Norio Kôno

On self-affine functions

We define a self-affine function whose typical example is the component function of the famous Peano curve (P1(t),P2(t)), 0≤t≤1. We obtain the Hausdorff and packing dimension of the graph of a self-affine function under some conditions. We also prove that the functionP1(t−P2(t) has a continuous occupation density with respect to time and space which will be the first example of a continuous deterministic function whose occupation density is continuous with respect to time and space.

On self-affine functions II

We parametrize continuous self-affine functions defined in [3] and represent them by the infinite sum. We also give a necessary and sufficient condition so that a self-affine function has absolutely continuous distribution with respect to the Lebesgue measure. Moreover we discuss surface filling functions defined by two self-affine functions and compare these with those constructed by D. Hilbert [4].

How Big Are the Increments of a Two-Parameter Gaussian Process?

Some limit theorems on the increments of a two-parameter Gaussian process are obtained via estimating large deviation probability inequalities on the suprema of the Gaussian process which is a generalization of a two-parameter Lévy Brownian motion.