Probability and Stochastic Processes
Random fields, noisy evolution, and Monte Carlo estimates require different kinds of probabilistic information. This chapter connects distributions and expectations with Gaussian structure, stochastic dynamics, and correlated-sample uncertainty, making clear which conclusions require independence, Gaussianity, or mixing.
Enter this chapter
Section titled “Enter this chapter”Begin with Probability Spaces, Random Variables, and Conditional Expectation to identify the probability law and the information being conditioned on. For numerical estimates, follow the limit-theorem and correlated-sample-error pages.
Topics
Section titled “Topics”- Probability Spaces, Random Variables, and Conditional Expectation — Defines probability laws, measurable quantities, and conditional averages.
- Probabilistic Convergence, Laws of Large Numbers, and Central Limit Theorems — Separates consistency from fluctuation laws and their hypotheses.
- Characteristic Functions, Moments, Cumulants, and Generating Functionals — Relates derivatives of generating functions to moments and connected cumulants.
- Gaussian Vectors, Processes, Random Distributions, and Wick Structure — Explains when mean and covariance determine all finite-probe distributions.
- Stochastic Processes and Correlation Functions — Distinguishes joint laws from the limited information in correlation functions.
- Brownian Motion, Stochastic Calculus, Langevin Equations, and Fokker–Planck Dynamics — Relates random trajectories to deterministic evolution of their probability density.
- Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error — Connects mixing and autocorrelation to uncertainty in sampled averages.
How the ideas fit together
Section titled “How the ideas fit together”A Gaussian process is determined by its mean and covariance, and its higher moments reduce to pairings. A general process is not: identical two-point functions can coexist with different higher cumulants. Knowing a covariance therefore does not justify Wick reduction unless Gaussianity is established.
A second distinction concerns sampling. A stationary Markov chain can have the desired invariant law without successive samples being independent. Under a suitable central limit theorem, the variance of its average includes the sum of time-lag covariances; long correlations reduce effective information. See Roberts and Rosenthal 2004, § 5. Invariance alone neither proves mixing nor supplies that central limit theorem.
Review the chapter
Section titled “Review the chapter”Does drawing twice as many consecutive correlated samples always halve the variance of an average?
Solution
No. That scaling is asymptotic and needs a finite long-run variance plus a central limit theorem. At a fixed run length, unresolved slow modes or insufficient equilibration can prevent the expected improvement; marginal variances alone omit the cross-covariances.
Where to go next
Section titled “Where to go next”Continue to Functional Integrals and Free Gaussian Fields for the relation between regulated Gaussian measures and quantum correlators.