Time-Homogeneous Markov Semigroups

• Time-homogeneous Markov semigroups are families of operators with constant transition laws that model probability evolution in time-invariant systems.
• They use infinitesimal generators and the semigroup property to establish well-defined dynamics in both finite and infinite-dimensional spaces.
• Their stability and ergodicity are demonstrated via contraction properties and Lyapunov conditions, underpinning applications from diffusion to quantum channels.

A time-homogeneous Markov semigroup is a family of operators describing the evolution of probability distributions or functionals under a stochastic process whose transition laws do not depend on time. These semigroups arise in probability theory, mathematical physics, analysis of partial differential equations, and quantum information theory, underpinning the evolution of Markov processes in discrete or continuous spaces, and finite or infinite-dimensional settings. Central objects of the theory include the semigroup property, infinitesimal generators, spectral and contraction properties, embedding criteria for Markov matrices, and ergodic and stability characteristics.

1. Formal Definition and Semigroup Structure

A time-homogeneous Markov semigroup $(P_t)_{t\ge0}$ is a family of operators acting on a function space or state space, satisfying:

• $P_0 = \mathrm{Id}$,
• $P_{s+t} = P_s P_t$ for all $s,t\ge0$ (Chapman–Kolmogorov property),
• $P_t$ is positivity preserving: $P_t f\ge0$ if $f\ge0$,
• $P_t1=1$ (preservation of constants, Markov property),
• strong continuity: in suitable topology, $t\mapsto P_t f$ is continuous for all $f$ in the domain.

When acting on $L^p$-spaces (or $C_b$ spaces), the semigroup may be expressed by

$P_t f(x) = \mathbb{E}_x[f(X_t)],$

where $X=(X_t)_{t\ge0}$ is a Markov process with generator $L$. The Markov property implies invariance under time translations, i.e., the future evolution depends only on the present state and not on the history or absolute time.

Semigroups may arise as integral operators defined via a kernel $p_t(x,dy)$: $P_t f(x) = \int_E f(y)\,p_t(x,dy).$

On abstract ordered Banach spaces ("abstract state spaces"), an operator semigroup $T=(T_t)_{t\ge0}$ is time-homogeneous Markov if $T_0=I$, $T_{t+s}=T_tT_s$, $T_t$ is positive and $T_t$ maps the state "base" $K = \{x\in X_+ : f(x) = 1\}$ to itself (Erkurşun-Özcan et al., 2018).

2. Generators and Infinitesimal Description

Every (strongly continuous) Markov semigroup is associated with an infinitesimal generator $(L,D(L))$: $$L f = \lim_{t\to0^+}\frac{P_t f - f}{t}, \quad f \in D(L).$$ Generation theorems (e.g., Hille–Yosida, Lumer–Phillips) provide analytic criteria ensuring that a (possibly unbounded) linear operator $L$ generates a strongly continuous contraction semigroup on a Banach or Hilbert space (Andrisani et al., 2011). Dissipativity and surjectivity conditions guarantee existence and uniqueness of the semigroup. The generator encodes the infinitesimal dynamics: for diffusions, $L$ often has the form $L = \Delta - \nabla V\cdot\nabla$, with $P_t=e^{tL}$, or more generally as a (pseudo-)differential or integral operator, such as for Lévy processes: $$[L_0f](x) = \frac12 \nabla\cdot A\nabla f(x) + \int_{y\neq0}\left[f(x+y) - f(x) - y\cdot\nabla f(x)\mathbf{1}_{|y|\le1}\right]\ell(dy).$$

For infinite-dimensional systems, $L$ acts on an appropriate function space equipped with a locally convex topology, such as the strict topology or mixed topology on $C_b(E)$. In the case of Gauss–Markov processes on separable Hilbert spaces, $L$ is an explicitly constructed second-order differential operator with domains and core functions reflecting the geometry of $H$ (Goldys et al., 2013).

3. Structural Properties: Reversibility, Embeddability, and Hypercontractivity

Reversibility and Markov Matrices

A finite Markov matrix $M$ is reversible if there exists $p>0$ such that $p_i M_{ij} = p_j M_{ji}$ for all $i,j$ (Baake et al., 27 Nov 2025). Embeddability into a continuous-time semigroup $M = e^Q$ is characterized by:

• $\sigma(M) \subset \mathbb{R}_+$,
• the principal matrix logarithm $Q=\log M$ has non-negative off-diagonals,
• $Q$ preserves detailed balance.

If negative eigenvalues occur with even multiplicity, non-reversible embeddings may exist, but never reversible ones. Reducible or weakly reversible cases are treated blockwise or with relaxed equilibrium constraints.

