A Generalized Telegraph Process with Resetting to the Origin Driven by Bernoulli Trials

Abstract We study a finite-velocity random motion on the real line with alternating velocities $$-v$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>v</mml:mi> </mml:mrow> </mml:math> and c , subject to stochastic resetting to the origin. At each velocity-switching epoch, a Bernoulli trial determines whether the particle is instantaneously reset to the origin with probability $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> or it continues its motion with the opposite velocity with probability $$1-\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>θ</mml:mi> </mml:mrow> </mml:math> . The resulting reset times form a renewal process. We derive the probability distributions of the position-velocity process when the intertimes between consecutive velocity changes have identical (i) exponential distribution, and (ii) Erlang-2 distribution. We also obtain explicit expressions for the moments of the position process and, in case (i), the mean-square distance between the process with resetting and an independent counterpart without resetting. We further characterize the stationary distribution, expressing it in terms of the asymptotic average of the local time of the position process. Finally, we investigate the limiting behavior of the position density under a Kac-type scaling. In both cases (i) and (ii), the position density, conditioned on random initial velocity, converges to that of the Brownian motion with Poisson resetting.