STOCHASTIC SOLUTIONS OF A CLASS OF HIGHER ORDER CAUCHY PROBLEMS IN ℝd

Author:

NANE ERKAN1

Affiliation:

1. Department of Mathematics and Statistics, 221 Parker Hall, Auburn University, Auburn, AL 36849, USA

Abstract

We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Allouba and Zheng [4], Baeumer, Meerschaert and Nane [10], Meerschaert, Nane and Vellaisamy [37], and Nane [42]. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index 0 < β < 1, or by the absolute value of a symmetric α-stable process with 0 < α ≤ 2, independent of the Markov process. In some special cases we represent the solutions by running composition of k independent Brownian motions, called k-iterated Brownian motion for an integer k ≥ 2. We make use of a connection between fractional-time diffusions and higher order partial differential equations established first by Allouba and Zheng [4] and later extended in several directions by Baeumer, Meerschaert and Nane [10].

Publisher

World Scientific Pub Co Pte Ltd

Subject

Modeling and Simulation

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