Moving randomly amid scattered obstacles

We study a planar random motion at finite velocity performed by a particle which, at even-valued Poisson events, changes direction (each time chosen with uniform law in [0, 2 pi]). In other words this model assumes that the time between successive deviations is a Gamma random variable. It can also be interpreted as the motion of particles that can hazardously collide with obstacles of different size, some of which are capable of deviating the motion. We obtain the explicit densities of the random position (X(t), Y(t)), t > 0 under the condition that the number of deviations N(t) is known. We express Pr{X(t) is an element of dx, Y(t) is an element of dy vertical bar N(t) = n} as suitable combinations of distributions of the motion (U(t), V(t)), t > 0 described by a particle changing direction at all Poisson events. The conditional densities of (U(t), V(t)) and (X(t), Y(t)) are connected by means of a new discrete-valued random variable, whose distribution is expressed in terms of Beta integrals. The technique used in the analysis is based on rather involved properties of Bessel functions, which are derived and explored in detail in order to make the paper self-contained.