By means of the unitary Gabor transform one can relate operators on signals to operators
on the space of Gabor transforms. In order to obtain a translation and modulation
invariant operator on the space of signals, the corresponding operator on the reproducing
kernel space of Gabor transforms must be left invariant, i.e. it should commute with the left regular action of the Heisenberg group. By using the left invariant vector fields on H3 and the corresponding left-invariant vector fields on a cross-section of the phase space H3/¿ inthe generators of our transport and diffusion equations on Gabor transforms we naturally employ the essential group structure on the domain of a Gabor transform. We shall use these evolutions for three different tasks. First, there is the task of enhancing Gabor transforms (and corresponding signals) by means of non-linear left invariant diffusion. Secondly, there is the task of non-linear adaptive left-invariant convection (reassignment) towards the most probable curves, while maintaining the original signal. Finally, there is the task of extracting the most probable curves in the Gabor domain.