In this article, we consider a family of real-valued diffusion processes on the time interval [0, 1] indexed by their prescribed initial value $$x \in \mathbb {R}$$ x ∈ R and another point in space, $$y \in \mathbb {R}$$ y ∈ R . We first present an easy-to-check condition on their drift and diffusion coefficients ensuring that the diffusion is pinned in y at time $$t=1$$ t = 1 . Our main result then concerns the following question: can this family of pinned diffusions be obtained as the bridges either of a Gaussian Markov process or of an Itô diffusion? We eventually illustrate our precise answer with several examples.

中文翻译：

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固定扩散和马尔可夫桥

在本文中，我们考虑在时间间隔 [0, 1] 上的一系列实值扩散过程，它们由它们规定的初始值 $$x \in \mathbb {R}$$ x ∈ R 和空间中的另一个点索引， $$y \in \mathbb {R}$$ y ∈ R 。我们首先对它们的漂移和扩散系数提出一个易于检查的条件，确保扩散在时间 $$t=1$$ t = 1 时固定在 y 中。然后，我们的主要结果涉及以下问题：这个固定扩散族能否作为高斯马尔可夫过程或 Itô 扩散的桥梁获得？我们最终用几个例子来说明我们的精确答案。