Book Review
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Book Information:
Author:
Ross G. Pinsky
Title:
Positive harmonic functions and diffusion: An integrated analytic and probabilistic approach
Additional book information:
Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge,
1995,
xvi + 474, vol. 45 pp.,
ISBN 0-521-47014-5,
$80.00$
1. M. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. Math. 121, 429- 461 (1985).
2. R. Bañuelos, Lifetime and heat kernel estimates in nonsmooth domains, Partial differential equations with minimal smoothness and applications, IMA Vol. Math. Appl. 42, 37-48, Springer, NY (1992).
3. Chung, K.L. and Zhao, Z., From Brownian motion to the Schrödinger equation, Springer-Verlag, 1995.
4. M. Cranston, On specifying invariant 
-fields, Sem. on Stoch. Proc. 29, Birkhauser (1992).
5. M. Cranston, A probabilistic approach to Martin boundaries for manifolds with end, PTRF, 96, 319-334 (1993).
6. R.D. deBlassie, The lifetime of conditional Brownian motion in certain Lipschitz domains, Probability Theory and Related Fields, 75, no.1, 55-65 (1987).
7. M. Donsker and S. Varadhan, On the principal eigenvalue of second order elliptic differential operators, Comm. Pure and App. Math, 29, 559-565.
8. J.L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85, 431-458 (1957).
9. Friedlin and Ventcel, Random Perturbations of Dynamical Systems, Springer-Verlag, New York (1984).
10. K. Itô, Differential equations determining Markov processes, Zenkoku Shij\={o} S\={u}gaku Danwakai, 244, No. 1077, 1352-1400 (1942).
11. R.S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc., 49, 137-172 (1941).
12. N.G. Meyers and J. Serrin, The exterior Dirichlet problem for second order elliptic partial differential equations, J. Math. Mech. 9, 513-538 (1960).
13. A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math. 142, 71-96 (1995).
14. J. Prat, Étude asymptotique et convergence angulaire du mouvement brownien sur une variété à courbure négative, C.R. Acad. Sc. Paris, 280, Serie A, 1539-1542 (1975).
15. B. Simon, Large time behavior of the 
norm of Schrödinger semigroups, J. Func. Anal. 35, 215-229 (1981).
16. D. Stroock, On the spectrum of Markov semigroups and the existence of invariant measures, Func. Anal. in Markov Processes, LNM, 923, Springer-Verlag.
17. D. Stroock and S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. 6th Berkeley Symp. Math. Stat. and Prob., 3, 333-360 (1970).
18. D. Stroock and S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag (1979).
19. N. Wiener, Differential space, J. Math. Physics, 2, 132-174 (1923).
20. Z. Zhao, Subcriticality, positivity and gaugeability of the Schrödinger operator, Bull. Amer. Math. Soc., 23, No. 2, 513-517 (1990).
- 1.
- M. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. Math. 121, 429- 461 (1985).
- 2.
- R. Bañuelos, Lifetime and heat kernel estimates in nonsmooth domains, Partial differential equations with minimal smoothness and applications, IMA Vol. Math. Appl. 42, 37-48, Springer, NY (1992).
- 3.
- Chung, K.L. and Zhao, Z., From Brownian motion to the Schrödinger equation, Springer-Verlag, 1995.
- 4.
- M. Cranston, On specifying invariant -fields, Sem. on Stoch. Proc. 29, Birkhauser (1992).
- 5.
- M. Cranston, A probabilistic approach to Martin boundaries for manifolds with end, PTRF, 96, 319-334 (1993).
- 6.
- R.D. deBlassie, The lifetime of conditional Brownian motion in certain Lipschitz domains, Probability Theory and Related Fields, 75, no.1, 55-65 (1987).
- 7.
- M. Donsker and S. Varadhan, On the principal eigenvalue of second order elliptic differential operators, Comm. Pure and App. Math, 29, 559-565.
- 8.
- J.L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85, 431-458 (1957).
- 9.
- Friedlin and Ventcel, Random Perturbations of Dynamical Systems, Springer-Verlag, New York (1984).
- 10.
- K. Itô, Differential equations determining Markov processes, Zenkoku Shij\={o} S\={u}gaku Danwakai, 244, No. 1077, 1352-1400 (1942).
- 11.
- R.S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc., 49, 137-172 (1941).
- 12.
- N.G. Meyers and J. Serrin, The exterior Dirichlet problem for second order elliptic partial differential equations, J. Math. Mech. 9, 513-538 (1960).
- 13.
- A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math. 142, 71-96 (1995).
- 14.
- J. Prat, Étude asymptotique et convergence angulaire du mouvement brownien sur une variété à courbure négative, C.R. Acad. Sc. Paris, 280, Serie A, 1539-1542 (1975).
- 15.
- B. Simon, Large time behavior of the norm of Schrödinger semigroups, J. Func. Anal. 35, 215-229 (1981).
- 16.
- D. Stroock, On the spectrum of Markov semigroups and the existence of invariant measures, Func. Anal. in Markov Processes, LNM, 923, Springer-Verlag.
- 17.
- D. Stroock and S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. 6th Berkeley Symp. Math. Stat. and Prob., 3, 333-360 (1970).
- 18.
- D. Stroock and S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag (1979).
- 19.
- N. Wiener, Differential space, J. Math. Physics, 2, 132-174 (1923).
- 20.
- Z. Zhao, Subcriticality, positivity and gaugeability of the Schrödinger operator, Bull. Amer. Math. Soc., 23, No. 2, 513-517 (1990).
Review Information:
Reviewer:
Michael Cranston
Affiliation:
University of Rochester
Email:
cran@db1.cc.rochester.edu
Journal:
Bull. Amer. Math. Soc.
34 (1997), 333-337
DOI:
https://doi.org/10.1090/S0273-0979-97-00722-2
Review copyright:
© Copyright 1997
American Mathematical Society
