Cosmic-ray escape from accelerators: the case of pulsar wind nebulae and TeV $\gamma$-ray halos

The escape of cosmic rays from their sources and their subsequent propagation in the surrounding medium remain central open issues in high-energy astrophysics, owing to their inherently multi- scale and nonlinear nature. The recent discovery of TeV gamma-ray halos and X-ray filaments around middle-aged pulsars provides direct evidence of particle escape and diffusion on scales of tens of parsecs, at multi-TeV energies where direct CR measurements currently offer no constraints on transport. So far, only three TeV halos have been firmly detected, but ongoing surveys are rapidly expanding the sample of candidates. The inferred strong suppression of the diffusion coefficient in these sources raises fundamental questions about its origin and implications for Galactic cosmic ray transport, the positron fraction, and diffuse gamma-ray emission. In this contribution, we review the current theoretical scenarios proposed to explain these observations—ranging from pre-existing turbulence to anisotropic or self-generated diffusion. In addition, we discuss how X-ray filaments may trace current-driven instabilities triggered by partially charge-separated escape of particles. Future observations with LHAASO, the ASTRI Mini-Array, and CTAO, combined with kinetic and MHD simulations, will be crucial to establish a consistent picture linking particle acceleration, escape, and Galactic propagation.