A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.

中文翻译：

---

在均匀空间上展开随机游走

如果一个局部紧群上的一个测度相对于 Haar 测度而言不是奇异的，则该测度被称为是展开的。使用马尔可夫链理论，我们对具有分散增量分布的均匀空间上的随机游走进行了详细分析。对于有限体积空间，我们得到了n步长分布：它们向 Haar 度量均匀分布，通常在起始位置呈指数级快速和局部均匀分布。此外，许多经典极限定理被证明是成立的。在无限体积的情况下，我们证明了对称散布随机游走在最多二次增长的均匀空间上的递归和比率极限定理。这在一个长期存在的猜想中确定了一个方向。