A conservative evolution of the Brownian excursion

Lorenzo Zambotti (University of Paris 6, France)


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.

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Pages: 1096-1119

Publication Date: July 9, 2008

DOI: 10.1214/EJP.v13-525


  1. L. Ambrosio, N. Gigli, G. SavarÈ (2005). Gradient flows in metric spaces and in the spaces of probability measures. Lectures in Mathematics ETH Z¸rich, Birkh‰user Verlag, Basel.
  2. L. Ambrosio, G. SavarÈ, L. Zambotti (2007). Existence and Stability for Fokker-Planck equations with log-concave reference measure, to appear in Probab. Theory Related Fields.
  3. E. CÈpa, (1998). ProblËme de Skorohod multivoque, Ann. Prob. 28, no. 2, 500-532.
  4. G. Da Prato, M. Rˆckner (2002). Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Related Fields 124, no. 2, 261--303.
  5. G. Da Prato, J. Zabczyk (2002). Second order partial differential equations in Hilbert spaces, London Mathematical Society Lecture Note Series, n. 293.
  6. A. Debussche, L.Zambotti (2007). Conservative Stochastic Cahn-Hilliard equation with reflection, Annals of Probability, vol. 35, no. 5, 1706-1739.
  7. R.T. Durrett, D.L. Iglehart, D.R. Miller (1977). Weak convergence to Brownian meander and Brownian excursion, Ann. Probability, 5, no. 1, pp. 117-129.
  8. D. Nualart, E. Pardoux (1992). White noise driven quasilinear SPDEs with reflection, Prob. Theory and Rel. Fields, 93, pp. 77-89.
  9. D. Revuz, M. Yor, (1991). Continuous Martingales and Brownian Motion, Springer Verlag.
  10. L. Zambotti (2002). Integration by parts on convex sets of paths and applications to SPDEs with reflection, Prob. Theory and Rel. Fields 123, no. 4, 579-600.
  11. L. Zambotti, (2004). Occupation densities for SPDEs with reflection, Annals of Probability, 32 no. 1A, 191-215.
  12. L. Zambotti (2008). Fluctuations for a conservative interface model on a wall, to appear in ALEA.

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