The probability research group is primarily focused on discrete probability topics. Random graphs and percolation models (infinite random graphs) are studied using stochastic ordering, subadditivity, and the probabilistic method, and have applications to phase transitions and critical phenomena in physics, flow of fluids in porous media, and spread of epidemics or knowledge in populations. Convergence rates to equilibrium in Markov chains are studied and applied to Markov Chain Monte Carlo simulation, and related algorithms for perfect sampling are created and analyzed.Various probabilistic and other techniques are used to analyze the performance of algorithms in computer science used for such purposes as sorting and searching. A plethora of interesting questions and applications allow us to involve both undergraduate and graduate students in valuable research in modern probability and stochastic processes.

## Background: Areas of Research

Probability theory aims to provide a mathematical framework to describe, model, analyze, and solve problems involving random phenomena and complex systems. While its original motivation was the study of gambling problems, probability has significant applications in finance, computer science, engineering, statistical mechanics, and biology. In the mathematical sciences, probability is fundamental for the analysis of statistical procedures, and the “probabilistic method” is an important tool for proving existence theorems in discrete mathematics.

Stochastic processes are probabilistic models for random quantities evolving in time or space. The evolution is governed by some dependence relationship between the random quantities at different times or locations. Major classes of stochastic processes are random walks, Markov processes, branching processes, renewal processes, martingales, and Brownian motion. Important application areas are mathematical finance, queuing processes, analysis of computer algorithms, economic time series, image analysis, social networks, and modeling biomedical phenomena. Stochastic process models are used extensively in operations research applications.

## Related Courses

Complete descriptions appear in the course catalog.

View the semester course schedule.

- EN.110.653 Stochastic Differential Equations: An Introduction with Applications
- EN.550.722 Introduction to Stochastic Calculus
- EN.550.627 Stochastic Processes and Applications to Finance
- EN.553.112 Statistical Analysis II
- EN.553.171 Discrete Mathematics
- EN.553.211 Probability and Statistics for the Life Sciences
- EN.553.310 Prob & Stats for the Physical and Information Sciences & Engineering
- EN.553.310 Probability & Statistics for the Physical Sciences & Engineering
- EN.553.311 Probability and Statistics for the Biological Sciences and Engineering
- EN.553.4/613 Applied Statistics and Data Analysis
- EN.553.4/614 Applied Statistics and Data Analysis II
- EN.553.4/616 Introduction to Statistical Learning, Data Analysis and Signal Processing
- EN.553.4/620 Introduction to Probability
- EN.553.4/626 Introduction to Stochastic Processes
- EN.553.4/627 Stochastic Processes and Applications to Finance
- EN.553.4/628 Stochastic Processes and Applications to Finance II
- EN.553.4/629 Introduction to Research in Discrete Probability
- EN.553.4/630 Introduction to Statistics
- EN.553.4/633 Monte Carlo Methods
- EN.553.4/644 Introduction to Financial Derivatives
- EN.553.4/645 Interest Rate and Credit Derivatives
- EN.553.4/692 Mathematical Biology
- EN.553.720 Probability Theory I
- EN.553.721 Probability Theory II
- EN.553.723 Markov Chains
- EN.553.727 Large Deviations Theory
- EN.553.729 Topics in Probability: Random Graphs and Percolation
- EN.553.730 Statistical Theory
- EN.553.731 Statistical Theory II
- EN.553.734 Introduction to Nonparametric Estimation
- EN.553.735 Topics in Statistical Pattern Recognition
- EN.553.738. High-Dimensional Approximation, Probability, and Statistical Learning
- EN.553.764 Modeling, Simulation, and Monte Carlo
- EN.553.782 Statistical Uncertainty Quantification
- EN.553.790 Topics In Applied Math

## Learn More

Research and academic opportunities in probability and stochastic processes