Extensions of functional LIL w.r.t. (r,p)—Capacities on Wiener space

Let {wt, t⩾0} be a d-dimensional Brownian motion and ξt(t)=ξtw(s)=w(ts)/2tLLt, 0⩽s⩽1, where LLt=loglogt. Let γ:R+→R. Under suitable conditions on γ, we generalize here functional law of the iterated logarithm (LIL) of Chung type to capacity Cr,p, that the limit set of γ(t)ξt(·) as t→∞ exists and is determined in a Hölderian topology or uniform topology w.r.t. capacity Cr,p-q.e. on Wiener space. A functional LIL of Strassen type in Hölder norm w.r.t. Cr,p is also derived.