Small deviations for two classes of Gaussian stationary processes and $L^p$-functionals, $0<p\le\infty$

Let $w(t)$ be a standard Wiener process, $w(0)=0$, and let $\eta_a(t)=w(t+a)-w(t)$, $t\ge0$, be increments of the Wiener process, $a>0$. Let $Z_a(t)$, $t\in[0,2a]$, be a zeromean Gaussian stationary a.s. continuous process with a covariance function of the form $\mathbf EZ_a(t)Z_a(s)=\frac12[a-|t-s|]$, $t,s\in[0,2a]$. For $0<p<\infty$, we prove results on sharp asymptotics as $\varepsilon\to0$ of the probabilities
$$ \mathbf P\Biggl\{\int_0^T|\eta_a(t)|^p\,dt\le\varepsilon^p\Biggr\}\quad\text{для}\ T\le a,\qquad\mathbf P\Biggl\{\int_0^T|Z_a(t)|^p\,dt\le\varepsilon^p\Biggr\}\quad\text{для}\ T<2a, $$
and compute similar asymptotics for the sup-norm. Derivation of the results is based on the method of comparing with a Wiener process. We present numerical values of the asymptotics in the case $p=1$, $p=2$, and for the sup-norm. We also consider application of the obtained results to one functional quantization problem of information theory.