Hypercontractivity and Functional Inequalities

Time-homogeneous Markov semigroups on $L^2(\mu)$ may exhibit hypercontractivity, with the semigroup improving regularity of distributions: $\|P_t f\|_{L^{\Phi_{u(t)}}} \le \|f\|_{L^{\Phi_0}}$ for a suitable family of Orlicz functions $\{\Phi_u\}$, equivalent to functional inequalities (e.g., log-Sobolev, F-Sobolev). Time-homogeneity simplifies the analysis, as the generator $L$ remains fixed and invariant measures are stationary (Roberto et al., 2021).

4. Stability, Contraction, and Ergodicity

Semigroup stability is quantified by contraction properties with respect to various distances—total variation, weighted norms, or Kantorovich/Wasserstein metrics (Moral et al., 11 Nov 2025): $$D_\phi(P_t(x,\cdot), P_t(y,\cdot)) \le \alpha(t) \phi(x,y)$$ under suitable Lyapunov-drift and minorization (Doeblin) conditions. A drift condition $P_{t_0}V(x) \le a V(x)+ b 1_K(x)$, together with local minorization, yields a "V-positive" semigroup with quantitative spectral gap (Moral et al., 2021).

On abstract state spaces, the Dobrushin coefficient $\delta(T)$ encodes contraction on the "mass-zero" subspace. Uniform asymptotic stability (exponential mixing to a unique stationary state) is equivalent to $\delta(T_{t_0})<1$. Lyapunov-type conditions and perturbation bounds yield robust error and sensitivity estimates for invariant states under small generator perturbations (Erkurşun-Özcan et al., 2018).

5. Topological Foundations and Infinite-Dimensional Extensions

Classical $C_0$-semigroup theory is adapted to non-normed or infinite-dimensional topologies. The mixed topology $\tau_1^{\mathscr{M}}$ on $C_\kappa(E)$ interpolates between norm and compact-uniform topologies, enabling (bi)continuous semigroup generation and analysis on Polish or Prohorov spaces (Goldys et al., 2022). The generator $(L,D(L))$ can be reconstructed via Euler formulas, and Markov core operators provide a criterion for uniqueness of the Fokker–Planck or martingale problem.

For Gauss–Markov semigroups on Hilbert spaces, the strict topology on $C_b(H_{\mathrm{bw}})$ (bounded-weak) is necessary to ensure strong continuity and the existence of appropriate cores of the generator (Goldys et al., 2013). The infinitesimal generator acts on cylinder test functions via infinite-dimensional Courrège-type formulas.

6. Special Cases, Limitations, and Non-Markovianity

Time-homogeneous Markov semigroups fail to arise from genuinely path-dependent stochastic equations unless special structure is present. In stochastic Volterra equations (SVEs) with Hölder coefficients, the time-homogeneous Markov property is lost unless the kernel $K(t)$ is exponentially decaying, i.e., $K(t)=c e^{-\lambda t}$, corresponding exactly to the case where the SVE reduces to an ordinary SDE (Friesen et al., 25 Oct 2025). For generic Volterra or rough models, only "local" (SDE-type) memoryless evolutions generate a time-homogeneous semigroup.

In infinite-dimensional hypocoercive models, such as kinetic Fokker–Planck structures on $(\mathbb{R}^m)^{\mathbb{Z}^d}$ or degenerate diffusions, strong smoothing and ergodic properties can be established via careful commutator and Lyapunov-type arguments within the semigroup framework (Kontis et al., 2013).

7. Applications and Further Directions

Time-homogeneous Markov semigroups provide the evolution mechanism for reversible and irreversible Markov processes, including diffusions, jump processes, quantum channels, and interacting particle systems. Extensions encompass semigroups in quantum probability, nonlinear (convex, order-preserving) semigroups relevant for stochastic control and viscosity solutions to Hamilton–Jacobi–Bellman equations (Goldys et al., 2022), and bilateral Markov semigroups associated to generalized Schrödinger equations via Doob transforms between Markov generators and self-adjoint Hamiltonians (Andrisani et al., 2011).

Contemporary research includes operator-theoretic unifications of stability and contractivity across classical and quantum models, investigation of the embedding problem for Markov matrices of arbitrary spectrum or structure, and analysis of ergodicity, spectral gaps, and mixing in unbounded domains and infinite-dimensional systems. Rigorous non-Markovianity results for path-dependent equations further delimit the scope of time-homogeneous semigroup representations (Friesen et al., 25 Oct 2025).