The Infinitesimal Generator

One operator summarizes a process: (Af)(x) is the instantaneous rate of change of E[f(X_t)], and for Brownian motion it is 12 d^2dx^2.

The idea

The infinitesimal generator is the compact description of a Markov process: an operator recording what the process does to averages over a single instant. The paths themselves are complicated and uncountably many; the generator carries the same information in one object.

Definition (Infinitesimal generator).

Let $X$ be a Markov process. Its infinitesimal generator is the operator $A$ that sends a smooth function $f$ of the state to the function $(Af)(x) = \lim_{t \to 0} \frac{\mathbb{E}\big[f(X_{t}) \mid X_{0} = x\big] - f(x)}{t},$ the rate at which the average of $f$ is changing at time zero when the process starts at $x$.

The generator $A$ is an operator rather than a number: given a function $f$, it returns another function $Af$, whose value at each starting point $x$ is that instantaneous rate. Note that it is a derivative in time of an expectation, not a derivative of $f$; and that it depends on the process as much as on $f$, which is how the process is encoded in it.

One instant suffices because a Markov process rebuilds its future one instant at a time from its current state. The generator therefore determines how averages evolve over long stretches, which partial differential equation they satisfy, and how long the process takes to leave a region.

Ways to work on it

• Walkthrough. Define the generator of a diffusion and compute it for Brownian motion.
• Proof. Dynkin's formula from Itô's lemma: the generator is the ds coefficient, and the dW-integral dies under expectation.
• Practice. Read generators off stochastic differential equations and apply them to simple functions.
• Hardest. Find the expected time for Brownian motion to leave an interval.

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