Markov Processes. Questo articolo non è disponibile.Markov Processes

The modem theory of Markov processes has its origins in the studies of A. A. MARKOV (1906-1907) on sequences of experiments "connected in a chain" and in the attempts to describe mathematically the physical phenomenon known as Brownian motion (L. BACHELlER 1900, A. EIN­ STEIN 1905). The first correct mathematical construction of a Markov process with continuous trajectories was given by N. WIENER in 1923. (This process is often called the Wiener process.) The general theory of Markov processes was developed in the 1930's and 1940's by A. N. KOL­ MOGOROV, W. FELLER, W. DOEBLlN, P. LEVY, J. L. DOOB, and others. During the past ten years the theory of Markov processes has entered a new period of intensive development. The methods of the theory of semigroups of linear operators made possible further progress in the classification of Markov processes by their infinitesimal characteristics. The broad classes of Markov processes with continuous trajectories be­ came the main object of study. The connections between Markov pro­ cesses and classical analysis were further developed. It has become possible not only to apply the results and methods of analysis to the problems of probability theory, but also to investigate analytic problems using probabilistic methods. Remarkable new connections between Markov processes and potential theory were revealed. The foundations of the theory were reviewed critically: the new concept of strong Markov process acquired for the whole theory of Markov processes great importance.

§ 1. Modern definition of a Markov process.- § 2. Shift operators. Infinitesimal and characteristic operators.- § 3. Diffusion processes. Probabilistic solution of differential equations.- § 4. Additive functionals.- § 5. Superharmonic and harmonic functions.- § 6. Transformations of Markov processes connected with additive functionals.- § 7. Generalized Brownian motion.- § 8. What is the structure of the most general continuous strong Markov process?.- § 9. Nonnegative harmonic functions and asymptotic behavior of paths of a Markov process.- One Contraction semigroups of linear operators on Banach spaces.- § 1. Banach spaces.- § 2. Contraction semigroups of linear operators and their infinitesimal operators.- § 3. Uniqueness theorems.- § 4. Construction of a semigroup from an infinitesimal operator.- § 5. Relationship between measurability properties and continuity properties of semigroups of operators.- § 6. The weak infinitesimal operator.- § 7. Excessive elements.- Two Infinitesimal operators of transition functions.- § 1. Transition functions and corresponding semigroups of operators.- § 2. Uniqueness theorems.- § 3. Examples.- § 4. Feller transition functions on compact spaces.- § 5. ?-Functions on semi-compacts.- Three Markov processes.- § 1. Definition of a Markov process.- § 2. Markov processes and transition functions.- § 3. Strong Markov processes.- Four First entrance and exit times and the intrinsic topology in the state space.- § 1. First entrance, contact and exit times.- § 2. The intrinsic topology in the state space.- § 3. Continuous functions in the intrinsic topology.- § 4. The intrinsic topology for the Wiener process.- Five Characteristic operators of Markov processes. Differential generators of diffusion processes.- § 1. General theorems on resolvents and infinitesimal operators of Markov processes.- § 2. Absorbing and stable states.- § 3. Definition and general properties of characteristic operators.- § 4. Characteristic operators of continuous processes.- § 5. Diffusion processes and their differential generators.- § 6. Construction of a diffusion process from the differential generator.- § 7. Characteristic operators in the intrinsic topology.- Six Functionals of Markov processes.- § 1. Basic definitions.- § 2. Operation of passage to the limit.- § 3. W-functionals.- § 4. Approximation of nonnegative, additive functionals by W-functionals.- § 5. Mathematical expectations of random variables, connected with additive functionals.- Seven Stochastic integrals.- § 1. Stochastic integrals as functionals of a Wiener random function.- § 2. A theorem on the transformation of integral functionals.- § 3. Stochastic integrals as functionals of a Wiener process.- Eight Nonnegative additive functionals of a Wiener process.- § 1. Integral representation of a W-function.- § 2. W-functionals.- § 3. S-functionals.- § 4. Functionals of one-dimensional Wiener processes.- Nine Transition functions, corresponding to almost multiplicative functionals.- § 1. Definition and examples.- § 2. Construction of a functional from a quasi-transition function.- § 3. Properties of trajectories of Markov processes, corresponding to transformations of transition functions.- § 4. Transformation of the resolvent and the infinitesimal operator.- Ten Transformations of Markov processes.- § 1. Curtailment of lifetimes and formation of parts of processes.- § 2. Stopped processes.- § 3. Transformation of the measures Px.- § 4. (?, ?)-subprocesses.- § 5. Random time change.- § 6. Transformation of the state space.- Eleven Stochastic integral equations and diffusion processes.- § 1. Stochastic integral equations for additive functionals of a Wiener random function.- § 2. Construction of diffusion processes.- § 3. Stopped diffusion processes.- Twelve Excessive, superharmonic and harmonic functions.- § 1. Excessive functions for transition functions.- § 2. Excessive functions for Markov processes.- § 3. Asymptotic behavior of excessive functions along trajectories of a process.- § 4. Superharmonic functions.- § 5. Harmonic functions.- Thirteen Harmonic and superharmonic functions associated with strong Feller processes. Probabilistic solution of certain equations.- § 1. Some properties of strong Feller processes.- § 2. The Dirichlet problem. Regular points of the boundary.- § 3. Harmonic and superharmonic functions associated with diffusion processes.- § 4. Solutions of the equation Af ? Vf = ?g.- § 5. Parts of a diffusion process and Green’s functions.- Fourteen The multi-dimensional Wiener process and its transformations.- § 1. Harmonic and superharmonic functions related to the Wiener process.- § 2. The mapping ?.- § 3. Additive functionals and Green’s functions.- § 4. Brownian motion with killing measure ? and speed measure v.- § 5. q-subprocesses.- § 6. Brownian motion with drift.- Fifteen Continuous strong Markov processes on a closed interval.- § 1. General properties of one-dimensional continuous strong Markov processes.- § 2. Characteristics of regular processes.- § 3. Computation of the characteristic and infinitesimal operators.- § 4. Superharmonic and harmonic functions connected with regular one-dimensional processes.- Sixteen Continuous strong Markov processes on an open interval.- § 1. Harmonic functions and behavior of trajectories.- § 2. S-functions and character of the motion along a trajectory.- § 3. Infinitesimal operators.- Seventeen Construction of one-dimensional continuous strong Markov processes.- § 1. Transformations of the state space. Canonical coordinate.- § 2. Construction of regular continuous strong Markov processes on an open interval.- § 3. Construction of regular continuous strong Markov processes on a closed interval.- § 4. Computation of the harmonic functions and the resolvents for regular processes.- § 1. Measurable spaces and measurable transformations.- § 2. Measures and integrals.- § 3. Probability spaces.- § 4. Martingales.- § 5. Topological measurable spaces.- § 6. Some theorems on partial differential equations.- § 7. Measures and countably additive set functions on the line and corresponding point functions.- § 8. Convex functions.- Historical-bibliographical note.- List of symbols.- List of symbols.