Brownian Sheet and Quasi-sure Analysis

We present a self-contained and modern survey of some existing quasi-sure results via the connection to the Brownian sheet. Among other things, we prove that quasi-every continuous function: (i) satisfies the local law of the iterated logarithm; (ii) has Lévy’s modulus of continuity for Brownian motion; (iii) is nowhere differentiable; and (iv) has a nontrivial quadratic variation. We also present a hint of how to extend (iii) to obtain a quasi-sure refinement of the M. Csörgő–P. Révész modulus of continuity for almost every continuous function along the lines suggested by M. Fukushima.