A trajectorial interpretation of Moser's proof of the Harnack inequality

In 1971 Moser published a simplified version of his proof of the parabolic Harnack inequality. The core new ingredient is a fundamental lemma due to Bombieri and Giusti, which combines an $L^p-L^\infty$-estimate with a weak $L^1$-estimate for the logarithm of supersolutions. In this note, we give a novel proof of this weak $L^1$-estimate. The presented argument uses parabolic trajectories and does not use any Poincar\'e inequality. Moreover, the proposed argument gives a geometric interpretation of Moser's result and could allow transferring Moser's method to other equations.