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}, $$ X(t) = \int_0^t r^\alpha \left( \int_r^t u^\beta (u-r)^\gamma\,du \right)dW_r, \qquad t\ge 0, $$ where $$ \alpha>-\frac12, \quad \gamma\in\left(-1,-\frac12\right), \quad H:=\alpha+\beta+\gamma+\frac32>0. $$ 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 $\gamma+\frac32$ 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.