Goldstein-Kac telegraph equations and random flights in higher dimensions

摘要

In this paper we deal with random motions in dimensions two, three, and five, where the governing equations are telegraph-type equations in these dimensions. Our methodology is first applied to the second-order telegraph equation and we refine well-known results found by other methods. Next, we show that the (2,λ)-Erlang distribution for sojourn times defines the underlying stochastic process for the three-dimensional Goldstein-Kac type telegraph equation and by finding the corresponding fundamental solution of this equation, we have obtained the approximated expression for the transition density of the three-dimensional movement, our results are more complete than previous ones, and this result may have important consequences in applications. We also obtain the 5-dimensional telegraph-type equation by assuming a random motion with an (4,λ)-Erlang distribution for sojourn times, and such equation can be factorized as the product of two telegraph-type equations where one of them is the Goldstein-Kac 5-dimensional telegraph equation. In our analysis the dimension n is related to the (n−1,λ)-Erlang distribution for sojourn times of the random walks.