Summary
We study path properties of two-parameter Gaussian processes {X(t,v),t ∈R} of the formX(t,v)=∫ ∞0 ∞−∞ Γ(t,v,x,y)dW(x,y), where the kernel function Γ(t, v, x, y) is assumed to be square integrable in (x, y) onR×R +, andW(x, y) is a standard two-parameter Wiener process.
References
Antoniadis, A., Carmona, R.: Eigenfunction expansions for infinite dimensional Ornstein-Uhlenbeck processes. Probab. Th. Rel. Fields74, 31–54 (1987)
Carmona, R.: Measurable norms and some Banach space valued Gaussian processes. Duke Math. J.44, 109–127 (1977)
Csáki, E., Csörgő, M.: Inequalities for increments of stochastic processes and moduli of continuity, Tech. Rep. Ser. Lab. Res. Stat. Probab. No. 143, Carleton University-University of Ottawa, Ottawa, 1990
Csáki, E., Csörgő, M., Lin, Z.Y., Révész, P.: On infinite series of independent Ornstein-Uhlenbeck processes, Tech. Rep. Ser. Lab. Res. Stat. Probab. No. 144, Carleton University-University of Ottawa, Ottawa, 1990
Csörgő, M., Lin, Z.Y.: On moduli of continuity for Gaussian andx 2 processes generated by Ornstein-Uhlenbeck processes. C.R. Math. Rep. Acad. Sci., Can.10, 203–207 (1988)
Csörgő, M. Lin, Z.Y.: Path properties of infinite dimensional Ornstein-Uhlenbeck processes. In: Révész, P. (ed.) Coll. Math. Soc. J. Bolyai 57. Limit theorems in probability and statistics, pp. 107–135, Amsterdam: North-Holland, 1989
Csörgő, M., Lin, Z.Y.: On moduli of continuity for Gaussian andl 2-norm squared processes generated by Ornstein-Uhlenbeck processes. Can. J. Math.42, 141–158 (1990)
Csörgő, M., Révész, P.: Strong approximations in probability and statistics. Budapest: Akadémiai Kiadó, New York: Academic Press, 1981
DaPrato, G., Kwapien, S., Zabczyk, J.: Regularity of solutions of linear stochastic equations in Hilbert space. Stochastics23, 1–23 (1987)
Dawson, D.A.: Stochastic evolution equations. Math. Biosci.15, 287–316 (1972)
Dawson, D.A.: Stochastic evolution equations and related measure processes. J. Multivariate Anal.5, 1–52 (1975)
Fernique, X.: Continuite des processus Gaussiens. C.R. Acad. Sci., Paris258, 6058–6060 (1964)
Fernique, X.: La regularité des fonctions aléatoires d'Ornstein-Uhlenbeck á valeurs dansl 2; le cas diagonal. C.R. Acad. Sci., Paris309, 59–62 (1989)
Holley, R., Stroock, D.: Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions. Publ. RIMS Kyoto Univ.14, 741–788 (1978)
Iscoe, I., McDonald, D.: Large deviations forl 2-valued Ornstein-Uhlenbeck processes. Ann. Probab.17, 58–73 (1989)
Iscoe, I., Marcus, M., McDonald, D., Talagrand, M., Zinn, J.: Continuity ofl 2-valued Ornstein-Uhlenbeck processes. Ann. Probab.18, 68–84 (1990)
Jain, N.C., Marcus, M.B.: Continuity of subgaussian processes. In: Kuelbs, J. (ed.) Probability on Banach spaces. Adv. Probab. Relat. Top.4, 81–196 (1978).
Kallianpur, G., Wolpert, R.: Infinite dimensional stochastic differential equation models for spatially distributed neurons. J. Appl. Math. Optimization12, 125–172 (1984)
Kotelenez, P.: A maximal inequality for stochastic integrals on Hilbert spaces and space-time regularity of linear stochastic partial differential equations. Stochastics21, 345–358 (1987)
Miyahara, Y.: Infinite dimensional Langevin equation and Fokker-Planck equation. Nagoya Math. J.81, 177–223 (1981)
Röckner, M.: Traces of harmonic functions and a new path space for the free quantum field. J. Funct. Anal.79, 211–249 (1988)
Schmuland, B.: Moduli of continuity for some Hilbert space valued Ornstein-Uhlenbeck processes. C.R. Math. Rep. Acad. Sci., Can.10, 197–202 (1998a)
Schmuland, B.: Some regularity results on infinite dimensional diffusions via Dirichlet forms. Stochastic Anal. Appl.6, 327–348 (1988b)
Schmuland, B.: An energy approach to reversible infinite dimensional Ornstein-Uhlenbeck processes. (Manuscript, 1989)
Slepian, D.: The one sided barrier problem for Gaussian noise. Bell Syst. