Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
[AhFuZa] N.U. Ahmed, M. Fuhrman and J. Zabczyk, On filtering equations in infinite dimensions, J. Func. Anal., 143 (1997), 180–204.
[Bi] P. Billingsley Probability and Measure, John Wiley and Sons, 1979
[CaDaPr] P. Cannarsa and G. Da Prato, Second-order Hamilton-Jacobi equations in infinite dimensions, SIAM J. Control and optimization, 29(1991), 474–492.
[CeGg] S. Cerrai and F. Gozzi, Strong solutions of Cauchy problems associated to weakly continuous semigroups, Differential and Integral Equations., 8 (1994), 465–486.
[ChojGo1] A. Chojnowska-Michalik and B. Goldys, On regularity properties of nonsymmetric Ornstein-Uhlenbeck semigroup in L p spaces, Stochastics Stochastics Reports 59 (1996), 183–209.
[ChojGo2] A. Chojnowska-Michalik and B. Goldys, Nonsymmetric Ornstein-Uhlenbeck semigroup as a second quantized operator, submitted.
[Chow] P.L. Chow, Infinite-dimensional Kolmogorov Equations in Gauss-Sobolev spaces, Stochastic Analysis and Applications, (to appear).
[ChowMe1] P.L. Chow and J.L. Menaldi, Variational inequalities for the control of stochastic partial differential equations, Proceedings of the Trento Conference 1988, LN in math., 1390 (1989).
[CrLi] M.G. Crandall and P.L. Lions. Hamilton-Jacobi equations in infinite dimensions, Part IV, Unbounded linear terms, J. Functional Analysis, 90(1900), 237–283.
[Dal] Yu. Daleckij, Differential equations with functional derivatives and stochastic equations for generalized random processes, Dokl. Akad. Nauk SSSR, 166(1966), 1035–38.
[DaPr1] G. Da Prato, Some results on elliptic and parabolic equations in Hilbert spaces, Rend. Mat. Acc. Lincei, 9(1996), 181–199.
[DaPr2] G. Da Prato, Parabolic equations in Hilbert spaces, Lecture Notes, May 1996.
[DaPr3] G. Da Prato, Stochastic control and Hamilton-Jacobi equations, Lecture Notes, February 1998.
[DaPrElZa] G. Da Prato, D. Elworthy and J. Zabczyk. Strong Feller property for stochastic semilinear equations, Stochastic Analysis and Applications, 13(1995), no.1, 35–45.
[DaPrGoZa] G.Da Prato, B. Goldys, and J. Zabczyk, Ornstein-Uhlenbeck semigroups in open sets of Hilbert spaces, Comptes Rendus Acad. Sci. Paris, Serie I, 325(1997), 433–438.
[DaPrZa1] G.Da Prato and J. Zabczyk, Smoothing properties of transition semigroups in Hilbert Spaces, Stochastic and Stochastics Reports, 35 (1991), 63–77.
[DaPrZa2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992.
[DaPrZa3] G.Da Prato and J. Zabczyk Regular densities of invariant measures in Hilbert spaces, J. Functional Analysis, 130(1995), 427–449.
[DaPrZa4] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996.
[Da] E.B. Davis, One-parameter Semigroups, Academic Press, 1980.
[Do] R. Douglas On majoration, factorization and range inclusion of operators on Hilbert spaces, Proc. Amer. Math. Soc., 17, 413–415.
[DS] N. Dunford and J.T. Schwartz Linear Operators. Part II: Spectral Theory, Interscience Publishers, New York, London 1963.
[Dy] E.B. Dynkin, Markov Processes, Vol I, Springer Verlag, 1965.
[GoMu] B. Goldys and M. Musiela, On partial differential differential equations related to term structure models, Working paper, University of New South Wales, 1996.
[Go] F. Gozzi, Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem, Commun. in Partial Differential Equations, 20(1995), 775–826.
[Gu] P. Guitto, Non-differentiability of heat semigroups in infinite dimensional Hilbert space, forthcoming in Semigroup Forum.
[Fr] A. Friedman, Partial Differential Equations of Parabolic Type, Englewood Cliffs, New York, Prentice Hall, 1964
[GiSk] I.I. Gikhman and A.V. Shorokhod, The Theory of Stochastic Processes, Vols. I, II and III. Springer-Verlag, 1974, 1975, 1979.
