The Fractal Dimensions Of The Fractional Brownian Motions

This paper studies the Dimensions of the fractional Brownian Motions.A series of new results are obtained. And some of them improve or extend the related results in the literatures.Chapter 1 introduces the background of the problem-researching and the recent development of the research in this field.Chapter 2 introduces the preliminary knowledge which include the concepts and basic characters of fractal measures and fractal dimensions.Chapter 3 focuses on the dimensions for image sets, graph sets, level sets and inverseimage sets of the fractional Brownian Motions.We obtain some new results as follow:(1) Let X = {X(t), t∈ R^N} be a d dimensions a order fractional Brownian Motions and E are a random set which in R^N,then(Ⅰ) dim(X(E)) = min(d,1/αdim(E)) a.s(Ⅱ) dim(Gr(X(E))) = min(1/α dim(E), dim(E) + (1 - α)d) a.s. (Ⅲ) About Lebesgue measures almost all x∈ R^d, dim(X^-1(x)) ≤ max{N - αd,0}.(2) Let X = {X(t), t ∈ R^N} be a d dimensions α order fractional Brownian Motions and E are a random set which in R^N,then(Ⅰ) Dim(X(E)) = min(d,1/αDim(E)) a.s(Ⅱ) Dim(Gr(X(E))) = min(1/αDim(E), Dim(E) + (1 - α)d) a.s.(Ⅲ) About Lebesgue measures almost all x∈R^d, Dim(X^-1(x))≤ max{(N -αd),0}.Chapter 4 focuses on the uniform Dimensions for image sets, graph sets, level sets, inverseimage sets, of the fractional Brownian Motions.We obtain some new results as follow:(1)Let X = {X(t), t ∈ R^N} be a d dimensions α order fractional Brownian Motions ,theP(dimGr(X(E,ω))≤1/αdimE,forall E∈B(R^N))=1In particular,if N ≤ αd,thenP(dimGr(X(E,ω))=1/αdimE,forall E∈B(R^N))=1(2) Let X = {X(t), t 6 RN} be a d dimensions a order fractional Brownian Motions ,thenP(DimGr(X(E,u)) > -Dim￡, farall E e B(RN)) = 1.aIn particular.if N < o;d,thenP(DimGr(X(E,uj)) = -DimE, farall E e B(RN)) - 1a...