Independent Gaussian Fields

The theory about local times and intersection local times is always concerned by scholars in probability theory and physics. It not only has profound theoretical background but also is widely applied in mathematics of finance, statistical mechan-ics, quantum field theory, string theory and many other fields. In this paper, we investigate the local times of additive Brownian sheets and the intersection local times of some independent Gaussian fields, the main results are as follows:(Ⅰ) Let X(t1,…,tk)=W1(t1)+…+Wk(tk), where {W1(t),t∈R+N1},…, {Wk(t),t∈R+Nk｝are independent of Brownain sheets,and call X＝{X(t1,…,tk), tn∈R+Nn,1≤n≤k} as (N1,…,Nk;d) additive Brownian sheets. it is proved that X possesses the local time L on Q which is a jointly continuous in (t, x)∈Q×Rd. and let L be jointly continuous local time of then there is a finite constant c1 such that for any there are two positive finite constants c1,c2, such that forⅠ-4) If N1=…=Nk, then for all Q(?)R+N1+…+Nk dim X(Q)=d∧2dimQ a.s. (Ⅱ)In Chapter 4,we investigate the existence of intersection local time for some independent Gaussian fields.(Ⅲ)Let X(t)=(W1(t1)-W2(t2),W2(t2)-W3(t3),…,WN-1(tN-1)-WN(tN)). where Wi(ti)=Bi1(ti1)+Bi2(ti2),i=1,…,N are independent of planar additive Brownian motions.Ⅲ-1)For any N∈N and any Q∈(?)(R+2N),X a.s.has a jointly continuous local time on Q.Ⅲ-2)Letτ∈R+2N,there exist a constant C1 not depending onτ,and an a.s.finite random variableξ(ω),which depends onτ,such that, L(Q,x)≤C1λ(Q)1/N(log|logλ(Q)|)2N a.s.Ⅲ-3)For all A(?)B(R2),Qi=[0,ui]×[0,ui](?)R+2,i=1,…,N.