We consider the classical problem of determining the stationary distribution of the semimartingale reflected Brownian motion (SRBM) in a two-dimensional wedge. Under standard assumptions on the parameters of the model (opening of the wedge, angles of the reflections, drift), we study the algebraic and differential nature of the Laplace transform of this stationary distribution. We derive necessary and sufficient conditions for this Laplace transform to be rational, algebraic, differentially finite or more generally differentially algebraic. These conditions are explicit linear dependencies between the angles of the model.

A complicated integral expression for this Laplace transform has recently been obtained by two authors of this paper. In the differentially algebraic case, we provide a simple, explicit integral-free expression in terms of a hypergeometric function. It specializes to earlier expressions in several classical cases: the skew-symmetric case, the orthogonal reflections case and the sum-of-exponential densities case (corresponding to the so-called Dieker-Moriarty conditions on the parameters). This paper thus closes, in a sense, the quest of all “simple” cases.

To prove these results, we start from a functional equation that the Laplace transform satisfies, to which we apply tools from diverse horizons. To establish differential algebraicity, a key ingredient is Tutte’s invariant approach, which originates in enumerative combinatorics. It allows us to express the Laplace transform (or its square) as a rational function of a certain canonical invariant, a hypergeometric function in our context. To establish differential transcendence, we turn the functional equation into a difference equation and apply Galoisian results on the nature of the solutions to such equations.

We consider homogenization of random surfaces and study the variational principle for graph homomorphisms from subsets of ${\mathbb{Z}}^m$ into $\mathbb{Z}$, where the underlying uniform measure is perturbed by a random potential. Motivated by the theories of random walks in random potentials, we assume that random potential is stationary, ergodic, and bounded in $L^1$. We show that the variational principle holds in probability and that the entropy functional homogenizes, i.e. is independent of the values taken by the random potential. The main ingredients in the argument are the existence of the quenched surface tension, the equivalence of the quenched and the annealed surface tension, and robustness of the surface tension under change in boundary data. These ingredients are deduced by a combination of a superadditive ergodic theorem and combinatorial results, especially the Kirszbraun theorem.