  Applying the resolution-scale relativity principle to develop a mechanics of
non-differentiable dynamical paths, we find that, in one dimension, stationary
motion corresponds to an Ito process driven by the solutions of a Riccati
equation. We verify that the corresponding Fokker-Planck equation is solved for
a probability density corresponding to the squared modulus of the solution of
the Schrodinger equation for the same problem. Inspired by the treatment of the
one-dimensional case, we identify a generalization to time dependent problems
in any number of dimensions. The Ito process is then driven by a function which
is identified as establishing the link between non-differentiable dynamics and
standard quantum mechanics. This is the basis for the scale relativistic
interpretation of standard quantum mechanics and, in the case of applications
to chaotic systems, it leads us to identify quantum-like states as
characterizing the entire system rather than the motion of its individual
constituents.
