Editing Wiener process (section)

==== 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>