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We derive bridges from general multidimensional linear non time-homogeneous processes by using only the transition densities of the original process giving their integral representations (in terms of a standard Wiener process) and their so-called anticipative representations. We derive a stochastic differential equation satisfied by the integral representation and we prove a usual conditioning property for general multidimensional linear process bridges. We specialize our results for the one-dimensional case; especially, we study one-dimensional Ornstein-Uhlenbeck bridges.

Abstract. Let X = {X(t)} t≥0 be an operator semistable Lévy process in R d with exponent E, where E is an invertible linear operator on R d and X is semi-selfsimilar with respect to E. By refining arguments given in Meerschaert and Xiao [17] for the special case of an operator stable (selfsimilar) Lévy process, for an arbitrary Borel set B ⊆ R + we determine the Hausdorff dimension of the partial range X(B) in terms of the real parts of the eigenvalues of E and the Hausdorff dimension of B.

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It is well known that certain fractional diffusion equations can be solved by the densities of stable Lévy motions. In this paper we use the classical semigroup approach for Lévy processes to define semi-fractional derivatives, which allows us to generalize this statement to semistable Lévy processes. A Fourier series approach for the periodic part of the corresponding Lévy exponents enables us to represent semifractional derivatives by a Grünwald-Letnikov type formula. We use this formula to calculate semi-fractional derivatives and solutions to semi-fractional diffusion equations numerically. In particular, by means of the Grünwald-Letnikov type formula we provide a numerical algorithm to compute semistable densities.