Recurrent Properties For The Sections Of Two-Parameter Generalized Brownian Sheet

It is well-known that d-dimensional Brownian Motion is (neighborhood-) re-current if and only if d≤2;an isotropic Levy process in R^d is recurrent if andonly if d≤αwithα∈(0, 2],extending the recurrent results of Brownian Motion.In1984,M.Fukushima and N. K(?)no have resolved the problem of recurrent properties forthe Sections Of two-parameter d-dimensional Brownian Sheet W=(W(s, t), s, t≥0}; 2004,R.C.Dalang and D.Khoshnevisan discussed the more general process- StableLévy Sheet's recurrent properties for the sections,gaining the following results:considertwo-parameter d-dimensional Stable Lévy Sheet X=(X(s,t),s,t≥0}, withα∈(0,2].Let L_d,α:=(?){s＞0,(?)t≥n,such that|X(s,t)|＜ε},(I)if d＞2α,thenL_d,α=φ, a.s.; (â…¡)if d∈(α, 2α],then with probability one, L_d,αis everywhere denseand dim_H L_d,α=2-d/α,almost surely, where dim_H denotes the Hausdorff dimen-sion.These results extend the results about recurrent properties for the sections oftwo-parameter d-dimensional Brownian Sheet.Two-parameter d-dimensional Generalize Brownian Sheet is a development gen-eralization of two-parameter d-dimensional Brownian Sheet,how about its recurrentproperties for the sections?because the deviation measure of the Generalize Brow-nian Sheet is Lebegue-Stieltjes measure, the Generalize Brownian Sheet may nothave the Scaling Property, so the problems of recurrent properties for the sectionsare much more complex. In this paper, we investigate the recurrent properties forthe sections of two-parameter d-dimensional Generalize Brownian Sheet under moregeneral conditions: Let L_d:=(?)(?){s＞0, (?)t≥n, such that|(?)(s,t)|＜ε},(?)b＞a＞0, c＞0, fix s＞0,let T_c(s)=inf{t:F(s,t)=c},Γ_c:={(s,t)∈R₊²:a≤s≤b,F(s,t)=c}.The main results are follows: (I)if d＞4 and W satisfy:The condition (C₁) Suppose W={W(s, t), s, t≥0} be two-parameter d-dimensional Generalize Brownian Sheet,its deviation measure F＜＜L,that is(?)_s,t～N(0, F(s, t)I_d×d), F(s,t)= integral from n=0 to s integral fromfo n=0 to t (f (u, v)) dvdu andexistsC(a,b) only relying on a,b,such that the F-measure of the-occupationenclosed by the high-equal lineΓ_c,s=a, s=b and s-axis: integral froma n=0 to b integral from n=0 to T_c(s) f(s, t)dtds≤C(a, b)c(log c)².Then L_d=φ, a. s.This proves: if d＞4, P{(?)s＞0, the process t→W(s,t) is recurrent}=0;that is for any s＞0,consistly, the process t→(?)(s, t) is not recurrent.(â…¡)if 2＜d＜4 and (?) satisfy:The condition (C₂) Let (?)= {(?)(s,t),s.t≥0} be two-parameter d-dimensionalGeneralize Brownian Sheet. its deviation measure F_i＜＜L(1≤i≤d) and (?)c₁, c₂＞0,for all rectangle A∈(0, +∞)²,such thatc₁|A|≤F_i(A)≤c₂|A|, 1≤i≤d,where |A| denotes the Lebegue measure of set A and F_i(A) denotes F_i measure ofset A.Then with probability one, L_d, is everywhere dense and dimH L_d, =2-d/2, almostsurely, where dimH denotes the Hausdorff dimension.This proves:if 2＜d≤4, then P{(?)_s＞0,the process t→(?)(s, t) is recur-rent}=1;that is for any s＞0,consistly, the process t→(?)(s, t) is recurrent.