Markov Processes

§ 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.