**Contemporary Mathematics**

Volume: 668;
2016;
221 pp;
Softcover

MSC: Primary 05; 15; 49; 54; 60;

**Print ISBN: 978-1-4704-1945-5
Product Code: CONM/668**

List Price: $108.00

AMS Member Price: $86.40

MAA Member Price: $97.20

**Electronic ISBN: 978-1-4704-3473-1
Product Code: CONM/668.E**

List Price: $108.00

AMS Member Price: $86.40

MAA Member Price: $97.20

# Probability on Algebraic and Geometric Structures

Share this page *Edited by *
*Gregory Budzban; Harry Randolph Hughes; Henri Schurz*

This volume contains the proceedings of the
International Research Conference “Probability on Algebraic and
Geometric Structures”, held from June 5–7, 2014, at
Southern Illinois University, Carbondale, IL, celebrating the careers
of Philip Feinsilver, Salah-Eldin A. Mohammed, and Arunava
Mukherjea.

These proceedings include survey papers and new research on a
variety of topics such as probability measures and the behavior of
stochastic processes on groups, semigroups, and Clifford algebras;
algebraic methods for analyzing Markov chains and products of random
matrices; stochastic integrals and stochastic ordinary, partial, and
functional differential equations.

#### Readership

Graduate students and research mathematicians interested in probability and stochastic processes.

# Table of Contents

## Probability on Algebraic and Geometric Structures

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface ix10
- The fine structure of the stationary distribution for a simple Markov process 114
- Superdegenerate hypoelliptic differential operators 1326
- Some remarks on the convolution equation 𝜇*𝛽=𝜇 and product semigroups 2134
- Limit distributions of products of I.I.D. random 2×2 stochastic matrices: an open problem 3144
- Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations 4356
- Matrices with zero row sums, tree theorems and a Markov chain on trees 6780
- Convolution of orbital measures on symmetric spaces: A survey 8194
- Exit times for some autoregressive processes with non-Gaussian noise distributions 111124
- Brief history of optimal control theory and some recent developments 119132
- 1. Introduction 119132
- Energy conservation 120133
- Element of arc 120133
- Total time 120133
- Functional 121134
- Euler-Lagrange equation 121134
- Legendre (1736–1813) 121134
- Hamilton (1805–1865) - Weierstrass (1815–1897) 122135
- Jacobi 122135
- Bellman 123136
- Question 123136
- Pontryagin 1908–1988 124137
- Deterministic 124137
- Stochastic 125138
- 2. Optimal Control Theory adapted to Abrupt Financial Markets Changes 126139
- Control problem with Markov regime switching with noise 126139
- 3. Application 127140
- 4. Conclusion 129142
- References 129142

- Krawtchouk matrices, Feynman path integral and the split quaternions 131144
- 1. What are Krawtchouk matrices 131144
- 2. Counting, controlled and erratic 133146
- 3. What is quantum computing? 134147
- 4. Ehrenfest Urn problem: Beyond Kac’s solution 136149
- 5. Topological interpretation of Krawtchouk matrices 138151
- 6. Feynman sum over paths interpretation 139152
- 7. Quaternions and related Lie groups and algebras 144157
- Appendix A. Krawtchouk matrices 157170
- Appendix B. Krawtchouk matrices from Hadamard matrices 159172
- Appendix C. Pascal-Krawtchouk pyramid 160173
- Appendix D. Subspaces of binary spaces 162175
- References 163176

- The Itô-Stratonovich formula for an operator of order four 165178
- Random walks on motion groups 171184
- Existence, uniqueness, and energy of modified stochastic sine-Gordon equation with multiplicative noise on one-dimensional domain 179192
- Squared Bessel process with delay 199212
- Kravchuk matrices and induced operators on Clifford algebras 207220
- Back Cover Back Cover1236