Abstract
In this article, we extend the integration by parts formulae (IbPF) for the laws of Bessel bridges recently obtained in [2] to linear functionals. Our proof relies on properties of hypergeometric functions, thus providing a new interpretation of these formulae.
Highlights
1.1 Bessel SPDEsRecently a family of stochastic PDEs which are infinite-dimensional analogues of Bessel processes were studied in [2] and [1]
We extend the integration by parts formulae (IbPF) for the laws of Bessel bridges recently obtained in [2] to linear functionals
While the Bessel SPDEs of parameter δ ≥ 3, which are reversible dynamics for the laws of Bessel bridges of dimension δ ≥ 3, had been introduced by Zambotti in the articles [7] and [8], an open problem for several years was to extend the construction to δ < 3: apart from the derivation of an integration by parts formula for the special value δ = 1 – see [9] and [3] – the extension to the whole regime δ < 3 had remained out of sight
Summary
A family of stochastic PDEs which are infinite-dimensional analogues of Bessel processes were studied in [2] and [1]. While the Bessel SPDEs of parameter δ ≥ 3, which are reversible dynamics for the laws of Bessel bridges of dimension δ ≥ 3, had been introduced by Zambotti in the articles [7] and [8], an open problem for several years was to extend the construction to δ < 3: apart from the derivation of an integration by parts formula for the special value δ = 1 – see [9] and [3] – the extension to the whole regime δ < 3 had remained out of sight This extension was a major challenge since, while the laws of Bessel bridges of dimension δ ≥ 3 can be represented as Gibbs measures with respect to the law of a Brownian bridge with an explicit, convex potential, such a representation fails for the laws of Bessel bridges of dimension δ < 3, see Chap. By exploiting the remarkable properties of Bessel bridges, the recent articles [2] and [1] have achieved this extension
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