Course description - After a brief overview of Markov processes and martingale theory, the course will explore a variety of key stochastic models. How these models are related to each other and to deterministic models will be discussed in details. Although originally motivated by biology, these processes are general invasion and competition models that are important in physics and sociology as well. The main objective is to learn how to use the stochastic framework to model phenomena that arise from applied sciences, and how to simulate and analyze these models. The lectures are adapted from recent research papers by the instructor and do not require any specific background in mathematics outside basic probability.

Basics about stochastic processes

• Poisson point process, number of points in a finite set, connections between Bernoulli, Binomial, Poisson, Geometric, and Exponential random variables.
• Discrete-time Markov chains, irreducibility, recurrence and transience, application to symmetric random walks, continuous-time Markov chains, embedded Markov chains, exponential holding times.
• Martingales, supermartingales, submartingales, number of excursions, Dubins' inequality, martingale convergence theorem.

Non-spatial models (no interaction)

• Birth and death process, critical value.
• Branching processes, connection with martingales, critical value, generating function.

Spatially implicit models (global interactions)

• Logistic growth model, carrying capacity, quasi-stationary distribution, time to extinction.
• Wright-Fisher model, Moran's model, connection with martingales, probability of fixation, Ewens' argument, rescaled Wright-Fisher model, diffusion approximation, time to fixation.
• Coalescent theory, Kingman's coalescent, time to the most recent common ancestor.

Spatially explicit static models (local interactions)

• Bond percolation in two dimensions, percolation probability, mean cluster size, planar duality, critical value.
• Site percolation in two dimensions, percolation probability, critical value, contour argument.
• Oriented site percolation in two dimensions, connection with Markov chains, percolation probability, critical value, contour argument.

Spatially explicit dynamic models (local interactions)

• Interacting particle systems on the regular lattice, generalization to connected graphs, regular tree, small world network, Harris' graphical representation, simulation in finite volume.
• Contact process, weak survival, strong survival, survival probability, coupling argument, critical birth rate, connection with the logistic growth model.
• Voter model, duality with coalescing random walks, clustering in two dimensions, cluster size, spatial correlations, coexistence in three dimensions, connection with the Wright-Fisher model.
• Multitype contact process, duality, tree structure, ancestor hierarchy, renewal points, clustering in two dimensions, coexistence in three dimensions.

Class time: Tuesday and Thursday, 3:00pm to 4:15pm
Location: PSA 307 (Tempe)
Class number: 77293
Office hours: Tuesday and Thursday, from 4:15pm to 5:00pm

Course description - Laws of probability, combinatorial analysis, random variables, probability distributions, expectations, moment-generating functions, transformations of random variables, law of large numbers, and central limit theorem.

Textbook: A first course in probability, Sheldon Ross, Pearson Prentice Hall, 2006.

Here is a summary of the main concepts that will be reviewed during the semester.

Homework assignments

1. Ex 3, 8, 10, 12, 15, 17, 20, 23, 29, 32 of chapter 1 (due Sept 8)
2. Ex 13, 16, 43, 45, 55 of chapter 2 (due Sept 22)
3. Ex 11, 18, 23, 30, 38, 47, 53, 74 of chapter 3 (due Oct 8)
4. Ex 7, 8, 22, 23, 28, 35 of chapter 4 (due Oct 29)
5. Ex 48, 50, 56, 71, 82 of chapter 4 (due Nov 10)
6. Ex 2, 6, 17, 34, 37 from chapter 5 (due Nov 24)