Some Notes On Nondegenerate Diffusion Process

There are some results' on the study of a N-dimensional Nondegenerate Diffusion Process. In [1], for N ≥ 2, the Hausdorff dimension of image set of its sample path is obtained, but for N=l, only its upper bound estimate (See [2]). In this paper, we obtained the Hausdorff dimension of image set for N≥1. However, the proving method is distinct from [1]. This is the first note on the Nondegenerate Diffusion Process.In [3], it was shown that the Hausdorff dimension of level set of one-dimensional Brownian Sample Paths is 1/2 with probability 1. This paper obtained the result on the Hausdorff dimension of level set of one-dimensional Nondegenerate Diffusion Process, which resembles that of the Brownian motion. Futhermore, we studied that the Hausdorff dimension of its inverse image set. This is the second note on the Nondegenerate Diffusion Process.It is well known that, the Brownian paths in N-space have no double points with probability 1 if N≥4, butforN≤3, there are double points with probability 1. Let {X(t, w): t∈RD, w∈Ω} be a path continuous centered Gaussian random field defined on a complete probability space (Ω,F, P) and take values in RN such that E[(X,. (t)-X, (s))2 ] = | t-s|2a 02D, then almost all sample paths {X(t, w): t∈RD, (w∈Ω} have no double points of any length. But, if aN<2D, there aredouble points with probability 1 (See [4]). From this, we begin to consider whether the N-dimensional Nondegenerate Diffusion Process possesses the same property. In fact, we discussed its double points problem. This is the third note on the Nondegenerate Diffusion Process.Concretely speaking, the main results are obtained as follows:1. Let a (x) and 13 (x) satisfy condition C, X(t) be a solution of equation (1-1). Let E be a bounded closed set in Rt. then,dimX(E)=min{ N, 2dimE} a. s.2. Let N=1, a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). Let E be a bounded closed set in Rf. then,dimX(E)=min{ 1, 2dimE} a. s.3. Let N=l, a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). Let E be a bounded closed set in Rt. dimE>1/2. then, dimX(E)= 1 a. s.Remark. The result 3 shows that when dimE>1/2, the Hausdorff dimension of image set of one-dimensional Nondegenerate Diffusion Process takes the maximum value.4. Let N=l, a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). then, dimX-1(x)=1/2 a. s. (x∈R)5. Let N=1, α (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). Let B be a borel set in R1. Let X-1 (B) = {t≥0: X(t)∈B}. then, dimX-1 (B) = 1 + dim(B)/2 a. s.?6. Let a (x) and β (x) satisfy condition C, X(t) be a solution of equation (1-1). If N≥5, then, L2=φ a. s. where L , ={x ∈ R N ; 3 distinct t , , t , ∈ R + , such thatX(t1, w)=X( t2,w)=x}...