Probability mass functions of the pre-absorption positions of a stochastically resetting random walker

Abstract

Consider a one-dimensional random walker that is allowed to reset its position stochastically to an absorbing state at the origin. Can we infer anything about the statistics of the walker's position prior to absorption? In this text, we provide insight by computing probability mass functions of pre-absorption positions under the effects of variable bias, initial position, resetting region, and resetting probability. We verify that negative bias promotes occurrences of recoveries by increasing the exposure of the particle to pre-absorption positions within the resetting region. For particles initially located at positions within the resetting region, resetting peaks at pre-absorption positions around the location of the initial position. For particles initially at positions outside the resetting region, instances of resetting peak at pre-absorption positions at the boundary of the resetting region. Lastly, increasing the resetting probability associated with each step increasingly confines pre-absorption positions to positions around the location of the initial position. These results are potentially relevant in the context of RNA cleavage during the transcription of genetic information.