[Gr] L. Gross, Potential theory on Hilbert spaces, J. Funct. Anal., 1(1967), 123–181.
[HeJaMo] D. Heath, R. Jarrow and A. Morton, Bond princing and term structure of interest rates: a new methodology for contingent claim valuation, Econometrica, 60(1992), 77–105.
[It] K TtÔ, On a stochastic integral equation, Proc. Imp. Acad., Tokyo, 22(1946), 32–35.
[Ko] A.N. Kolmogorov, Uber die analitischen methoden in der Wahrscheinlichkeitsrechnung, Math. Ann. 104(1931), 415–458.
[Kr] N.V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, AMS, Providence, 1996.
[Ku] H.H. Kuo, Gaussian measures in Banach spaces, Lect. Notes in Math., 463(1975), Springer Verlag.
[Le] P. Levy, Problems Concrets d’Analyse Fonctionelle, Gauthier-Villar., Paris, 1951.
[MaRa] Z.M.Ma and M.Rockner, Introduction to the Theory of (Non Symmetric) Dirichlet Forms Springer Verlag, 1992.
[Mu] M. Musiela, Stochastic PDEs and the term structure models, Journees Internationales de Finance, IGR-AFFI. La Baule, June 1993.
[MuRu] M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling Springer, 1997.
[NeZa] J.M.A.M. Neerven and J. Zabczyk, Norm discontinuity of Ornstein-Uhlenbeck semigroups, forthcoming in Semigroup Forum.
[NuUs] D. Nualart and A.S. Ustunel, Une extension du laplacian sur l’espace de Wiener et la formule d’ItÔ associee, C.R. Acad. Sci. Paris, t. 309, Serie I, p. 383–386, 1984.
[Pa] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, 1967.
[Pe] S. Peszat, Large deviation principle for stochastic evolution equations, PTRF, 98 (1994), pp. 113–136.
[PeZa] S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces, Annals of Probability, 23(1995), 157–172.
[Ph] R.R. Phelps Gaussian null sets and differentiability of Lipschitz map on Banach spaces, Pac. J. Math. 77(1978), 523–531.
[Pie] A. Piech A fundamental solution of the parabolic equation on Hilbert space, J. Func. Anal., 3 (1969), 85–114.
[Pr] E. Priola, π-semigroups and applications, Preprint, 9, Scuola Normale Superiore, Pisa, 1998.
[Sw] A. Swiech, “Unbounded” second order partial differential equations in infinite dimensional Hilbert spaces, Commun.in Partial Differential Equations, 19(1994), 1999–2036.
[TeZa] G. Tessitore and J. Zabczyk Comments on transition semigroups and stochastic invariance, Preprint, Scuola Normale Superiore, Pisa, 1998.
[Yo] K. Yosida, Functional Analysis, Springer, 1965.
[Va] T. Vargiolu, Invariant measures for a Langevin equation describing forward rates in an arbitrage free market, forthcoming in Finance Stochastic.
[Za1] J. Zabczyk, Linear stochastic systems in Hilbert spaces: spectral properties and limit behavior, Report No. 236, Institute of Mathematics, Polish Academy of Sciences, 1981. Also in Banach Center Publications, 41(1985), 591–609.
[Za2] J.Zabczyk, Mathematical Control Theory. An Introduction, Birkhauser, 1992.
[Za3] J. Zabczyk, Bellman’s inclusions and excessive measures Preprint 8 Scuola Normale Superiore, March 1998.
[Za4] J. Zabczyk, Infinite dimensional diffusions in Modelling and Analysis, Jber. d Dt.Math.-Verein., (to appear).
Editor information
Rights and permissions
Copyright information
© 1999 Springer-Verlag
About this chapter
Cite this chapter
Zabczyk, J. (1999). Parabolic equations on Hilbert spaces. In: Da Prato, G. (eds) Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions. Lecture Notes in Mathematics, vol 1715. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092419
Download citation
DOI: https://doi.org/10.1007/BFb0092419
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66545-8
Online ISBN: 978-3-540-48161-4
eBook Packages: Springer Book Archive