Telegraph Process with Elastic Boundary at the Origin

Abstract

We investigate the one-dimensional telegraph random process in the presence of an elastic boundary at the origin. This process describes a finite-velocity random motion that alternates between two possible directions of motion (positive or negative). When the particle hits the origin, it is either absorbed, with probability α, or reflected upwards, with probability 1 − α. In the case of exponentially distributed random times between consecutive changes of direction, we obtain the distribution of the renewal cycles and of the absorption time at the origin. This investigation is performed both in the case of motion starting from the origin and non-zero initial state. We also study the probability law of the process within a renewal cycle.

Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics

“…However, this is at odds with many observations as well as with common sense [16]: since the body of animals of all species has the front end and the rear end, they are more likely to choose the new movement direction (following re-orientation) close to the movement direction at the preceding moment. The corresponding movement pattern is known as the correlated random walk (CRW) [17] and the corresponding microscopic stochastic process as the telegraph process [18], and their mean-field counterpart is known as a telegraph equation [19][20][21][22][23]:…”

Some Results on the Telegraph Process Confined by Two Non-Standard Boundaries

“…Over the years, some results have been obtained on the telegraph process subject to reflecting or absorbing boundaries, whereas the case of hard reflection at the boundaries (with random switching to full absorption) seems, to our knowledge, quite new. Along the line of a previous paper [12] concerning the one-dimensional telegraph process under a single boundary, in the present contribution we investigate the case of a random motion confined by two boundaries of the above described type. The main results obtained here are related to the expected values of the renewal cycles and of the time till the absorption.…”

“…The analysis of the telegraph process in the presence of an elastic boundary is a quite new research topic. Various results on the related absorption time and renewal cycles have been obtained by Di Crescenzo et al [12]. A similar problem has been investigated by Smirnov [33], where a telegraph equation confined by two endpoints is studied from an analytical point of view.…”

“…which is equal to the mean return time to the origin of the telegraph process in the presence of a single boundary at 0 (cf. Proposition 3 of [12]).…”

“…. Similarly, we get X(C H,0 (D)) = 0 if and only if C H,0 (D) = 2 k i=1 U i + H for a given k. In addition, in this case one has We point out that the above approach, based on the auxiliary compound Poisson process Y (t) and its first-crossing time through suitable boundaries, has been adopted successfully in some papers concerning the telegraph process (see, for instance, [5], [11], [12]).…”