Markov Processes, Feller Semigroups and Evolution Equations (Series on Concrete and Applicable Mathematics) (Series on Concrete and Applicable Mathematics, 12)

The book provides a systemic treatment of time-dependent strong Markov processes with values in a Polish space. It describes its generators and the link with stochastic differential equations in infinite dimensions. In a unifying way, where the square gradient operator is employed, new results for backward stochastic differential equations and long-time behavior are discussed in depth. The book also establishes a link between propagators or evolution families with the Feller property and time-inhomogeneous Markov processes. This mathematical material finds its applications in several branches of the scientific world, among which are mathematical physics, hedging models in financial mathematics, and population models. 1. Introduction : Stochastic differential equations. 1.1. Weak and strong solutions to stochastic differential equations. 1.2. Stochastic differential equations in the infinite-dimensional setting. 1.3. Martingales. 1.4. Operator-valued Brownian motion and the Heston volatility model. 1.5. Stopping times and time-homogeneous Markov processes -- 2. Strong Markov processes on Polish spaces. 2.1. Strict topology. 2.2. Strong Markov processes and Feller evolutions. 2.3. Strong Markov processes : Main results. 2.4. Dini's lemma, Scheffe's theorem, and the monotone class theorem -- 3. Strong Markov processes : Proof of main results. 3.1. Proof of the main results : Theorems 2.9 through 2.13 -- 4. Space-time operators and miscellaneous topics. 4.1. Space-time operators. 4.2. Dissipative operators and maximum principle. 4.3. Korovkin property. 4.4. Continuous sample paths. 4.5. Measurability properties of hitting times -- 5. Feynman-Kac formulas, backward stochastic differential equations and Markov processes. 5.1. Introduction. 5.2. A probabilistic approach : Weak solutions. 5.3. Existence and uniqueness of solutions to BSDE's. 5.4. Backward stochastic differential equations and Markov processes -- 6. Viscosity solutions, backward stochastic differential equations and Markov processes. 6.1. Comparison theorems. 6.2. Viscosity solutions. 6.3. Backward stochastic differential equations in finance. 6.4. Some related remarks -- 7. The Hamilton-Jacobi-Bellman equation and the stochastic Noether theorem. 7.1. Introduction. 7.2. The Hamilton-Jacobi-Bellman equation and its solution. 7.3. The Hamilton-Jacobi-Bellman equation and viscosity solutions. 7.4. A stochastic Noether theorem -- 8. On non-stationary Markov processes and Dunford projections. 8.1. Introduction. 8.2. Kolmogorov operators and weak-continuous semigroups. 8.3. Kolmogorov operators and analytic semigroups. 8.4. Ergodicity in the non-stationary case. 8.5. Conclusions. 8.6. Another characterization of generators of analytic semigroups. 8.7. A version of the Bismut-Elworthy formula -- 9. Coupling methods and Sobolev type inequalities. 9.1. Coupling methods. 9.2. Some ergodic theorems. 9.3. Spectral gap. 9.4. Some related stability results. 9.5. Notes -- 10. Invariant measure. 10.1. Markov chains : Invariant measure. 10.2. Markov processes and invariant measures. 10.3. A proof of Orey's theorem. 10.4. About invariant (or stationary) measures This book provides a systemic treatment of time-dependent strong Markov processes with values in a Polish space. It describes its generators and the link with stochastic differential equations in infinite dimensions. In a unifying way, where the square gradient operator is employed, new results for backward stochastic differential equations and long-time behavior are discussed in depth. This mathematical material finds its applications in several branches of the scientific world among which mathematical physics, hedging models in financial mathematics, population models This volume provides a systematic treatment of time-dependent strong Markov processes with values in a Polish space. It describes its generators and the link with stochastic differential equations in infinite dimensions Provides a systemic treatment of time-dependent strong Markov processes with values in a Polish space. This book describes its generators and the link with stochastic differential equations in infinite dimensions.