Description:
<jats:p>Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$\rho (x,t)$" id="E1"><mml:mi>ρ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> denote a family of probability density functions parameterized by time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$t$" id="E2"><mml:mi>t</mml:mi></mml:math>. We show the existence of a family <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$\{\tau_{t}:t>0\}$" id="E3"><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>τ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>:</mml:mo><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math> of deterministic nonlinear (chaotic) point transformations whose invariant probability density functions are precisely <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$\rho (x,t)$" id="E4"><mml:mi>ρ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>. In particular, we are interested in the densities that arise from the diffusions. We derive a partial differential equation whose solution yields the family of chaotic maps whose density functions are precisely those of the diffusion.</jats:p>