Cosine families, invariant subspaces, and boundary conditions for a class of diffusions on star graphs
Abstract
This paper explores the interplay between boundary conditions and invariant subspaces for one-dimensional Laplacians, extending these concepts to Walsh’s spider process on a star-like graph. We establish a precise correspondence between the transmission condition characterizing this process and a specific subspace within a larger function space. This correspondence is facilitated by relating the cosine family associated with the spider process to the basic cosine family of unrestricted Brownian motion. Furthermore, we introduce a complementary subspace, leading to a novel decomposition of the function space that generalizes known results for simpler boundary conditions. This decomposition reveals a fundamental relationship between two distinct transmission conditions, highlighting their complementary nature. Our findings provide new insights into the structure of Walsh’s spider process and offer a framework for further analysis, including the study of its limiting behavior as the stickiness parameter varies.
