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Time-like Graphical Models

About this Title

Tvrtko Tadić, University of Washington, Department of Mathematics, Box 354350, Seattle, WA 98195-4350, USA ; University of Zagreb, Department of Mathematics, Bijenička cesta 30, 10000 Zagreb, Croatia

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 261, Number 1262
ISBNs: 978-1-4704-3685-8 (print); 978-1-4704-5416-6 (online)
Published electronically: November 6, 2019
Keywords: Stochastic processes indexed by graphs, graphical models, time-like graphs, martingales indexed by directed sets, stochastic heat equation
MSC: Primary 60G20, 60G60, 60H15, 60J65, 60J80, 62H05; Secondary 05C99

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Table of Contents


  • Introduction

1. Construction and properties

  • 1. Geometry of time-like graphs
  • 2. Processes indexed by time-like graphs
  • 3. Markov properties of processes indexed by TLG’s
  • 4. Filtrations, martingales and stopping times

2. Natural Brownian motion and the stochastic heat equation

  • 5. Maximums of Gaussian processes
  • 6. Random walk and stochastic heat equation reviewed
  • 7. Limit of the natural Brownian motion on a rhombus grid

3. Processes on general and random time-like graphs

  • 8. Non-simple TLG’s
  • 9. Processes on non-simple TLG’s
  • 10. Galton-Watson time-like trees and the Branching Markov processes

Open questions and appendix

  • 11. Open questions
  • A. Independence and processes
  • Acknowledgments


We study continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a continuous version of graphical models indexed by graphs with an embedded time structure—so-called time-like graphs. We extend the notion of time-like graphs and find properties of processes indexed by them. In particular, we solve the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. We provide a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, our treatment of time-like graphical models reveals connections to Markov random fields, martingales indexed by directed sets and branching Markov processes.

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