Asymptotic behaviour of a family of time-inhomogeneous diffusions

Abstract: Let X a solution of the time-inhomogeneous stochastic differential equation driven by a Brownian motion with drift coefficient b(t, x) = ρ sgn(x) |x| α tβ. This process can be viewed as a dis-torted Brownian motion in a potential possibly singular, depending on time. After obtaining results on the existence and the uniqueness of solution, we study its asymptotic behaviour and made a precise description, in terms of parameters ρ, α and β, of the recurrence, transience and convergence. More precisely, asymptotic distributions, iterated logarithm type laws and rates of transience are proved for such processes. Key words: time-inhomogeneous diffusions, singular stochastic differential equations, explosion times, scaling transformations and changes of time, recurrence and transience, iterated logarithm type laws, asymptotic distribution