Integration by parts on \boldsδ-Bessel bridges, \boldsδ>3, and related SPDEs

• Lorenzo Zambotti ^[1]
  1. [1] Scuola Normale Superiore, Pisa
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 31, Nº. 1, 2003, págs. 323-348
• Idioma: inglés
• DOI: 10.1214/aop/1046294313
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• Resumen

  • We study a white-noise driven semilinear partial differential equation on the spatial interval [0,1] with Dirichlet boundary condition and with a singular drift of the form cu−3, c>0. We prove existence and uniqueness of a non-negative continuous adapted solution u on [0,∞)×[0,1] for every nonnegative continuous initial datum x, satisfying x(0)=x(1)=0. We prove that the law πδ of the Bessel bridge on [0,1] of dimension δ>3 is the unique invariant probability measure of the process x↦u, with c=(δ−1)(δ−3)/8 and, if δ∈N, that u is the radial part in the sense of Dirichlet forms of the Rδ-valued solution of a linear stochastic heat equation. An explicit integration by parts formula w.r.t. πδ is given for all δ>3.