Applications of the graphs to the Generalized Ornstein-Uhlenbeck process

TL;DR

This paper explores the application of graph theory to the generalized Ornstein-Uhlenbeck process, specifically constructing Lévy noise and representing the process using rooted trees with two leaf types.

Contribution

It introduces a novel graph-based representation of the generalized Ornstein-Uhlenbeck process and constructs the Lévy noise explicitly.

Findings

Lévy noise is explicitly constructed for the process.

The process is represented using rooted trees with two types of leaves.

The graph-based approach provides new insights into the structure of the process.

Abstract

We consider the generalized Ornstein- Uhlenbeck equation $\partial_t X=-m X_t+\eta$. In this paper We construct the L\'evy noise $\eta$. The generalized Ornstein- Uhlenbeck process $X_t$ will be represented by a special types of graphs called rooted trees with two types of leaves.