Diffusion Processes

Core Idea

A diffusion process is a continuous-path strong Markov process, typically the solution of an SDE dX = μ(x)dt + σ(x)dW. Its behavior is characterized by the infinitesimal generator Lf = μf' + (1/2)σ²f'', the scale function s(x) = ∫exp(-2∫μ/σ² dy)dx (which determines the direction of drift), and the speed measure m(dx) = 2dx/(σ²(x)s'(x)) (which determines how long the process spends near each point). Together, scale and speed classify every one-dimensional diffusion's boundary behavior and long-run properties.

Explainer

A diffusion process is the continuous-time, continuous-path Markov process that arises as the solution of an SDE dX = μ(x)dt + σ(x)dW with σ(x) > 0. The term "diffusion" refers to both the process and the PDE framework (Fokker-Planck equation) that describes its density evolution. Diffusions are the natural continuous-state extension of continuous-time Markov chains: where CTMCs jump between discrete states with exponential holding times, diffusions move continuously through the real line (or higher-dimensional space) driven by noise.

The infinitesimal generator Lf(x) = μ(x)f'(x) + (1/2)σ²(x)f''(x) is the fundamental operator associated with the diffusion. For any sufficiently smooth function f, the process f(X(t)) - ∫₀ᵗ Lf(X(s))ds is a local martingale — the generator L computes the expected instantaneous rate of change of f along the process. The backward Kolmogorov equation ∂u/∂t = Lu governs expectations; the forward (Fokker-Planck) equation ∂ρ/∂t = L*ρ governs the density evolution. The generator is the single object from which all probabilistic and analytical information about the diffusion can be extracted.

In one dimension, the theory is remarkably complete thanks to two functions: the scale function s(x) and the speed measure m(dx). The scale function s(x) = ∫exp(-2∫₀ˣ μ(y)/σ²(y) dy)dx transforms the diffusion into a local martingale: s(X(t)) has no drift. It determines hitting probabilities — the probability of reaching level a before level b, starting from x, is (s(x)-s(b))/(s(a)-s(b)). The speed measure m(dx) = 2dx/(σ²(x)s'(x)) determines how long the process spends near each point. Together, s and m classify the boundary behavior (Feller's classification into regular, exit, entrance, and natural boundaries) and determine the stationary distribution (proportional to m when it has finite total mass).

Feller's boundary classification is the capstone of one-dimensional diffusion theory. Each boundary point is classified as: regular (reached in finite time, from which the process can return), exit (reached in finite time but not returned from), entrance (not reached from the interior but can be a starting point), or natural (inaccessible from either direction). For the OU process, both ±∞ are natural boundaries — the mean-reverting drift prevents escape. For Brownian motion with drift μ > 0, +∞ is natural but -∞ is also natural (the drift pushes rightward, but the process is recurrent only if μ = 0). These classifications, computed entirely from the scale function and speed measure, determine the long-run behavior and the need for boundary conditions.