Brownian subordinators and fractional Cauchy problems

By Boris Baeumer, Mark M. Meerschaert, Erkan Nane. 2007

109 citations recorded by the source corpus.

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, involves 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.

Subfields: Applied Mathematics, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/047e9d60d0b34511ff644298ce7d0bc76008452d

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In Applied Mathematics

1. Palaniappan Vellaisamy
2. Erkan Nane
3. Mark M. Meerschaert
4. Fubo Li
5. Chuang Chen
6. René L. Schilling
7. Miao Li
8. Anatoly N. Kochubei

In Mathematical Physics

1. Mark M. Meerschaert
2. Palaniappan Vellaisamy
3. Erkan Nane
4. René L. Schilling
5. Marcin Magdziarz
6. Fubo Li
7. Chuang Chen
8. Miao Li

In Modeling and Simulation

1. Palaniappan Vellaisamy
2. Mark M. Meerschaert
3. Erkan Nane
4. Marcin Magdziarz
5. René L. Schilling
6. Fubo Li
7. Chuang Chen
8. Miao Li

Citations in the corpus

• Distributed-order fractional diffusions on bounded domains (2009, background): “[3] proved strong solutions for the case where L is a uniformly elliptical operator on Rd , and Meerschaert et al.”
• On fractional powers of generators of fractional resolvent families (2010, background): “6) without any further assumption on the operator A and a counterexample was given in [6].”
• Asymptotic properties of Brownian motion delayed by inverse subordinators (2013, background): “In this case, the operator Φ t is the Riemann-Liouville fractional derivative 0 D 1 − α t (cf. [52]), Then, (5) becomes the fractional diffusion equation [2, 35, 36, 43, 50].”
• Mild solutions for abstract fractional differential equations (2013, unknown): “[1–11] and the references therein).”
• The Fractional Poisson Process and the Inverse Stable Subordinator (2010, background): “2), can be obtained by arguments similar to those used in [3] to connect the inverse stable subordinator to iterated Brownian motion.”