Existence and asymptotic behaviour of some time-inhomogeneous diffusions

TL;DR

This paper investigates the existence, uniqueness, and long-term behavior of solutions to a class of time-inhomogeneous stochastic differential equations driven by Brownian motion, revealing conditions for recurrence, transience, and convergence based on parameters.

Contribution

It provides new results on the existence, uniqueness, and detailed asymptotic analysis of solutions to a specific class of time-inhomogeneous diffusions with singular, time-dependent drifts.

Findings

Conditions for recurrence, transience, and convergence based on parameters

Asymptotic distributions and laws of iterated logarithm

Rates of transience and explosion

Abstract

Let us consider a solution of the time-inhomogeneous stochastic differential equation driven by a Brownian motion with drift coefficient $b(t,x)=\rho\,{\rm sgn}(x)|x|^\alpha/t^\beta$. This process can be viewed as a distorted Brownian motion in a potential, possibly singular, depending on time. After obtaining results on existence and uniqueness of solution, we study its asymptotic behaviour and made a precise description, in terms of parameters $\rho,\alpha$ and $\beta$, of the recurrence, transience and convergence. More precisely, asymptotic distributions, iterated logarithm type laws and rates of transience and explosion are proved for such processes.