Random Evolutions Are Driven by the Hyperparabolic Operators

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.