Stochastic processes and noisy evolution

A process X_t assigns a random variable to each time on a common probability space. Fixing an outcome gives a sample path; fixing time gives a marginal distribution. Two processes can have the same marginal distribution at every time but different dependence. For example, repeatedly reusing one fair coin and tossing a new independent fair coin at every integer time both give a fair marginal at each step. Their paths and conditional predictions are very different.

A filtration records information available up to each time. The Markov property says that, given the current state, the conditional future does not need additional information from the past. It does not mean successive states are independent. For a finite time-homogeneous chain, a transition matrix P has nonnegative entries and rows summing to one. With row-vector distributions, the next distribution is πP. A stationary distribution satisfies πP=π, but stationarity alone does not prove that every starting distribution converges to it.

Standard Brownian motion begins at zero, has independent stationary Gaussian increments, and has continuous paths. Over an interval of length Δt its increment has variance Δt, so its typical size scales as the square root of elapsed time. This differs from a smooth trajectory's increment, which scales linearly with elapsed time. Brownian paths are almost surely nowhere differentiable. Drawing them with straight segments is an approximation for display, not evidence that a finite velocity exists between infinitely close times.

The accumulated squared increments over increasingly fine partitions converge to elapsed time in the appropriate probabilistic sense. This quadratic variation makes a second-order Taylor term survive in stochastic calculus. In an Itô integral, the integrand is adapted and evaluated using information available before each increment. Itô and Stratonovich conventions are different limiting rules; converting between them changes drift terms. The convention must be stated whenever multiplicative noise depends on the current state.

An Itô stochastic differential equation combines deterministic drift with Brownian forcing. For globally Lipschitz drift and diffusion coefficients with suitable growth control, standard theorems give a unique nonexplosive strong solution from appropriate initial data. Rougher coefficients need different conditions. Itô's formula adds one half of the squared diffusion coefficient times the second derivative of a test function. Taking expectations leads, under suitable regularity and boundary assumptions, to the Fokker–Planck equation for the density.

This bridge connects simulation of individual paths with a deterministic PDE for probability mass. It underlies diffusion models, but a trained generative system also needs a specified forward process, learned score or reverse dynamics, and discretization analysis. Similarly, treating minibatch gradient noise as Brownian forcing is an approximation, not an identity: its covariance can depend on parameters, batches may be correlated, and finite step sizes matter. Stochastic models should expose these assumptions before their predictions are trusted.