A conservative evolution of the Brownian excursion

TL;DR

This paper investigates a constrained stochastic PDE related to the Brownian excursion, focusing on existence issues due to the double Laplacian and lack of maximum principle, with implications for stochastic analysis.

Contribution

It introduces a novel approach to analyze a constrained SPDE involving a double Laplacian, addressing existence challenges without relying on the maximum principle.

Findings

Established existence of solutions under new conditions

Developed techniques for SPDEs with double Laplacian

Provided insights into constrained stochastic processes

Abstract

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.