Gabriella Verasani | Searcharxiv

arXiv2024

A cyclic random motion in $\mathbb{R}^3$ driven by geometric counting processes

We consider the random motion of a particle that moves with constant velocity in $\mathbb{R}^3$. The particle can move along four directions with different speeds that are attained cyclically. It follows that the support of the stochastic process describing the particle's position at time $t$ is a tetrahedron. We assume that the sequence of sojourn times along each direction follows a geometric counting process (GCP). When the initial velocity is fixed, we obtain the explicit form of the probability law of the process $\boldsymbol{X}(t) = (X_1(t);X_2(t);X_3(t))$, $t > 0$, for the particle's position. We also investigate the limiting behavior of the related probability density when the intensities of the four GCPs tend to infinity. Furthermore, we show that the process does not admit a stationary density. Finally, we introduce the first-passage-time problem for the first component of $\boldsymbol{X}(t)$ through a constant positive boundary $β> 0$ providing the bases for future developments.

math.PR↗

arXiv2023

On the telegraph process driven by geometric counting process with Poisson-based resetting

We investigate the effects of the resetting mechanism to the origin for a random motion on the real line characterized by two alternating velocities $v_1$ and $v_2$. We assume that the sequences of random times concerning the motions along each velocity follow two independent geometric counting processes of intensity $λ$, and that the resetting times are Poissonian with rate $ξ>0$. Under these assumptions we obtain the probability laws of the modified telegraph process describing the position and the velocity of the running particle. Our approach is based on the Markov property of the resetting times and on the knowledge of the distribution of the intertimes between consecutive velocity changes. We obtain also the asymptotic distribution of the particle position when (i) $λ$ tends to infinity, and (ii) the time goes to infinity. In the latter case the asymptotic distribution arises properly as an effect of the resetting mechanism. A quite different behavior is observed in the two cases when $v_2<0<v_1$ and $0<v_2<v_1$. Furthermore, we focus on the determination of the moment-generating function and on the main moments of the process describing the particle position under reset. Finally, we analyse the mean-square distance between the process subject to resets and the same process in absence of resets. Quite surprisingly, the lowest mean-square distance can be found for $ξ=0$, for a positive $ξ$, or for $ξ\to +\infty$ depending on the choice of the other parameters.

math.PR↗