Continuous sample space reducing stochastic process

Abstract

We propose a simple model for sample space reducing (SSR) stochastic process, where the dynamical variable denoting the size of the state space is continuous. In general, one can view the model as a multiplicative stochastic process, with a constraint that the size of the state space cannot be smaller than a visibility parameter $\hat{I}\mu$. We study the survival time statistics that reveal a subtle difference from the discrete version of the process. A straightforward generalization can explain the noisy SSR process, characterized by a tunable parameter $\hat{I}»\in [0,1]$. We also examine the statistics of the size of the state space that follows a power-law distributed probability ${\mathbb{P}}_{\hat{I}}\mu (z\le \hat{I}\mu )\sim z^{-\hat{I}\pm }$, with a nontrivial value of the exponent as a function of the tunable parameter $\hat{I}\pm =1+\hat{I}»$.

7 pages, 6 figures