AI 中文总结
该研究针对带幂型核的高斯Volterra过程，推导其小球概率双边估计，进而证明各固定点、原点及无穷远处的Chung重对数律，明确参数对过程特性的决定作用。
AI 中文摘要
考虑Mishura和Shklyar引入的高斯Volterra过程，其表达式为$$X(t) = \text{积分从0到t} r^\beta \times (\text{积分从r到t} u^\beta (u-r)^\beta \text{d}u) \text{d}W_r, \text{其中} t\text{大于等于0}$$，参数满足$$\text{α大于负二分之一}, \text{γ属于(-1,-二分之一)}, \text{H等于α加β加γ加二分之三大于0}$$。我们得到X的小球概率的双边估计，作为应用，证明了在每个固定点t>0、原点和无穷远处的Chung重对数律（Chung's LILs）。固定时间结果来自小球估计和Lamperti变换，而原点和无穷远处的结果来自Talagrand的下界准则。这些结果表明，γ加二分之三决定局部粗糙度和小球指数，α加β决定固定正时间的局部波动尺度，H控制原点和无穷远处的自相似标度。
英文摘要
Consider the Gaussian Volterra process introduced by Mishura and Shklyar \cite{MS22a,MS22b}, $$ X(t) = \int_0^t r^α\left( \int_r^t u^β(u-r)^γ\,du \right)dW_r, \qquad t\ge 0, $$ where $$ α>-\frac12, \quad γ\in\left(-1,-\frac12\right), \quad H:=α+β+γ+\frac32>0. $$ 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 $γ+\frac32$ determines the local roughness and the small ball exponent, $α+β$ determines the scale of local fluctuations at fixed positive times, and $H$ governs the self-similar scaling at the origin and infinity.
