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A conservative evolution of the Brownian excursion. (English) Zbl 1187.60063
Summary: We consider the problem of conditioning the Brownian excursion to have a fixed time average over the interval $$[0,1]$$ and we study an associated stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space-time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution.

##### MSC:
 60J65 Brownian motion 60G15 Gaussian processes 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H07 Stochastic calculus of variations and the Malliavin calculus 37L40 Invariant measures for infinite-dimensional dissipative dynamical systems
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