Context :

Aims : 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 .

Methods : 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 .

Results : 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 \sim 1.4 ^ { \circ } below 10 GeV to a radius of \sim 0.8 ^ { \circ } above 100 GeV , with a shift in the centroid location .

Conclusions : 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 .