Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions

Baudoin, Fabrice; Ouyang, Cheng; Tindel, Samy

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 111-135.

Dans ce papier nous étudions des bornes supérieures pour la densité d’une solution déquation différentielle conduite par un mouvement brownien fractionnaire d’indice de Hurst $H>1/3$. Nous montrons, que sous certaines conditions géomètriques, dans le cas régulier $H>1/2$, la densité de la solution satisfait l’inégalité de log-Sobolev, l’inégalité de concentration gaussienne et admet une borne supérieure gaullienne. Dans le cas $H>1/3$ et sous la même condition géomètrique, nous montrons que la densité est infiniment différentiable et admet une borne supérieure sous-gaussienne.

In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/3$. We show that under some geometric conditions, in the regular case $H>1/2$, the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case $H>1/3$ and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound.

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