Strong atmospheric escape has been detected in several close-in exoplanets. As these planets consist mostly of hydrogen, observations in hydrogen lines, such as Ly-alpha and H-alpha, are powerful diagnostics of escape. Here, we simulate the evolution of atmospheric escape of close-in giant planets and calculate their associated Ly-alpha and H-alpha transits. We use a one-dimensional hydrodynamic escape model to compute physical properties of the atmosphere and a ray-tracing technique to simulate spectroscopic transits. We consider giant (0.3 and 1M_jup) planets orbiting a solar-like star at 0.045au, evolving from 10 to 5000 Myr. We find that younger giants show higher rates of escape, owing to a favourable combination of higher irradiation fluxes and weaker gravities. Less massive planets show higher escape rates (1e10 -- 1e13 g/s) than those more massive (1e9 -- 1e12 g/s) over their evolution. We estimate that the 1-M_jup planet would lose at most 1% of its initial mass due to escape, while the 0.3-M_jup planet, could lose up to 20%. This supports the idea that the Neptunian desert has been formed due to significant mass loss in low-gravity planets. At younger ages, we find that the mid-transit Ly-alpha line is saturated at line centre, while H-alpha exhibits transit depths of at most 3 -- 4% in excess of their geometric transit. While at older ages, Ly-alpha absorption is still significant (and possibly saturated for the lower mass planet), the H-alpha absorption nearly disappears. This is because the extended atmosphere of neutral hydrogen becomes predominantly in the ground state after ~1.2 Gyr.
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