Integration by parts formulae for the laws of Bessel bridges, and Bessel stochastic PDEs

Abstract : In this thesis, we derive integration by parts formulae (IbPF) for the laws of Bessel bridges of dimension δ > 0, thus extending previous formulae obtained by Zambotti in the case δ ≥ 3. This allows us to identify the structure of some stochastic PDEs (SPDEs) having the law of a Bessel bridge of dimension δ < 3 as invariant measure, and which extend in a natural way the family of SPDEs previously considered by Zambotti for δ ≥ 3. We call these equations Bessel SPDEs, and write them using renormalized local times. In the particular cases δ = 1, 2, using Dirichlet forms, we construct a solution to a weak version of these SPDEs. We also provide several partial results suggesting that the SPDEs associated with δ < 3 should have several important properties: strong Feller property, existence of local times. Finally, we consider dynamical critical wetting models, in the discrete and in the continuum, and prove a tightness result. We conjecture that these models have a common limit in law which should be described by the Bessel SPDE associated with δ = 1.
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Henri Elad Altman. Integration by parts formulae for the laws of Bessel bridges, and Bessel stochastic PDEs. Probability [math.PR]. Sorbonne Université, 2019. English. ⟨tel-02284974⟩

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