Journal of Statistical Physics, Vol. 119, Nos. 1/2, April 2005, p. 165-196.
Unlike Brownian motion, which propagates diffusively and whose sample-patn trajectories are continuous, non-Brownian Levy motions propagate via jumps (nights) and their sample-path trajectories are purely discontinuous. When analyzing systems involving non-Brownian Levy motions, the common practice is to use either spectral or fractional-calculus methods. In this manuscript we suggest an alternative analytical approach: using the Poisson-superposition jump structure of non-Brownian Levy motions. We demonstrate this approach in two exemplary topics: (i) systems governed by Levy-driven Ornstein-Uhlenbeck dynamics; and, (ii) systems subject to temporal Levy subordination. We show that this approach yields answers and insights that are not attainable using spectral methods alone.