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Transition Semigroups and Infinitesimal Generators for PDEs

Learn how a diffusion’s generator gives the instantaneous rate of change of expected test functions and connects that evolution to a differential operator.

Advanced 9 min read

What You'll Learn

  • Define a diffusion’s transition kernel and transition semigroup.
  • Relate the infinitesimal generator to the drift and diffusion coefficients.
  • Prove the semigroup-generator differentiation identity under explicit domain assumptions.
  • Compute semigroup actions and generator actions for three standard diffusions.
  • Distinguish the operator identity from the additional regularity needed for a classical PDE solution.
  • Identify the common error of applying the generator to a function outside its domain.

From Transition Probabilities to Expected Test Functions

A diffusion describes random motion, while a partial differential equation describes how quantities vary across space and time. The link between them is expectation: start the diffusion at \(x\), evaluate a test function \(f\) at a later time, and take its expected value. The transition semigroup packages all these expectations into operators. Its infinitesimal generator describes their instantaneous change.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\). Consider the time-homogeneous diffusion \(dX_t=\mu(X_t)\,dt+\sigma(X_t)\,dB_t\), with \(X_0=x\). Assume the coefficients are globally Lipschitz and have at most linear growth, so the SDE has a unique, nonexplosive strong solution. The stochastic integral here is an Ito integral: on each finite time interval its progressively measurable integrand \(\sigma(X_s)\) is in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T \sigma(X_s)^2\,ds<\infty\). Write \(\mathbb{E}_x\) for expectation when the initial state is \(x\).

Definition: The transition kernel is \(P_t(x,A)=\mathbb{P}_x(X_t\in A)\), for measurable sets \(A\). For a measurable test function \(f\) for which the expectation exists, the transition semigroup acts by \(P_t f(x)=\mathbb{E}_x[f(X_t)]=\int f(y)\,P_t(x,dy)\). The Markov property gives \(P_0f=f\) and \(P_{s+t}f=P_s(P_tf)\).

For the operator viewpoint, take \(C_0\) to be the continuous functions on the state space that vanish at infinity, with the supremum norm. Assume the diffusion semigroup maps \(C_0\) into \(C_0\), is strongly continuous there, and is a contraction: \(\|P_t f\|_\infty\le\|f\|_\infty\). Strong continuity means \(\|P_t f-f\|_\infty\to0\) as \(t\downarrow0\). These are the Feller semigroup assumptions used below. They hold for the standard examples in this tutorial.

Definition: The infinitesimal generator \(A\) has domain \(D(A)\) consisting of those \(f\in C_0\) for which the limit \(\lim_{h\downarrow0}(P_hf-f)/h\) exists in the supremum norm. For such \(f\), \(Af\) is that limit. For a one-dimensional diffusion, on twice continuously differentiable functions where this generator action is valid, Ito's formula identifies \(Af(x)=\mu(x)f'(x)+\tfrac12\sigma(x)^2f''(x)\).

The last expression is the diffusion operator: the drift contributes the first-derivative term, and the diffusion contributes the second-derivative term. The formula does not assert that every twice differentiable function belongs to \(D(A)\); the norm limit and the chosen state space still matter.

The Semigroup-Generator Differentiation Identity

Theorem: Let \((P_t)_{t\ge0}\) be a strongly continuous contraction semigroup on \(C_0\), with generator \(A\) defined by the limit above. If \(f\in D(A)\), then \(P_t f\in D(A)\) and \(\frac{d}{dt}P_t f=P_tAf=AP_tf\) in the \(C_0\) norm for \(t\ge0\), with a right derivative at \(t=0\).

Proof. Fix \(f\in D(A)\). By the semigroup law, for \(h>0\), \((P_hP_tf-P_tf)/h=P_t(P_hf-f)/h\). As \(h\downarrow0\), the right side converges in \(C_0\) to \(P_tAf\), since \(P_t\) is a bounded operator. Thus \(P_tf\in D(A)\) and \(AP_tf=P_tAf\). The same computation gives the right derivative of \(P_tf\) at every \(t\ge0\).

For \(t>0\), the left difference quotient is \((P_tf-P_{t-h}f)/h=P_{t-h}(P_hf-f)/h\), for \(0<h<t\). The factor \((P_hf-f)/h\) converges to \(Af\). Also \(P_{t-h}Af\to P_tAf\), by strong continuity and the semigroup law. To justify the product limit, write its difference from \(P_tAf\) as \(P_{t-h}((P_hf-f)/h-Af)+(P_{t-h}Af-P_tAf)\); the first term tends to zero by contraction, and the second by strong continuity. Hence the left derivative agrees with the right derivative. This proves differentiability and both operator identities. \(\square\)

For \(u(t,x)=P_tf(x)\), the identity says that the expected test function evolves at rate \(P_tAf\). If \(u(t,\cdot)\) has the required spatial derivatives and the generator acts on it through the diffusion formula, then \(AP_tf(x)=\mu(x)\partial_xu(t,x)+\tfrac12\sigma(x)^2\partial_{xx}u(t,x)\). This is the operator form of the associated evolution equation. The semigroup identity itself is a statement in the function space; by itself, it does not establish classical spatial differentiability or boundary conditions.

Worked Examples: The Operator in Three Diffusions

Worked Example: Brownian Motion and a Quadratic Test Function

Let \(X_t=x+B_t\), and use \(f(x)=x^2\). Compute the semigroup action and compare its time derivative with the generator action.

