  We study foliations by non-smooth chord-arc Jordan curves of the twice
punctured Riemann sphere $\mathbb C \smallsetminus \{0\}$ using the
Loewner-Kufarev equation. We associate to such a foliation a function on the
plane that describes the "local winding" along each leaf. Our main theorem is
that this function has finite Dirichlet energy if and only if the Loewner
driving measure $\rho$ has finite Loewner-Kufarev energy, defined by $$S(\rho)
= \frac{1}{2}\int_{S^1 \times \mathbb{R}} \nu_t'(\theta)^2 \, d \theta d t$$
whenever $\rho$ is of the form $\nu_t(\theta)^2 d\theta dt$, and set to
$\infty$ otherwise. Moreover, if either of these two energies is finite they
are equal up to a constant factor and in this case the foliation leaves are
Weil-Petersson quasicircles.
  This duality between energies has several consequences. The first is that the
Loewner-Kufarev energy is reversible, that is, invariant with respect to
inversion and time-reversal of the foliation. Furthermore, the Loewner energy
of a Jordan curve can be expressed using the minimal Loewner-Kufarev energy of
those measures that generate the curve as a leaf. This provides a new and
quantitative characterization of Weil-Petersson quasicircles. Finally, we
consider conformal distortion of the foliation, and show that the
Loewner-Kufarev energy satisfies an exact transformation law involving the
Schwarzian derivative.
  The proof of our main result uses an isometry between the Dirichlet energy
space and $L^2(2\rho)$ that we construct using a version of Hadamard's formula
expressed by means of the Loewner-Kufarev equation. While our results and
methods use no probability theory, there are links to probabilistic models. In
particular, $\kappa$-parameter duality and large deviations of Schramm-Loewner
evolutions coupled with Gaussian random fields.
