On the sample paths of diagonal Brownian motions on the infinite dimensional torus

Abstract

We study the intrinsic regularity of the sample paths of certain Brownian motions on the infinite dimensional torus T ∞ . Here, intrinsic regularity means regularity with respect to the intrinsic distance d associated to the Brownian motion in question. We prove that some Brownian motions on T ∞ satisfy the classical law of iterated logarithm, that is, lim sup t→0 d(X 0,X t) 4t loglog(1/t) =1 whereas others do not. For instance, for any slowly varying function ψ such that ψ( t)⩾loglog t at infinity, we give examples of Brownian motions on T ∞ such that 0< lim sup t→0 d(X 0,X t) 4tψ(1/t) <∞. We prove similar results concerning the uniform modulus of continuity of Brownian paths on the time interval [0,1]. Namely, we prove that there are Brownian motions on T ∞ satisfying the classical Lévy-type result but, for any slowly varying function ψ such that ψ( t)⩾log t at infinity, we give examples of Brownian motions on T ∞ such that 0< lim ε→0 sup 0<s<t⩽1 t−s⩽ε d(X s,X t) 4tψ(1/t) <∞. We also obtain partial results concerning the behavior of lim inf t→0 d(X 0,X t) h(t) for appropriate functions h.