Asymptotics of average case approximation complexity for tensor products of Euler integrated processes

We consider random fields that are tensor products of $d$ Euler integrated processes. The average case approximation complexity for a given random field is defined as the minimal number of values of continuous linear functionals that is needed to approximate the field with relative $2$-average error not exceeding a given threshold $\varepsilon$. In the paper we obtain logarithmic asymptotics of the average case approximation complexity for such random fields for fixed $\varepsilon$ and $d\to\infty$ under rather weak assumptions for the smoothness parameters of the marginal processes.