Continuous sample space reducing stochastic process

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 $\epsilon$. 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 $\lambda \in [0, 1]$. We also examine the statistics of the size of the state space that follows a power-law distributed probability $\mathbb{P}_{\epsilon}(z\le \epsilon) \sim z^{-\alpha}$, with a nontrivial value of the exponent as a function of the tunable parameter $\alpha = 1+\lambda$.