Abstract
This paper presents a theory of stochastic evolution equations for nuclear-space-valued processes and provides a unified treatment of several examples from the field of applications. (C 0 , 1) reversed evolution systems on countably Hilbertian nuclear spaces are also investigated.
Similar content being viewed by others
References
Babalola V (1974) Semigroups of operators on locally convex spaces. Trans Amer Math Soc 199:163–179
Choe YH (1985) Co-semigroups on a locally convex space. J Math Anal Appl 106:293–320
Christensen SK, Kallianpur G (1986) Stochastic differential equations for neuronal behavior. In: Van Ryzin J (ed) Adaptive Statistical Procedures and Related Topics. Institute of Mathematical Statistics, Lecture Notes—Monograph Series, pp 400–416
Curtain RF, Pritchard AJ (1978) Infinite Dimensional Linear Systems Theory. Lecture Notes in Control and Information Sciences. Springer-Verlag, New York
Hitsuda M, Mitoma I (1986) Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions. J Multivariate Anal 19:311–328
Kallianpur G (1980) Stochastic Filtering Theory. Springer-Verlag, New York
Kallianpur G (1986) Stochastic differential equations in duals of nuclear spaces with some applications. IMA Preprint Series No. 244. Institute for Mathematics and Its Applications, University of Minnesota
Kallianpur G, Wolpert R (1984) Infinite dimensional stochastic differential equation models for spatially distributed neurons. J Appl Math Optim 12:125–172
Kotelenez P (1986) Gaussian approximation to the nonlinear reaction-diffusion equation. Report No. 146, University of Bremen
Kotelenez P, Curtain RF (1982) Local behavior of Hilbert space-valued stochastic integrals and the continuity of mild solutions of stochastic evolution equations. Stochastics 6:239–257
Kunita H (1982) Stochastic differential equations and stochastic flows of diffeomorphisms. On cours au Ecole d'ete de Probabilites de Saint-Flour XII. Lecture Notes in Mathematics, vol 1097. Springer-Verlag, New York
McKean HP (1967) Propagation of Chaos for a Class of Nonlinear Parabolic Equations. Lecture Series in Differential Equations, vol 2. Van Nostrand, New York
Mitoma I (1981) Martingales of random distributions. Mem Fac Sci Kyushu Univ ser A 35:120–185
Mitoma I (1981) On the norm continuity ofL'-valued Gaussian processes. Nagoya Math J 82:209–220
Mitoma I (1985) An ∞-dimensional inhomogeneous Langevin equation. J Funct Anal 61:342–359
Miyadera I (1959) Semigroups of operators in Fréchet space and applications to partial differential equations. Tohoku Math J 11:162–183
Pazy A (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York
Tanabe H (1979) Equations of Evolution. Pitman, London
Tanaka H, Hitsuda M (1981) Central limit theorem for a simple diffusion model of interacting particles. Hiroshima Math J 11:415–423
Yosida K (1965) Time dependent evolution equations in a locally convex space. Math Ann 162:83–96
Yosida K (1980) Functional Analysis, 6th edn. Springer-Verlag, New York
Author information
Authors and Affiliations
Additional information
Research supported by the Air Force Office of Scientific Research Grant No. F49620 85 C 0144. The research of V. Perez-Abreu was also supported by CONACYT Grants PCEXCNA-040651 and PCMTCNA-750220 (Mexico).
Rights and permissions
About this article
Cite this article
Kallianpur, G., Perez-Abreu, V. Stochastic evolution equations driven by nuclear-space-valued martingales. Appl Math Optim 17, 237–272 (1988). https://doi.org/10.1007/BF01448369
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01448369