Feller Processes and Semigroups

• Feller processes are strong Markov processes whose transition semigroups act as strongly continuous contraction operators on C0(E), ensuring well-posed evolution.
• They are defined through analytic properties such as the Hille–Yosida theorem, resolvent relations, and equivalence theorems that rigorously characterize their generators.
• Advanced construction techniques, including state-space-dependent mixtures of Lévy processes and Euler-type Monte Carlo simulations, enhance both theoretical insights and practical applications.

A Feller process is a strong Markov process whose transition semigroup acts as a strongly continuous contraction semigroup on the Banach space of real-valued continuous functions vanishing at infinity, denoted $C_0(E)$ for a locally compact separable metric space $E$. The associated Feller semigroups, generators, and their analytic and probabilistic properties form the mathematical foundation for a vast array of spatially inhomogeneous Markovian dynamics, with direct connections to stochastic analysis, partial differential equations, functional analysis, and applications in finite and infinite dimensions.

1. Classical Definition and Characterization of Feller Semigroups

Let $E$ be a locally compact separable metric space, and $C_0(E)$ the Banach space of real-valued continuous functions vanishing at infinity, with $\|f\|_{\infty} = \sup_{x \in E}|f(x)|$. A family $(T_t)_{t \ge 0}$ of bounded linear operators on $C_0(E)$ is a Feller semigroup if:

• (i) $T_0 = \mathrm{Id}$, and $T_{s+t}=T_s T_t$ for all $s, t \ge 0$,
• (ii) $E$0 (contractivity),
• (iii) $E$1 is positivity-preserving: $E$2,
• (iv) For every $E$3, $E$4 as $E$5 (strong continuity).

The infinitesimal generator $E$6 of $E$7 is defined as: $E$8 where the domain $E$9 consists of those $E$0 for which the above limit exists in $E$1 (Kostrykin et al., 2011, Böttcher, 2010). Generators of Feller semigroups are characterized by the Hille–Yosida theorem: $E$2 generates a Feller semigroup if and only if it is dissipative, preserves the positive cone, and $E$3 for all $E$4, together with appropriate resolvent bounds (Böttcher, 2010).

The associated Feller process is a càdlàg Markov process $E$5 with state space $E$6 whose transition operators $E$7 form a Feller semigroup (Böttcher, 2010).

2. Feller Resolvents and Equivalence Theorems

Given a Feller semigroup, its resolvent family is defined by

$E$8

A family $E$9 is called a Feller resolvent if it satisfies:

• (a) $C_0(E)$0 for all $C_0(E)$1,
• (b) $C_0(E)$2,
• (c) For every $C_0(E)$3, $C_0(E)$4 as $C_0(E)$5 (Kostrykin et al., 2011).

A comprehensive equivalence theorem establishes that various analytic and pointwise continuity properties—on the semigroup or the resolvent—are all characterizations of the Feller property. For instance, strong continuity in norm may be replaced by pointwise right-continuity plus invariance of $C_0(E)$6 under the semigroup or resolvent (Kostrykin et al., 2011).

3. Strong Feller Property and Analytic Regularity

The strong Feller property strengthens the mapping property of a semigroup: For $C_0(E)$7, $C_0(E)$8 maps bounded measurable functions ($C_0(E)$9) into continuous functions ($\|f\|_{\infty} = \sup_{x \in E}|f(x)|$0): $\|f\|_{\infty} = \sup_{x \in E}|f(x)|$1 (Schilling et al., 2010, Budde et al., 2022). For Markov semigroups on Polish or locally compact spaces,

• Strong Feller $\|f\|_{\infty} = \sup_{x \in E}|f(x)|$2 Feller $\|f\|_{\infty} = \sup_{x \in E}|f(x)|$3 weak Feller, but neither implication is reversible in general (Budde et al., 2022).

Two characterizations of the strong Feller property are equivalent:

• Locally uniform absolute continuity: For each $\|f\|_{\infty} = \sup_{x \in E}|f(x)|$4, there is a Radon measure $\|f\|_{\infty} = \sup_{x \in E}|f(x)|$5 such that for every compact $\|f\|_{\infty} = \sup_{x \in E}|f(x)|$6 and $\|f\|_{\infty} = \sup_{x \in E}|f(x)|$7,

$\|f\|_{\infty} = \sup_{x \in E}|f(x)|$8

• Local Orlicz-ultracontractivity: For each $\|f\|_{\infty} = \sup_{x \in E}|f(x)|$9, there are $(T_t)_{t \ge 0}$0 and a Young function $(T_t)_{t \ge 0}$1 such that for compact $(T_t)_{t \ge 0}$2 and $(T_t)_{t \ge 0}$3,

$(T_t)_{t \ge 0}$4

These conditions are both necessary and sufficient and clarify that the strong Feller property can often be deduced from kernel estimates rather than structural properties of the process (Schilling et al., 2010).

A key result establishes that under analyticity (sectoriality) of the semigroup on $(T_t)_{t \ge 0}$5, the strong Feller property for the resolvent implies the strong Feller property for the semigroup (Kusuoka et al., 2021). Such analytic semigroups arise for symmetric Dirichlet forms and a broad class of elliptic generators.

