Large fluctuations of the area under a constrained Brownian excursion

We study large fluctuations of the area A under a Brownian excursion x(t) on the time interval |t| ≤ T , constrained to stay away from a moving wall x0(t) such that x ₀ (-T) = x ₀ (T) = 0 and x ₀ (|t| < T) > 0. We focus on wall functions described by a family of generalized parabolas x ₀ (t) = Tγ[1 - (t/T )2k], where k ≥ 1. Using the optimal fluctuation method (OFM), we calculate the large deviation function (LDF) of the area at long times. The OFM provides a simple description of the area fluctuations in terms of optimal paths, or rays, of the Brownian motion. We show that the LDF has a jump in the third derivative with respect to A at a critical value of A. This singularity results from a qualitative change of the optimal path, and it can be interpreted as a third-order dynamical phase transition. Although the OFM is not applicable for typical (small) area fluctuations, we argue that it correctly captures their power-law scaling of A with T, with an exponent that depends continuously on γ and on k. We also consider the cosinewall x ₀ (t) = Tγ cos[πt/(2T)] to illustrate a different possible behavior of the optimal path and of the scaling of typical fluctuations. For some wall functions additional phase transitions, which result from a coexistence of multiple OFM solutions, should be possible.