Editing Wiener process (section)

=== Self-similarity ===
[[File:Wiener process animated.gif|thumb|500px|A demonstration of Brownian scaling, showing <math>V_t = (1/\sqrt c) W_{ct}</math> for decreasing ''c''. Note that the average features of the function do not change while zooming in, and note that it zooms in quadratically faster horizontally than vertically. <!-- Feel free to rewrite this... -->]]

==== Brownian scaling ====
For every {{math|''c'' > 0}} the process <math> V_t = (1 / \sqrt c) W_{ct} </math> is another Wiener process.

==== Time reversal ====
The process <math> V_t = W_{1-t} - W_{1} </math> for {{math|0 ≤ ''t'' ≤ 1}} is distributed like {{math|''W<sub>t</sub>''}} for {{math|0 ≤ ''t'' ≤ 1}}.

==== Time inversion ====
The process <math> V_t = t W_{1/t} </math> is another Wiener process.

==== Projective invariance ====
Consider a Wiener process <math>W(t)</math>, <math>t\in\mathbb R</math>, conditioned so that <math>\lim_{t\to\pm\infty}tW(t)=0</math> (which holds almost surely) and as usual <math>W(0)=0</math>.  Then the following are all Wiener processes {{harv|Takenaka|1988}}:
<math display="block">
\begin{array}{rcl}
W_{1,s}(t) &=& W(t+s)-W(s), \quad s\in\mathbb R\\
W_{2,\sigma}(t) &=& \sigma^{-1/2}W(\sigma t),\quad \sigma > 0\\
W_3(t) &=& tW(-1/t).
\end{array}
</math>
Thus the Wiener process is invariant under the projective group [[PSL(2,R)]], being invariant under the generators of the group.  The action of an element <math>g = \begin{bmatrix}a&b\\c&d\end{bmatrix}</math> is
<math>W_g(t) = (ct+d)W\left(\frac{at+b}{ct+d}\right) - ctW\left(\frac{a}{c}\right) - dW\left(\frac{b}{d}\right),</math>
which defines a [[group action]], in the sense that <math>(W_g)_h = W_{gh}.</math>

==== Conformal invariance in two dimensions ====
Let <math>W(t)</math> be a two-dimensional Wiener process, regarded as a complex-valued process with <math>W(0)=0\in\mathbb C</math>.  Let <math>D\subset\mathbb C</math> be an open set containing 0, and <math>\tau_D</math> be associated Markov time:
<math display="block">\tau_D = \inf \{ t\ge 0 |W(t)\not\in D\}.</math>
If <math>f:D\to \mathbb C</math> is a [[holomorphic function]] which is not constant, such that <math>f(0)=0</math>, then <math>f(W_t)</math> is a time-changed Wiener process in <math>f(D)</math> {{harv|Lawler|2005}}.  More precisely, the process <math>Y(t)</math> is Wiener in <math>D</math> with the Markov time <math>S(t)</math> where
<math display="block">Y(t) = f(W(\sigma(t)))</math>
<math display="block">S(t) = \int_0^t|f'(W(s))|^2\,ds</math>
<math display="block">\sigma(t) = S^{-1}(t):\quad t = \int_0^{\sigma(t)}|f'(W(s))|^2\,ds.</math>