One may show that the Wiener measure (for curves in $\mathbb R^n$) is concentrated on the Hölder-continuous curves of Hölder index $< \frac 1 2$. What happens to the curves of Hölder index precisely $\frac 1 2$, are they Wiener-null?
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2$\begingroup$ This is correct: by the law of the iterated logarithm, paths of the Wiener process are not Hölder continuous with index $\tfrac{1}{2}$ with probability $1$. So no matter what you mean by "index precisely $\tfrac{1}{2}$", the answer to your question is likely "yes". $\endgroup$– Mateusz KwaśnickiMay 22, 2018 at 16:16
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$\begingroup$ @MateuszKwaśnicki: By "index precisely $\frac 1 2$" I meant "index $\frac 1 2$"; it is just for emphasis. $\endgroup$– Alex M.May 22, 2018 at 16:36
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1 Answer
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To take a different view of the question, what you are asking is whether the sample paths of this Wiener process $W_0$ can be "more smooth" in terms of Hölder continuity. The answer is NO.
To make @Mateusz's idea more precise, what we said is that Wiener process $W_0$ can be decomposed into summation of random walks and $${\displaystyle \limsup _{n\to \infty }{\frac {W_{n}}{\sqrt {2n\log \log n}}}=1,\quad {\text{a. s.}},}$$
and the resulting $\sqrt {2n\log \log n}$ is of Hölder modulus $1/2$. Though this is a strong evidence, it is not enough since what you were asking is essentially whether the sample paths are all Hölder at $1/2$.
It is worth pointing out that Wiener processes have the self-similarity property $$\forall\lambda > 0, W_t\overset{d}{=} \frac{1}{\sqrt{\lambda }}W_{\lambda t}$$, which fails to hold for any other power of $\lambda^\alpha$ for $\alpha<-1/2$. This gives a nice image about the Hölder continuity of resulting sample path because we can always regard the sample path as sort of iterated contractions.
Following this route, [PH] Corollary 2.14 indicated that the answer to your question is NO. It even fails to be continuous locally. In case of exactly $1/2$, I will say it is not based on the constructive proof in [FP]. If you want to know more about smoothness of sample paths, I wrote another answer [MO].
A side commment is that "measure concentration" seems to have slightly different meanings across communities of analysts and probabilists. In terms of analysis, people usually expect certain measures being supported on a certain set, as your example indicated, $W_0$ supported on curves of certain Hölder continuous smooth curves. In terms of probability, another (slightly different) meaning of "measure concentration" refers to the tail-behavior of the (probability, or at least bounded) measure under consideration. For example, Hoeffding inequality tells you about that there is a high probability that the probability measure "concentrates" around its mean. I guess this is worth of clarifying, at the very least.
[PH]Peter Hansen, BROWNIAN MOTION AND HAUSDORFF DIMENSION http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf
[MO]Is there a differentiable random walk?
[FP]Friz, Peter K., and Nicolas B. Victoir. Multidimensional stochastic processes as rough paths: theory and applications. Vol. 120. Cambridge University Press, 2010.
