Sojourn Times of Brownian Sheet

In a canonical way, one can think of B as “two-parameter Brownian motion”. In this article, we address the following question: “Given a measurable function υ : R → R+, what can be said about the distribution of ∫ [0,1]2 υ(Bs) ds?” The one-parameter variant of this question is both easy-to-state and well understood. Indeed, if b designates standard Brownian motion, the Laplace transform of ∫ 1 0 υ(bs+x) ds often solves a Dirichlet eigenvalue problem (in x), as prescribed by the Feynman–Kac formula; cf. Revuz and Yor [6], for example. While analogues of Feynman-Kac for B are not yet known to hold, the following highlights some of the unusual behavior of ∫ [0,1]2 υ(Bs) ds in case υ = 1[0,∞) and, anecdotally, implies that finding explicit formulæ may present a challenging task.