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Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection
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  • Published: August 2002

Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection

  • Lorenzo Zambotti1 

Probability Theory and Related Fields volume 123, pages 579–600 (2002)Cite this article

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Abstract.

 We consider the law ν of the Bessel Bridge of dimension 3 on the convex set K 0 of continuous non-negative paths on [0,1]. We prove an integration by parts formula on K 0 w.r.t. to ν, where an explicit infinite-dimensional boundary measure σ appears. We apply this to the solution (u,η) of a white-noise driven stochastic partial differential equation with reflection introduced by Nualart and Pardoux, where u:[0, ∞) ×[0,1] ↦ℝ+ is a random non-negative function and η is a random positive measure on [0, ∞) × (0,1). Indeed, we prove that u is the radial part in the sense of Dirichlet Forms of the ℝ3-valued solution of a linear stochastic heat equation, and that η has the following structure: s ↦ 2η([0,s],(0,1)) is the Additive Functional of u with Revuz measure σ for η(ds,(0,1))-a.e. s, there exists a unique r(s)\in(0,1) s.t. u(s,r(s))=0, and η(ds,dθ)=δ r(s) (dθ)η(ds,(0,1)), where δ a is the Dirac mass at a ?(0,1). This gives a complete description of (u, η) as solution of a Skorohod Problem in the infinite-dimensional non-smooth convex set K 0 .

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Authors and Affiliations

  1. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy. e-mail: zambotti@sns.it, , , , , , IT

    Lorenzo Zambotti

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  1. Lorenzo Zambotti
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Received: 7 October 2000 / Revised version: 30 January 2002 / Published online: 12 July 2002

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Zambotti, L. Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection. Probab Theory Relat Fields 123, 579–600 (2002). https://doi.org/10.1007/s004400200203

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  • Issue Date: August 2002

  • DOI: https://doi.org/10.1007/s004400200203

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Keywords

  • Reflection
  • Differential Equation
  • Partial Differential Equation
  • Positive Measure
  • Stochastic Partial Differential Equation
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