**Memoirs of the American Mathematical Society**

2019;
170 pp;
Softcover

MSC: Primary 60;
Secondary 05

**Print ISBN: 978-1-4704-3685-8
Product Code: MEMO/261/1262**

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

**Electronic ISBN: 978-1-4704-5416-6
Product Code: MEMO/261/1262.E**

List Price: $81.00

AMS Member Price: $48.60

MAA Member Price: $72.90

# Time-Like Graphical Models

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*Tvrtko Tadić*

The author studies 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.
The author extends the notion of time-like graphs and finds properties
of processes indexed by them. In particular, the author solves the conjecture
of uniqueness of the distribution for the process indexed by graphs
with infinite number of vertices.

The author provides a new result showing the stochastic heat
equation as a limit of the sequence of natural Brownian motions on
time-like graphs. In addition, the author's treatment of time-like
graphical models reveals connections to Markov random fields,
martingales indexed by directed sets and branching Markov
processes.

#### Table of Contents

# Table of Contents

## Time-Like Graphical Models

- Cover Cover11
- Title page i2
- Introduction 110
- Part 1 . Construction and properties 514
- Chapter 1. Geometry of time-like graphs 716
- 1.1. Basic definitions 716
- 1.2. TLG* family 918
- 1.3. Consistent representation of a TLG*-tower, spines and (re)construction 1322
- 1.4. Interval TLG*’s 1625
- 1.5. Topology on TLG’s 1827
- 1.6. TLG* as a topological lattice 2029
- 1.7. Cell collapse transformation and the stingy algorithm 2231
- 1.8. TLG’s with infinitely many vertices 2837

- Chapter 2. Processes indexed by time-like graphs 3140
- Chapter 3. Markov properties of processes indexed by TLG’s 4958
- Chapter 4. Filtrations, martingales and stopping times 6372

- Part 2 . Natural Brownian motion and the stochastic heat equation 7786
- Chapter 5. Maximums of Gaussian processes 8190
- Chapter 6. Random walk and stochastic heat equation reviewed 8796
- 6.1. Modification of the Local Limit Theorem 8796
- 6.2. Approximations of the classical heat equation solution 9099
- 6.3. Euler method for the stochastic heat equation 97106
- 6.4. Convergence of interpolation of the Euler method 105114
- 6.5. Euler method with initial value condition and no external noise 108117

- Chapter 7. Limit of the natural Brownian motion on a rhombus grid 111120

- Part 3 . Processes on general and random time-like graphs 119128
- Open questions and appendix 149158
- Chapter 11. Open questions 151160
- Appendix A. Independence and processes 155164
- A.1. Conditional independence and expectations 155164
- A.2. Construction of a conditional sequence 155164
- A.3. Markov and Brownian bridges 156165
- A.4. Markov random fields 159168
- A.5. White noise 160169
- A.6. The stochastic heat equation 161170
- A.7. Crump - Mode - Jagers trees 163172
- A.8. Branching Markov processes and branching Brownian motion 164173

- Acknowledgments 165174
- Bibliography 167176
- Index 169178

- Back Cover Back Cover1184