It is proved that genuinely infinite dimensional Brownian motions on $\ell^p$ sequence spaces have natural rates of escape, provided the coordinates are independent. An analogous result holds for separable Hilbert space. Computations of Brownian rates of escape and further properties are considered.
"On the Existence of Natural Rate of Escape Functions for Infinite Dimensional Brownian Motions." Ann. Probab. 10 (3) 623 - 638, August, 1982. https://doi.org/10.1214/aop/1176993772