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Brownian Motion, 2nd Edition by Björn Böttcher, Lothar Partzsch, René L. Schilling

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10 Regularity of Brownian paths

We have seen in the previous chapter that Brownian paths are not Hölder continuous of any order α > 1/2. In particular, they are not Lipschitz continuous or differentiable. In this chapter we continue the study of smoothness properties and show that the paths are actually a. s. nowhere (in t) differentiable.

10.1 Hölder continuity

We begin with a rather general criterion for the continuity of a stochastic process. Later on, in connection with stochastic differential equations, we need the following version for random fields, i. e. stochastic processes with a multi-dimensional index set.

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10.1 Theorem (Kolmogorov ...

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