Some theorems on stable processes

Abstract

Here t and x are points of Rn, dx is n-dimensional Lebesgue measure, (Q, x) is the ordinary inner product in Rn, and I 1 2 = (, t). We will be mainly interested in properties of the sample functions of these processes. Our main theorems (?4) extend some results of McKean [10; 11] on the Hausdorff-Besicovitch dimension of the range of the sample functions, and some results of Bochner [2, p. 127] on the variation of the sample functions. Some of these extensions are immediate, while for others the methods of Bochner and McKean are not immediately available. Our main tool is the notion of subordination. ??2 and 3 contain preliminary material that will be needed in ?4. Finally, in ?5, we obtain the asymptotic distribution of the eigenvalues for certain operators that are naturally associated with the symmetric stable processes. 2. Preliminaries. Let fa(t, x -y) be the continuous transition density defined by (1.1). Then fa(t, X) =t-nlafa(1, t-l/ax), where fa(1, x) is a continuous strictly positive function on Rn depending on x only through I x| . Using the Fourier inversion theorem for radial functions [3, Chapter II, ?7] we find