Reversible birth-and-death dynamics in continuum: a de Bruijn-type identity for free-energy dissipation

We investigate free-energy dissipation in a continuous-time birth-and-death dynamics in $\mathbb{R}^d$. For these Markov processes, the class of reversible measures coincides with the infinite-volume Gibbs point processes for some sufficiently nice Hamiltonian. For a wide class of initial distributions, we derive a de~Bruijn-type identity that relates the time evolution of the specific relative entropy along trajectories to the Fisher information, in particular establishing the thermodynamic limit of the latter. Along the way, we analyze some fine properties of the considered dynamics, such as the existence and regularity of local densities, obtain a spatial ergodic theorem for the entropy production per unit volume, and derive a small-time exponential series expansion of the dynamics.