Brownian subordinators and fractional Cauchy problems

Abstract

A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involve subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes, and consequently, an equivalence between these two families of partial differential equations.

18 pages, minor spelling corrections

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• Space-time duality for fractional diffusion
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• Large deviations for local time fractional Brownian motion and applications
• Laws of the iterated logarithm for a class of iterated processes
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• Fractal Dimensions for Continuous Time Random Walk Limits
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• Asymptotic properties of Brownian motion delayed by inverse subordinators
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• From Brownian-time Brownian sheet to a fourth order and a Kuramoto-Sivashinsky-variant interacting PDEs systems
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• Asymptotic Behavior of Polynomially Bounded Solutions of Linear Fractional Differential Equations