Selfsimilar stochastic differential equations

Abstract

Diffusion in a logarithmic potential (DLP) attracted significant interest in physics recently. The dynamics of DLP are governed by a Langevin stochastic differential equation (SDE) whose underpinning potential is logarithmic, and that is driven by Brownian motion. The SDE that governs DLP is a particular case of a selfsimilar SDE: one that is driven by a selfsimilar motion, and that produces a selfsimilar motion. This paper establishes the pivotal role of selfsimilar SDEs via two novel universality results. I) Selfsimilar SDEs emerge universally, on the macro level, when applying scaling limits to micro-level SDEs. II) Selfsimilar SDEs emerge universally when applying the Lamperti transformation to stationary SDEs. Using the universality results, this paper further establishes: a novel statistical-analysis approach to selfsimilar Ito diffusions; and the focal importance of DLP.

Weird Brownian motion

“…Due to its weird behaviors, WBM is a rather surprising diffusion model that defies common 'Brownian intuition'. Random motions that emerge-via scaling limits-over macroscopic time scales are selfsimilar processes [93]. Thus, from a macroscopic perspective of random motions, only selfsimilar such processes should be considered.…”

“…Namely, DLP is a Markovian and selfsimilar model of diffusion that is not Gaussian. In fact, DLP is the only selfsimilar model of diffusion whose evolution is governed by an Ito stochastic differential equation [93,95]. DLP attracted vast statistical-physics interest in recent years [45,[96][97][98][99][100][101][102][103][104][105].…”

Entropy of sharp restart

“…This intersection is of principal importance due to the following fact. When elevating from the microscopic level to the macroscopic level via scaling limits, the only Ito diffusions that emerge on the macro level are selfsimilar diffusions [106].…”