  The main purpose of this paper is to propose an ergodic theoretic approach to
the study of entire holomorphic curves. Brody curves are one-Lipschitz
holomorphic maps from the complex plane to the complex projective space. They
naturally form a dynamical system, and "random Brody curves" in the title
refers to invariant probability measures on it. We study their geometric and
dynamical properties. Given an invariant probability measure $\mu$ on the space
of Brody curves, our first main theorem claims that its rate distortion
dimension is bounded by the integral of a "potential function" over $\mu$. This
result is analogous to the Ruelle inequality of smooth ergodic theory. Our
second main theorem claims that there exists a rich variety of invariant
probability measures attaining equality in this "Ruelle inequality for Brody
curves". The main tools of the proofs are the deformation theory of Brody
curves and the variational principle for mean dimension with potential. This
approach is motivated by the theory of thermodynamic formalism for Axiom A
diffeomorphisms.
