Feller process

Feller process
Name	Feller process
Field	Probability theory
Introduced	1950s
Mathematician	William Feller
Related concepts	Markov process, semigroup, generator, Hunt process, Lévy process

Feller process A Feller process is a class of time-homogeneous stochastic processes with strong continuity and regularity properties, introduced in the mid-20th century by William Feller. It appears at the intersection of functional analysis and probability and links the work of Norbert Wiener, Andrey Kolmogorov, Kiyoshi Itô, Paul Lévy, and Edward Nelson. Feller processes underpin modern developments related to E. B. Dynkin, Shizuo Kakutani, Joseph Doob, Kai Lai Chung, and Ronald Getoor.

Definition and basic properties

A Feller process is defined via a strongly continuous contraction semigroup on C_0 of a locally compact separable metric space, following ideas from Marshall Stone and Israel Gelfand. The definition connects to the Hille–Yosida theorem as used by Willem van der Waerden and later formalized in works by Kurt Friedrichs and John von Neumann. Basic properties include right-continuous paths with left limits under additional hypotheses, a strong Markov property in the spirit of results by W. A. W. Courant and Lars Hörmander, and invariance under suitable Feller resolvents considered by J. L. Doob and E. B. Dynkin. Classic regularity statements reference compactification techniques related to Marshall H. Stone and boundary classifications analogous to those used by G. H. Hardy in analytic settings.

Feller semigroups and generators

The central analytic object is the Feller semigroup, a family of linear operators satisfying positivity and contraction properties, reminiscent of semigroup theory developed by Einar Hille and Kurt Yosida. Generators of Feller semigroups are closed operators on C_0 characterized by variants of the Hille–Yosida theorem, with dissipativity conditions highlighted by Kato and spectral considerations paralleling work by Harold Kesten and Israel Gelfand. The Lumer–Phillips theorem, used by R. S. Phillips and Jerrold L. Lions, provides alternative generator characterizations exploited in the study of infinitesimal generators appearing in classic texts by E. B. Dynkin and T. E. Harris. Domain descriptions often employ core techniques attributed to A. M. Lyapunov and techniques of approximation related to J. von Neumann.

Examples and classes of Feller processes

Important examples include diffusion processes generated by elliptic operators as in the work of Kiyoshi Itô and Henry McKean, jump processes related to Paul Lévy and Kurt Gödel (via probability on discontinuous paths), and Lévy processes with Feller property treated by Sato. Diffusions on manifolds draw on methods from Marcel Berger and Shiing-Shen Chern, while birth–death processes and queueing models connect to analyses by Andrei Kolmogorov and Agner Krarup Erlang. Stable processes, subordinators, and censored stable processes relate to studies by Benoît Mandelbrot and Gennady Samorodnitsky. More specialized classes include Feller processes with killing or sticky boundaries considered by Gilbert Hunt and by Daniel W. Stroock with S. R. S. Varadhan.

Boundary behavior and extensions

Boundary classification and extensions for Feller processes draw on boundary theory developed by F. Riesz and Marcel Riesz, and on boundary condition frameworks introduced by Richard Courant and D. Hilbert. Feller boundary conditions for one-dimensional diffusions relate to classical Sturm–Liouville theory associated with E. C. Titchmarsh and David Hilbert. Extensions via Martin boundary methods connect to the Martin compactification developed by Robert J. Martin and potential-theoretic techniques of Lars Ahlfors and Emile Borel. Reflecting, absorbing, and elastic boundaries have been studied in probabilistic settings by W. Feller contemporaries and by researchers such as K. L. Chung, J. L. Doob, and R. K. Getoor.

Connections to Markov processes and potential theory

Feller processes form a subclass of strong Markov processes as formalized by Joseph Doob and E. B. Dynkin, and they are tightly linked to the Hunt process framework introduced by Gilbert Hunt. Potential theory connections trace to classical work by Norbert Wiener and modern probabilistic potential theory developed by Ronald Getoor and Michael B. Marcus. Excessive functions, resolvents, and Green kernels for Feller processes reference techniques from Riesz and from analytic frameworks employed by T. E. Harris and S. R. S. Varadhan. Duality, time reversal, and symmetrization ties reflect contributions by Kiyoshi Itô and Edward Nelson in stochastic analysis.

Applications and significance in probability theory

Feller processes underpin large areas of probability theory, including stochastic differential equations popularized by Kiyoshi Itô and Kunita Hiroshi, limit theorems influenced by Andrey Kolmogorov and Paul Lévy, and ergodic theory developments related to George D. Birkhoff and John von Neumann. Applications include mathematical finance traditions traced to Robert C. Merton and Fischer Black, population dynamics modeled using branching processes studied by J. H. C. Whitehead and Theodore E. Harris, and statistical mechanics contexts influenced by Ludwig Boltzmann and Josiah Willard Gibbs. Feller processes also provide foundational tools for modern work in stochastic partial differential equations advanced by Giuseppe Da Prato and Jerzy Zabczyk.

Category:Stochastic processes