General solution of a hyperbolic equation arising in the study of motion with random acceleration. (English. Russian original) Zbl 0927.60096

Glushak, A. V.; Orsingher, E.

Theory Probab. Math. Stat. 55, 49-53 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 50-54 (1996).

The authors consider the motion of a particle with random acceleration \(A(t)=A(0)(-1)^{N(t)}\), where \(N(t)\) represents the number of random events up to time \(t\) of a homogeneous Poisson process with parameter \(\lambda>0\), and the initial acceleration \(A(0)\) is a random variable independent of \(N(t)\) for all \(t>0\), \(P(A(0)=a)=P(A(0)=-a)=1/2\). The joint distribution of velocity and position of the particle is investigated. The system of partial differential equations for joint densities is presented. The method of solution of the equations is proposed.

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

60K99

Special processes

Keywords:

random acceleration; Poisson process; telegraph process; hyperbolic equation; Bessel function