A Description of stochastic systems using chaotic maps

Let ρ(x, t) denote a family of probability density functions parameterized by time t. We show the existence of a family {τt : t > 0} of deterministic nonlinear (chaotic) point transformations whose invariant probability density functions are precisely ρ(x, t). 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.