Since \(B_t\) has mean zero and variance \(t\), \(\mathbb{E}_x[X_t^2]=\mathbb{E}[(x+B_t)^2]=x^2+t\). Ito's formula makes the generator calculation explicit: \(f'(x)=2x\), \(f''(x)=2\), and the correction term is \(\tfrac12 f''(X_t)\,dt=dt\). Thus the drift is zero and \(Af(x)=0\cdot2x+\tfrac12\cdot1^2\cdot2=1\). Therefore \(\partial_tP_tf(x)=1=P_tAf(x)\).

Check: differentiating the directly computed semigroup value \(x^2+t\) gives \(1\), agreeing with the generator. This check uses a function outside \(C_0\); here the calculation is valid as a moment computation, but the \(C_0\)-domain theorem above is not being applied to \(x^2\).

Worked Example: Constant Drift and Diffusion

Let \(X_t=x+\tfrac12t+B_t\), and take \(f(x)=x\). Find \(P_tf(x)\), \(Af(x)\), and the evolution rate.

Taking expectations gives \(P_tf(x)=\mathbb{E}_x[X_t]=x+\tfrac12t\). For \(f(x)=x\), \(f'(x)=1\) and \(f''(x)=0\); Ito's formula has correction term \(\tfrac12 f''(X_t)\,dt=0\). The generator is \(Af(x)=\tfrac12\cdot1+ \tfrac12\cdot1^2\cdot0=\tfrac12\). Hence \(P_tAf(x)=\tfrac12\), while differentiating \(x+\tfrac12t\) in time also gives \(\tfrac12\).

Check: at \(t=2\), the mean is \(x+\tfrac12(2)=x+1\), and its rate of change remains \(\tfrac12\). This function too is not in \(C_0\); the expectation and generator computations are valid for this integrable affine function, but do not invoke the \(C_0\)-domain theorem.

Worked Example: Ornstein–Uhlenbeck Motion

For \(dX_t=-X_t\,dt+dB_t\), \(X_0=x\), find the semigroup action on \(f(x)=x\), then check the generator identity.

Multiplying the SDE by the integrating factor \(e^t\) gives \(d(e^tX_t)=e^t\,dB_t\), so \(X_t=e^{-t}x+\int_0^t e^{-(t-s)}\,dB_s\). The Ito integrand is progressively measurable and square-integrable since \(\int_0^t e^{-2(t-s)}\,ds=(1-e^{-2t})/2<\infty\). The Ito integral has mean zero, giving \(P_tf(x)=e^{-t}x\).

For \(f(x)=x\), \(f'(x)=1\), \(f''(x)=0\), so Ito's formula has correction term \(\tfrac12 f''(X_t)\,dt=0\), and \(Af(x)=-x\). Applying the semigroup to \(Af\) gives \(P_tAf(x)=-e^{-t}x\), which equals the time derivative of \(e^{-t}x\).

Check: at \(t=0\), the mean is \(x\) and its instantaneous rate is \(-x\), exactly the drift at the initial state. The affine function \(f(x)=x\) is not in \(C_0\); this direct calculation is justified by the explicit solution and finite first moment, not by applying the \(C_0\)-domain theorem to \(f\).

What the Identity Does—and Does Not—Say

The generator is local: for a diffusion it uses the coefficients and derivatives at the current state. The semigroup is global in time: \(P_tf(x)\) averages the test function over the full distribution of \(X_t\) given \(X_0=x\). The differentiation identity connects these views. It says that the instantaneous change in the expected test function is the expected generator action, \(P_tAf\), and, when the domain is preserved, is also the generator applied to the evolved function, \(AP_tf\).

This is useful when studying expected payoffs, moments, and diffusion-driven evolution. It also tells you what must be checked before writing a differential equation: the test function must be in the generator domain for the theorem, and spatial regularity must be established before interpreting \(AP_tf\) through classical derivatives. In unbounded state spaces, functions such as \(x\) and \(x^2\) often require a weighted function space or a separate integrability argument instead of the \(C_0\) framework.

Common pitfall: writing \(Af=\mu f'+\tfrac12\sigma^2f''\) and immediately asserting the semigroup differentiation identity for any smooth \(f\). For example, \(f(x)=x^2\) is smooth but does not vanish at infinity, so it is not in the \(C_0\) space used in the theorem. The Brownian calculation above is correct because its moments can be computed directly; it is not a license to ignore the domain in other problems. The correction is to check the relevant function space and domain, or to prove the required integrability and differentiation facts by another method.

Check Your Understanding

For each question, state any domain or integrability condition needed for your argument.

  1. For Brownian motion and \(f(x)=x^2\), compute \(Af(x)\) directly from the diffusion coefficients and compare it with the derivative of \(P_tf(x)\).
  2. For \(dX_t=\mu\,dt+\sigma\,dB_t\) and \(f(x)=x\), what are \(P_tf(x)\) and \(P_tAf(x)\)?
  3. State the definition of \(D(A)\) for a strongly continuous semigroup on \(C_0\).
  4. In the semigroup-generator theorem, where is the semigroup law used to identify \(AP_tf\)?
  5. Why does smoothness alone not justify applying the \(C_0\)-domain theorem to \(f(x)=x\)?

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