Local continuity modulus for wiener and infinite dimensional OU processes

Abstract

In this paper, we gave a proof for the local continuity modulus theorem of the Wiener process, i.e., $$\mathop {\lim }\limits_{t \to 0} \mathop {\sup }\limits_{0 \leqslant s \leqslant t} |W(s)|/(2s\log \log (1/s))^{1/2} = 1$$ a.s. This result was given by Csorgo and Revesz (1981), but the proof gets them nowhere. We also gave a similar local continuity modulus result for the infinite dimensional OU processes.