In a bounded open region of the $d$ dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Dirichlet Laplacian centered at the point of return. We calculate the resolvent in closed form, study its spectral properties and determine explicitly the spectrum in dimension one. Two proofs of the exponential ergodicity are given, one using the inverse Laplace transform and properties of analytic semigroups, and the other based on Doeblin's condition. Both methods admit generalizations to a wide class of processes.
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机译：在$ d $维空间的有界开放区域中，我们考虑布朗运动，该运动一旦到达边界便在固定的内部点重生。相对于以返回点为中心的Dirichlet Laplacian的Green函数等于常数模的密度，演化是不变的。我们以封闭形式计算分解剂，研究其光谱特性，并明确确定一维光谱。给出了两种关于指数遍历性的证明，一种使用逆Laplace变换和解析半群的性质，另一种基于Doeblin条件。两种方法都可以归纳为多种过程。
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