Asymptotic behaviour of a family of one-dimensional time-inhomogeneous diffusions

Mihai Gradinaru (1) , Yoann Offret (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: time-inhomogeneous diffusions; singular stochastic differential equations; explosion times; scaling transformations and changes of time; recurrence and transience; iterated logarithm type laws; asymptotic distributions

Domaines: Probabilités [math.PR]

Mihai Gradinaru, Yoann Offret. Asymptotic behaviour of a family of one-dimensional time-inhomogeneous diffusions. 2009. ⟨hal-00433274v2⟩