  Taking advantage of more than 11 years of Fermi-LAT data, we perform a new
and deep analysis of the pulsar wind nebula (PWN) HESS J1825-137. Combining
this analysis with recent H.E.S.S. results we investigate and constrain the
particle transport mechanisms at work inside the source as well as the system
evolution. The PWN is studied using 11.6 years of Fermi-LAT data between 1 GeV
and 1 TeV. In particular, we present the results of the spectral analysis and
the first energy-resolved morphological study of the PWN HESS J1825-137 at GeV
energies, which provide new insights into the gamma-ray characteristics of the
nebula. An optimised analysis of the source returns an extended emission region
larger than 2$^{\circ}$, corresponding to an intrinsic size of about 150 pc,
making HESS J1825-137 the most extended gamma-ray PWN currently known. The
nebula presents a strong energy dependent morphology within the GeV range,
moving from a radius of $\sim1.4^\circ$ below 10 GeV to a radius of
$\sim$0.8$^\circ$ above 100 GeV, with a shift in the centroid location. Thanks
to the large extension and peculiar energy-dependent morphology, it is possible
to constrain the particle transport mechanisms inside the PWN HESS J1825-137.
Using the variation of the source extension and position, as well as the
constraints on the particle transport mechanisms, we present a scheme for the
possible evolution of the system. Finally, we provide an estimate of the
electron energy density and we discuss its nature in the PWN and TeV halo-like
scenario.