4. Construction of Feller Processes: Mixtures, Simulation, and Approximation

A central methodology for constructing Feller processes with spatially inhomogeneous dynamics is state-space-dependent mixing of Lévy processes (Böttcher, 2010). Given independent Lévy processes $(T_t)_{t \ge 0}$6 and bounded Lipschitz mixing functions $(T_t)_{t \ge 0}$7, the generator

$(T_t)_{t \ge 0}$8

yields a spatially inhomogeneous Feller process. Existence results follow from martingale problem/SDE techniques, pseudo-differential operator theory, and the Hille–Yosida framework (Böttcher, 2010).

Numerical simulation is facilitated by an Euler-type Monte Carlo algorithm: Given the symbol $(T_t)_{t \ge 0}$9, at each discrete time $C_0(E)$0, an increment $C_0(E)$1 is drawn from the law with characteristic function $C_0(E)$2, and $C_0(E)$3. Uniform growth bounds on $C_0(E)$4 ensure convergence of the discrete process to the true Feller process as $C_0(E)$5 (Böttcher, 2010).

Examples include:

• Brownian–Poisson–Cauchy mixtures,
• Stable-like processes with variable jump index,
• Variable-coefficient normal inverse Gaussian and Meixner processes, giving rise to a wide class of analytically tractable, spatially inhomogeneous models (Böttcher, 2010).

5. Feller Semigroups in Diverse Settings: Manifolds, Time-changes, Generalized Theories

On Riemannian manifolds of bounded geometry, sufficient conditions for second order elliptic operators to generate Feller semigroups are given by the closure of operators of the form

$C_0(E)$6

where the vector fields $C_0(E)$7 are smooth and (uniformly) bounded. The associated Feller semigroups can be approximated by Chernoff-type product formulas involving shift operators, leading to convergence of random walks to manifold diffusions (Mazzucchi et al., 2020).

Generalized Feller theory extends the concept to weighted spaces or infinite-dimensional settings. Here, the Banach space $C_0(E)$8 consists of functions controlled at infinity by an admissible weight $C_0(E)$9, and the semigroup properties are adapted with corresponding continuity and positivity assumptions (Cuchiero et al., 2023). This framework is essential for the analysis of infinite-dimensional stochastic processes (e.g., SPDEs, signature SDEs) and the construction of processes on non locally compact spaces.

For time-changed processes, a "substitute Feller property" describes continuity and invariance conditions on an appropriate subspace $T_0 = \mathrm{Id}$0, associated to the fine-support of the time-change measure. This framework provides sufficient conditions for Feller properties and uniform convergence of semigroups for time-changed Markov processes, with applications to Dirichlet-to-Neumann semigroups and Kreĭn-Feller operators (BenAmor et al., 2024).

6. Regularity Theory: Schauder Estimates and Harmonic Analysis

Hölder and Schauder estimates for the Poisson equation $T_0 = \mathrm{Id}$1, where $T_0 = \mathrm{Id}$2 is the infinitesimal generator of a Feller process, underlie the regularity theory for both local and non-local operators. For stable-like Feller processes with variable index $T_0 = \mathrm{Id}$3, the associated semigroups admit estimates of the form

$T_0 = \mathrm{Id}$4

with further estimates in variable-order Hölder-Zygmund spaces $T_0 = \mathrm{Id}$5 (Kühn, 2019). The Favard space, defined by finiteness of $T_0 = \mathrm{Id}$6, coincides with the domain $T_0 = \mathrm{Id}$7 of the extended generator for which $T_0 = \mathrm{Id}$8 (Kühn, 2019). Such estimates enable Schauder-type theorems adapted to stable-like or more general pseudo-differential generators.

7. Counterexamples and Necessity of Compactness-Type Hypotheses

It is not sufficient to assume strong Feller continuity and joint space-time continuity of a semigroup of Markov kernels to guarantee the existence of an associated càdlàg or right Markov process on the same state space. A prototypical counterexample is the restriction of the Brownian motion semigroup to $T_0 = \mathrm{Id}$9: the semigroup is strong Feller and jointly continuous but cannot arise from any right process on this state space, regardless of the Polish topology imposed (Beznea et al., 2022).

Such pathologies highlight the necessity of additional compactness, Lyapunov function existence, or quasi-regularity conditions. On locally compact spaces, these are provided by the structure of $T_{s+t}=T_s T_t$0, the existence of tight Lyapunov functions, or compact containment conditions, ensuring the classic Hunt process correspondence (Beznea et al., 2022).

8. Advanced Topics: Feynman Formulae, Time-fractional Dynamics, and Perturbation Theory

Chernoff product formulas provide Lagrangian and Hamiltonian path integral representations for Feller semigroups and the Cauchy–Dirichlet problem for killed processes. Such Feynman–Kac-type path integral approximations give analytical and simulation tools, including for non-local and time-fractional evolution equations, by representing solutions as limits of iterated integrals (Butko et al., 2012, Butko, 2017).

Perturbation theory for strong Feller semigroups—using Miyadera-Voigt type results—enables the construction of transition semigroups and analysis of ergodicity, invariant measures, and well-posedness for semilinear stochastic equations in infinite dimensions, provided suitable gradient bounds or irreducibility (Kunze, 2011).

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The theory of Feller processes and semigroups, including their analytic, probabilistic, and computational aspects, forms a robust, unifying framework for Markovian evolution in both homogeneous and inhomogeneous settings, and underpins much of modern research in stochastic analysis and its applications.