Sufficient conditions for an operator-valued Feynman-Kac formula
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- by Michael Dale Grady
- Trans. Amer. Math. Soc. 223 (1976), 181-203
- DOI: https://doi.org/10.1090/S0002-9947-1976-0423552-5
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Abstract:
Let E be a locally compact, second countable Hausdorff space and let $X(t)$ be a Markov process with state space E. Sufficient conditions are given for the existence of a solution to the initial value problem, $\partial u/\partial t = Au + V(x) \cdot u,u(0) = f$, where A is the infinitesimal generator of the process X on a certain Banach space and for each $x \in E,V(x)$ is the infinitesimal generator of a ${C_0}$ contraction semigroup on another Banach space.References
- Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York, Inc., New York, 1967. MR 0230022, DOI 10.1007/978-3-642-46066-1
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- E. B. Dynkin, Markovskie protsessy, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0193670
- Jerome A. Goldstein, Abstract evolution equations, Trans. Amer. Math. Soc. 141 (1969), 159–185. MR 247524, DOI 10.1090/S0002-9947-1969-0247524-5 R. Griego, Dual random evolutions, Univ. of New Mexico Tech. Report No. 301, September, 1974.
- Richard Griego and Reuben Hersh, Theory of random evolutions with applications to partial differential equations, Trans. Amer. Math. Soc. 156 (1971), 405–418. MR 275507, DOI 10.1090/S0002-9947-1971-0275507-7
- Richard J. Griego and Alberto Moncayo, Random evolutions and piecing out of Markov processes, Bol. Soc. Mat. Mexicana (2) 15 (1970), 22–29. MR 365723
- L. L. Helms, Markov processes with creation of mass, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7 (1967), 225–234. MR 220346, DOI 10.1007/BF00532639
- R. Hersh and G. Papanicolaou, Non-commuting random evolutions, and an operator-valued Feynman-Kac formula, Comm. Pure Appl. Math. 25 (1972), 337–367. MR 310940, DOI 10.1002/cpa.3160250307
- M. Kac, On some connections between probability theory and differential and integral equations, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley-Los Angeles, Calif., 1951, pp. 189–215. MR 0045333 S. Khalili, Bochner measurability, Univ. of Pittsburgh Research Report No. 73-04, 1973.
- Talma Leviatan, On Markov processes with random starting time, Ann. Probability 1 (1973), 223–230. MR 359018, DOI 10.1214/aop/1176996975
- Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210528
- Frank S. Scalora, Abstract martingale convergence theorems, Pacific J. Math. 11 (1961), 347–374. MR 123356, DOI 10.2140/pjm.1961.11.347
- Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824
- Adriaan Cornelis Zaanen, Integration, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. Completely revised edition of An introduction to the theory of integration. MR 0222234
Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 223 (1976), 181-203
- MSC: Primary 60J35
- DOI: https://doi.org/10.1090/S0002-9947-1976-0423552-5
- MathSciNet review: 0423552