Small ball probabilities and Chung's law of the iterated logarithm for Gaussian Volterra processes with power-type kernels

Abstract

Consider the Gaussian Volterra process introduced by Mishura and Shklyar \cite{MS22a,MS22b}, $$X(t)={\int }_0^t{r}^{\alpha }\left({\int }_r^t{u}^{\beta }(u-r{)}^{\gamma }du\right)d{W}_r,t\ge 0,$$ where $$\alpha >-\frac{1}{2},\gamma \in \left(-1,-\frac{1}{2}\right),H:=\alpha +\beta +\gamma +\frac{3}{2}>0.$$ We obtain two-sided estimates for the small ball probabilities of $X$. As applications, we prove Chung's laws of the iterated logarithm (Chung's LILs) at every fixed point $t>0$, at the origin, and at infinity. The fixed-time result follows from the small ball estimates and the Lamperti transformation, whereas the results at the origin and infinity follow from Talagrand's lower-class criteria \cite{talagrand1996lower}. These results show that $\gamma +\frac{3}{2}$ determines the local roughness and the small ball exponent, $\alpha +\beta$ determines the scale of local fluctuations at fixed positive times, and $H$ governs the self-similar scaling at the origin and infinity.