Editing Wiener process (section)

==== Qualitative properties ====
* For every ε > 0, the function ''w'' takes both (strictly) positive and (strictly) negative values on (0, ε).
* The function ''w'' is continuous everywhere but differentiable nowhere (like the [[Weierstrass function]]).
* For any <math>\epsilon > 0</math>, <math>w(t)</math> is almost surely not <math>(\tfrac 1 2 + \epsilon)</math>-[[Hölder continuous]], and almost surely <math>(\tfrac 1 2 - \epsilon)</math>-Hölder continuous.<ref>{{Cite book |last1=Mörters |first1=Peter |title=Brownian motion |last2=Peres |first2=Yuval |last3=Schramm |first3=Oded |last4=Werner |first4=Wendelin |date=2010 |publisher=Cambridge University Press |isbn=978-0-521-76018-8 |series=Cambridge series in statistical and probabilistic mathematics |location=Cambridge |pages=18}}</ref>
* Points of [[Maxima and minima|local maximum]] of the function ''w'' are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if ''w'' has a local maximum at {{mvar|t}} then <math display="block">\lim_{s \to t} \frac{|w(s)-w(t)|}{|s-t|} \to \infty.</math> The same holds for local minima.
* The function ''w'' has no points of local increase, that is, no ''t'' > 0 satisfies the following for some ε in (0, ''t''): first, ''w''(''s'') ≤ ''w''(''t'') for all ''s'' in (''t'' − ε, ''t''), and second, ''w''(''s'') ≥ ''w''(''t'') for all ''s'' in (''t'', ''t'' + ε). (Local increase is a weaker condition than that ''w'' is increasing on (''t'' − ''ε'', ''t'' + ''ε'').) The same holds for local decrease.
* The function ''w'' is of [[bounded variation|unbounded variation]] on every interval.
* The [[quadratic variation]] of ''w'' over [0,''t''] is ''t''. 
* [[root of a function|Zeros]] of the function ''w'' are a [[nowhere dense set|nowhere dense]] [[perfect set]] of Lebesgue measure 0 and [[Hausdorff dimension]] 1/2 (therefore, uncountable).