The stationary distribution of reflected Brownian motion in a wedge: differential properties

We consider a 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 on the axes, drift), we study the algebraic and differential nature of the Laplace transform of its 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 among the angles involved in the definition of the model. A complicated integral expression is known for this Laplace transform. In the differentially algebraic case, we go further and provide an explicit, integral-free expression. In particular, we obtain new derivations of the Laplace transform in several well known cases, namely 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). 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.