Analytic and stochastic description of Brownian motions on star graphs

Abstract

We provide a detailed description of all possible Feller processes on infinite} star graphs with finite number of edges, processes that while away from the graph's center behave like a one-dimensional Brownian motion. The description can be seen as a continuation of the seminal paper by Itô and McKean (devoted to Brownian motions on the half-line), recast from the perspective of the theory of multi-armed bandits.

Figures (3)

• Figure 1: Infinite star graph $K_{1,{\mathpzc k}}$ with ${\mathpzc k}=9$ edges and graph center $0$.
• Figure 2: 9999 Generalized inverses. If a $u\ge 0$ is a point of continuity of $\psi$ and $\psi (u) =t$ then $\psi^{-1} (t) =u$, and so $\psi \circ \psi^{-1} (t)=t$. If, however, at a $u\ge 0$ there is a jump of $\psi$, that is, $\psi(u-) < \psi (u)$ (by default $\psi(0-)=0$), then $\psi^{-1} (t) = u$ for all $t\in [\psi(u-), \psi(u)]$, and so, for such $t$, we have $\psi \circ \psi^{-1}(t)=\psi (u)$.
• Figure 3: A Brownian motion on the nonnegative half-line with jumps from the boundary where $x=0$. When the local time at $x=0$ exceeds an independent random variable, the process starts all over again at a random point.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Proposition 5.1
• proof
• Theorem 6.1
• Lemma 6.2
• proof
• proof : Proof of uniqueness in Theorem \ref{['thm:wba']}