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.
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