Functional Analysis for Probability and Stochastic Processes: An Introduction
This text is designed both for students of probability and stochastic processes, and for students of functional analysis. It presents some chosen parts of functional analysis that can help understand ideas from probability and stochastic processes. The subjects range from basic Hilbert and Banach spaces, through weak topologies and Banach algebras, to the theory of semigroups of bounded linear operators. Numerous standard and non-standard examples and exercises make the book suitable as a course textbook as well as for self-study.
|
Contents
4
|
23 |
5
|
29 |
Basic notions in functional analysis
|
37 |
3
|
63 |
23
|
70 |
5548
|
76 |
4
|
115 |
Brownian motion and Hilbert spaces
|
121 |
2
|
127 |
Dual spaces and convergence of probability measures
|
147 |
The Gelfand transform and its applications
|
201 |
Semigroups of operators and Lévy processes
|
234 |
Markov processes and semigroups of operators
|
294 |
Appendixes
|
363 |
385 | |
Other editions - View all
Common terms and phrases
algebraic subspace Banach algebra Banach space BM(R BM(S Borel measures bounded linear operator Brownian motion BUC(R C(R+ Co(R Co(S continuous functions Convergence Theorem converges weakly convolution semigroup Corollary defined definition denote dense density distribution elements equals equation Example Let Exercise Let exists follows formula functional F given H₁ Hence Hilbert space Hint to Exercise implies independent inequality integrable interval isometrically isomorphic L¹(N L¹(R L¹(R+ L²(N Lebesgue measure Lemma Lévy processes limit limn linear functional locally compact Markov operator martingale matrix measurable functions measure space metric Moreover multiplicative functionals non-negative o-algebra parameter particular Poisson probability measure probability space Proof random variables satisfies semigroup stochastic strongly continuous semigroup subset suffices to show Suppose t₁ topological space topology vector zero