diffusielimiet

Diffusielimiet, or diffusion limit, is a concept in probability and applied mathematics describing the limiting behavior of a stochastic system when time and space are rescaled so that the discrete randomness converges to a continuous diffusion process. In this limit, complex or high‑dimensional models can be approximated by a stochastic differential equation or a reflected diffusion, capturing the average drift together with random fluctuations.

Formally, one considers a family of processes Xn(t) and, after centering and scaling (for example, speeding up time by a factor n and scaling space by a factor sqrt(n)), shows that Xn converges in distribution to a diffusion process X that solves an equation of the form dX(t) = a(X(t),t) dt + b(X(t),t) dW(t), where W is a standard Brownian motion. In many applied settings, the limit process may be a reflected diffusion to account for boundary constraints.

Diffusion limits are especially common in queueing theory under heavy-traffic conditions, where the queue length or workload can be approximated by a reflected Brownian motion. They also appear in birth–death processes, population genetics, and financial mathematics, where large-scale or high-activity regimes yield diffusion approximations such as the Ornstein–Uhlenbeck process or general diffusion models.

Applications of diffusielimiet include obtaining tractable approximations for performance measures (e.g., waiting times, queue lengths, or hitting times) and providing insight into the qualitative behavior of systems when exact analysis is intractable. It is distinct from fluid limits, which yield deterministic trajectories, and from large deviations, which focus on rare events and exponential rates.