Stochastic processes describe systems that evolve randomly over time — stock prices, particle motion, queues, populations. The mathematics is genuinely advanced, resting on measure-theoretic probability, and this is a field where reading order is not a nicety but a necessity. Approach Brownian motion or stochastic calculus without the probabilistic foundations and you will simply be lost.
The path builds rigorously: the probability bedrock first, then discrete processes, then continuous ones, and finally advanced topics.
The probabilistic foundation
Start with Probability with martingales by David Williams, a beloved, rigorous introduction to measure-theoretic probability that makes a hard subject genuinely enjoyable. Reinforce it with Probability by Rick Durrett, the comprehensive graduate text that is the standard reference. Do not skip this stage — everything downstream assumes this machinery.
Markov chains
The most intuitive processes come next. Markov Chains by Norris is the clear, standard introduction to systems where the future depends only on the present, covering both discrete and continuous time. Markov Chains and Mixing Times then dives into the deep, modern question of how fast a chain reaches equilibrium — a rich and active area. These build strong intuition before the harder continuous theory.
Jump and continuous processes
Now the classic processes. Poisson processes by Kingman is the elegant, definitive treatment of the fundamental model for random events in time. Levy Processes And Infinitely Divisible Distributions generalizes this to processes with jumps, and Brownian motion by Morters and Peres introduces the continuous process at the heart of the field. Together they cover the essential zoo of stochastic behavior.
Advanced theory
Finally, the deep end. Stochastic Differential Equations by Oksendal is the accessible gateway to stochastic calculus and its applications, while Continuous martingales and Brownian motion by Revuz and Yor is the rigorous, advanced reference. Interacting particle systems by Liggett and Ergodic theory by Petersen round out the frontier for those going furthest. Read these last, once the foundations and classic processes are secure.
This is graduate-level mathematics, and progress here is measured in months, not weekends — but followed in order, the path turns an intimidating subject into a coherent, beautiful structure.
