What are Probability Theory and Stochastic Processes?

Mathematicians think of probabilities as numbers in the interval from 0 to 1 assigned to "events" whose occurrence or failure to occur is random. Probabilities P(E) are assigned to events E according to the probability axioms.

In the mathematics of probability, a stochastic process can be thought of as a random function. In practical applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a time series in applications) or a region of space (a stochastic process being called a random field). Familiar examples of time series include stock market and exchange rate fluctuations, signals such as speech, audio and video; medical data such as a patient's EKG, EEG, blood pressure or temperature; and random movement such as Brownian motion or random walks. Examples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogeneous material.

In the Mathematics Department at UCDavis, we currently have research focusing in areas of

• Stochastic optimization
• Random matrix theory
• Random growth models
• Control and estimation of non-linear dynamical systems and stochastic processes
• Random point processes
• Cellular automata theory
• Percolation models
• Mixing times for Markov chains
• High-dimensional convex geometry
• Applications of the probabilistic method in computer science