How small are the increments of a Wiener process?

Let , 0<aT[less-than-or-equals, slant]T<[infinity], and {W(t);0[less-than-or-equals, slant]t<[infinity]} be a standard Wiener process. This exposition studies the almost sure behaviour of inf0[less-than-or-equals, slant]t[less-than-or-equals, slant]T-aTsup0[less-than-or-equals, slant]s[less-than-or-equals, slant]aT [gamma]TW(t+s)-W(t) as T -->[infinity], under varying conditions on aT and T/aT. The following analogue of Lévy's modulus of continuity of a Wiener Process is also given: and this may be viewed as the exact "modulus of non-differentiability" of a Wiener Process.