# Reference Query – Difference Quotient for EDD Solutions and Liouville Equation

Assume that $$Phi$$ is the solution of
$$begin {cases} frac {d} {dt} Phi (x, t) = f ( Phi (x, t), t) quad t> 0 \ Phi (x, 0) = x quad x in mathbb {R} ^ N end {cases}$$

How can we prove that
$$tilde Phi (x, y, t) = left ( Phi (x, t), frac { Phi (x + ry, t) – Phi (x, t)} {r} right)$$
is the flow of ODE with
$$tilde {f} _r (x, y, t) = left (f (x, t), frac {f (x + ry, t) – f (x, t)} {r} right )$$
as a vector field?

In addition, in a response to Demonstrate that the flow of a divergent vector field preserves the measurements, it has been proved that if $$mu_t = ( Phi ( cdot, t)) _ { sharp} mu$$ denote the image of the measure $$mu$$ by the flow of $$f$$, then the family of measurements $${ mu_t } _ {t in mathbb R}$$ satisfies Liouville's equation
$$begin {cases} partial_t mu_t + operatorname {div ,} (f mu_t) = 0 \ mu_0 = mu end {cases}$$
in the sense of distributions.

What does the PDE do? $$tilde mu_t = ( tilde Phi_t) _ { sharp} mu$$ solve?