Existence and asymptotic behaviour of some time-inhomogeneous diffusions

Gradinaru, Mihai; Offret, Yoann

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 182-207.

Nous considérons la solution d’une équation différentielle stochastique, dirigée par un mouvement brownien linéaire standard, dont le terme de dérive varie avec le temps ${\rho sgn\left(x\right)\left|x\right|}^{\alpha }/t^{\beta }$. Ce processus peut être vu comme un mouvement brownien évoluant dans un potentiel dépendant du temps, éventuellement singulier. Nous montrons des résultats d’existence et d’unicité et nous étudions le comportement asymptotique de la solution. Les propriétés de récurrence ou de transience de cette diffusion sont décrites en fonction des paramètres $\rho$, $\alpha$ et $\beta$, et nous donnons les vitesses de transience et d’explosion. Des résultats de convergence en loi et des lois de type logarithme itéré sont également obtenus.

Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient ${\rho sgn\left(x\right)\left|x\right|}^{\alpha }/t^{\beta }$. This process can be viewed as a Brownian motion evolving in a potential, possibly singular, depending on time. We prove results on the existence and uniqueness of solution, 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.

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