Probabilistic and average linear widths of weighted Sobolev spaces on the ball equipped with a Gaussian measure

Let $L_{q,\mu}$, $1\leq q\leq\infty$, denotes the weighted $L_q$ space of functions on the unit ball $\Bbb B^d$ with respect to weight $(1-\|x\|_2^2)^{\mu-\frac12},\,\mu\ge 0$, and let $W_{2,\mu}^r$ be the weighted Sobolev space on $\Bbb B^d$ with a Gaussian measure $\nu$. We investigate the probabilistic linear $(n,\delta)$-widths $\lambda_{n,\delta}(W_{2,\mu}^r,\nu,L_{q,\mu})$ and the $p$-average linear $n$-widths $\lambda_n^{(a)}(W_{2,\mu}^r,\mu,L_{q,\mu})_p$, and obtain their asymptotic orders for all $1\le q\le \infty$ and $0