A time-homogeneous Markov process converts elapsed time into a semigroup of positive contraction operators. The infinitesimal generator contains its local drift and variation, while the semigroup contains its full transition law. This operator viewpoint connects stochastic processes with evolution equations, spectral theory, and numerical approximation.
Transition functions and semigroups
Definition. A transition function is a family of kernels \(p_t(x,\mathrm{d} y)\), \(t\ge0\), satisfying \[ p_0(x,\cdot)=\delta_x,\qquad p_{t+s}(x,A)=\int p_s(y,A)p_t(x,\mathrm{d} y). \] Its operators are \[ P_tf(x)=\int f(y)p_t(x,\mathrm{d} y)=\mathbb E_xf(X_t). \]
Theorem (semigroup property). For a time-homogeneous Markov process, \[ P_0=I,\qquad P_{t+s}=P_tP_s. \] Each \(P_t\) is positive, preserves constants, and is a contraction on bounded functions: \(\|P_tf\|_\infty\le\|f\|_\infty\).
Proof. The Markov property and tower rule give \[ \begin{aligned} P_{t+s}f(x) &=\mathbb E_xf(X_{t+s})\\ &=\mathbb E_x[ \mathbb E(f(X_{t+s})\mid\mathcal F_t)]\\ &=\mathbb E_x[P_sf(X_t)] =P_t(P_sf)(x). \end{aligned} \] The remaining properties follow because \(P_t\) integrates against a probability kernel.
Definition (Feller semigroup). On a locally compact state space \(S\), a Markov semigroup is Feller if \[ P_t:C_0(S)\to C_0(S) \] and \[ \|P_tf-f\|_\infty\to0\qquad(t\downarrow0) \] for every \(f\in C_0(S)\). It is then a strongly continuous contraction semigroup on the Banach space \(C_0(S)\).
The infinitesimal generator
Definition. The generator \(A\) of a strongly continuous semigroup is the generally unbounded operator \[ Af=\lim_{t\downarrow0}\frac{P_tf-f}{t}, \] with domain \[ D(A)=\left\{f:\text{the limit exists in the Banach-space norm}\right\}. \]
The domain is part of the operator. Writing a differential expression without specifying its domain does not fully specify a generator; boundary behavior can produce different Markov processes from the same formal differential operator.
Theorem (backward equation). If \(f\in D(A)\), then \(P_tf\in D(A)\) and \[ \frac{\mathrm{d}}{\mathrm{d}t}P_tf =AP_tf=P_tAf. \]
Proof. The semigroup identity gives \[ \frac{P_{t+h}f-P_tf}{h} =P_t\frac{P_hf-f}{h}\to P_tAf \] by boundedness of \(P_t\). Also \[ \frac{P_h(P_tf)-P_tf}{h} =P_t\frac{P_hf-f}{h}\to P_tAf, \] showing \(P_tf\in D(A)\) and \(AP_tf=P_tAf\).
Dynkin's formula and the martingale problem
Theorem (Dynkin martingale). For suitable \(f\in D(A)\), \[ M_t^f = f(X_t)-f(X_0)-\int_0^tAf(X_s)\,\mathrm{d} s \] is a martingale.
Proof. For \(s<t\), partition \([s,t]\) and use the Markov property: \[ \mathbb E[f(X_{u+h})-f(X_u)\mid\mathcal F_u] =(P_hf-f)(X_u). \] Divide by \(h\), sum the conditional increments, and pass to the limit using \((P_hf-f)/h\to Af\) and the assumed integrability. This gives \[ \mathbb E[f(X_t)-f(X_s)\mid\mathcal F_s] = \mathbb E\!\left[\int_s^tAf(X_u)\,\mathrm{d} u \middle|\mathcal F_s\right], \] which is the martingale identity.
Definition. A law solves the martingale problem for \((A,\mathcal D)\) if \(M_t^f\) is a martingale for every \(f\in\mathcal D\). Existence and uniqueness of this problem can define a Markov process even when no explicit transition density is available.
Diffusion generators
Proposition. For the diffusion \[ \mathrm{d}X_t=b(X_t)\,\mathrm{d}t+\sigma(X_t)\,\mathrm{d}W_t \] in \(\mathbb R^d\), Ito's formula identifies the formal generator \[ Af(x) = b(x)^\top\nabla f(x) +\frac12\operatorname{tr} \!\left(a(x)\nabla^2f(x)\right), \qquad a(x)=\sigma(x)\sigma(x)^\top. \]
Proof. Ito's formula gives \[ f(X_t)-f(X_0) = \int_0^tAf(X_s)\,\mathrm{d} s +\int_0^t\nabla f(X_s)^\top\sigma(X_s)\,\mathrm{d}W_s. \] The stochastic integral is a local martingale and, under boundedness or localization assumptions, a true martingale. Comparing with Dynkin's form identifies \(A\).
Example (Brownian motion). For \(X_t=x+W_t\), \[ Af=\frac12\Delta f,\qquad P_tf(x)=\int_{\mathbb R^d}f(y) \frac{e^{-\|y-x\|^2/(2t)}}{(2\pi t)^{d/2}}\,\mathrm{d} y. \] Hence \(u(t,x)=P_tf(x)\) solves the heat equation \(\partial_tu=\frac12\Delta u\) with \(u(0,x)=f(x)\).
Continuous-time Markov chains
Proposition. On a countable state space, a conservative rate matrix \(Q=(q_{ij})\) satisfies \(q_{ij}\ge0\) for \(i\ne j\) and \(q_{ii}=-\sum_{j\ne i}q_{ij}\). Formally, \[ Af(i)=\sum_jq_{ij}f(j). \] When the semigroup is nonexplosive, its transition matrix solves the Kolmogorov equations \[ \frac{\mathrm{d}}{\mathrm{d}t}P_t=QP_t=P_tQ, \qquad P_0=I, \] on the appropriate domain.
For finite matrices, \(P_t=e^{tQ}\). In infinite dimensions the generator is usually unbounded, and semigroup theory replaces the formal matrix exponential; this is one of the main interfaces with functional analysis.