caa a22 4500 445392010 CHVBK 20180317143050.0 cr unu---uuuuu 170323e20110201xx s 000 0 eng 10.1007/s10955-011-0131-0 doi (NATIONALLICENCE)springer-10.1007/s10955-011-0131-0 Random Evolutions Are Driven by the Hyperparabolic Operators [Elektronische Daten] [Alexander Kolesnik, Mark Pinsky] We resolve the long-standing problem of describing the multidimensional random evolutions by means of the telegraph equations. This problem was posed by Mark Kac more than 50 years ago and has become the subject of intense discussion among researchers on whether the multidimensional random flights could be described by the telegraph equations similarly to the one-dimensional case. We give the exhaustive answer to this question and show that the multidimensional random evolutions are driven by the hyperparabolic operators composed of the telegraph operators and their integer powers. The only exception is the 2D random flight whose transition density is the fundamental solution to the two-dimensional telegraph equation. The reason of the exceptionality of the 2D-case is explained. We also show that, under the standard Kac's condition, the governing hyperparabolic operator turns into the generator of the Brownian motion. Springer Science+Business Media, LLC, 2011 Random evolution nationallicence Random flight nationallicence Persistent random walk nationallicence Transport process nationallicence Telegraph process nationallicence Telegraph equation nationallicence Hyperparabolic operator nationallicence Fundamental solution nationallicence Generalized function nationallicence Kac's condition nationallicence Brownian motion nationallicence Kolesnik Alexander Institute of Mathematics and Computer Science, Academy Street 5, 2028, Kishinev, Moldova aut Pinsky Mark Department of Mathematics, Northwestern University, 2033 Sheridan Road, 60208, Evanston, IL, USA aut Journal of Statistical Physics Springer US; http://www.springer-ny.com 142/4(2011-02-01), 828-846 0022-4715 142:4<828 2011 142 10955 https://doi.org/10.1007/s10955-011-0131-0 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-011-0131-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Kolesnik Alexander Institute of Mathematics and Computer Science, Academy Street 5, 2028, Kishinev, Moldova aut NATIONALLICENCE 700 1- Pinsky Mark Department of Mathematics, Northwestern University, 2033 Sheridan Road, 60208, Evanston, IL, USA aut NATIONALLICENCE 773 0- Journal of Statistical Physics Springer US; http://www.springer-ny.com 142/4(2011-02-01), 828-846 0022-4715 142:4<828 2011 142 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer