abstract 89

Bulletin of Computational Applied Mathematics (Bull CompAMa) 

89

Random evolutions, telegraph and Klein-Gordon equations, and Brownian motion (Research Paper)

Henryk Gzyl

Here we provide a probabilistic way to compute the transition probabilities of the telegraph process. For that, we review a connection between random evolutions, the telegraph and the Klein-Gordon equations. This equation appears when one decouples the two equations that determine the transition probability of the telegraph process. Next we show how to integrate the Klein-Gordon equation by invoking a Brownian motion process. This connection comes up after the transformation of the hyperbolic initial value problem either into an elliptic problem by means of a Laplace transform, or into a parabolic problem by means of a transmutation operator.

Keywords: Random evolutions; telegraph equation; Klein-Gordon equation; Brownian motion; transmutation operators.

Cite this paper:

Gzyl H.

Random evolutions, telegraph and Klein-Gordon equations, and Brownian motion

Bull. Comput. Appl. Math. (Bull CompAMa)

Vol. 12, No.1 pp.35-46, 2024.