LEADER 04971nam 22005175 450 001 9910765490903321 005 20231116002731.0 010 $a3-031-33296-2 024 7 $a10.1007/978-3-031-33296-8 035 $a(MiAaPQ)EBC30949260 035 $a(Au-PeEL)EBL30949260 035 $a(DE-He213)978-3-031-33296-8 035 $a(EXLCZ)9928861500500041 100 $a20231116d2023 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aContinuous Parameter Markov Processes and Stochastic Differential Equations$b[electronic resource] /$fby Rabi Bhattacharya, Edward C. Waymire 205 $a1st ed. 2023. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2023. 215 $a1 online resource (502 pages) 225 1 $aGraduate Texts in Mathematics,$x2197-5612 ;$v299 311 08$aPrint version: Bhattacharya, Rabi Continuous Parameter Markov Processes and Stochastic Differential Equations Cham : Springer International Publishing AG,c2023 9783031332944 327 $a1. A review of Martingaels, stopping times and the Markov property -- 2. Semigroup theory and Markov processes.-3. Regularity of Markov process sample paths -- 4. Continuous parameter jump Markov processes -- 5. Processes with independent increments -- 6. The stochastic integral -- 7. Construction of difficusions as solutions of stochastic differential equations -- 8. Itô's Lemma -- 9. Cameron-Martin-Girsanov theorem -- 10. Support of nonsingular diffusions -- 11. Transience and recurrence of multidimensional diffusions -- 12. Criteria for explosion -- 13. Absorption, reflection and other transformations of Markov processes -- 14. The speed of convergence to equilibrium of discrete parameter Markov processes and Diffusions -- 15. Probabilistic representation of solutions to certain PDEs -- 16. Probabilistic solution of the classical Dirichlet problem -- 17. The functional Central Limit Theorem for ergodic Markov processes -- 18. Asymptotic stability for singular diffusions -- 19. Stochastic integrals with L2-Martingales -- 20. Local time for Brownian motion -- 21. Construction of one dimensional diffusions by Semigroups -- 22. Eigenfunction expansions of transition probabilities for one-dimensional diffusions -- 23. Special Topic: The Martingale Problem -- 24. Special topic: multiphase homogenization for transport in periodic media -- 25. Special topic: skew random walk and skew Brownian motion -- 26. Special topic: piecewise deterministic Markov processes in population biology -- A. The Hille-Yosida theorem and closed graph theorem -- References -- Related textbooks and monographs. 330 $aThis graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications. The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples. After a review of some background material, the reader is introduced to semigroup theory, including the Hille?Yosida Theorem, used to construct continuous parameter Markov processes. Illustrated with examples, it is a cornerstone of Feller?s seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes, and processes with independent increments, or Lévy processes. The greater part of the book is devoted to Itô?s fascinating theory of stochastic differential equations, and to the study of asymptotic properties of diffusions in all dimensions, such as explosion, transience, recurrence, existence of steady states, and the speed of convergence to equilibrium. A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes. Among Special Topics chapters, two study anomalous diffusions: one on skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media. 410 0$aGraduate Texts in Mathematics,$x2197-5612 ;$v299 606 $aProbabilities 606 $aMathematics 606 $aProbability Theory 606 $aApplications of Mathematics 615 0$aProbabilities. 615 0$aMathematics. 615 14$aProbability Theory. 615 24$aApplications of Mathematics. 676 $a519.233 700 $aBhattacharya$b Rabi$0102761 701 $aWaymire$b Edward C$0103517 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910765490903321 996 $aContinuous Parameter Markov Processes and Stochastic Differential Equations$93644970 997 $aUNINA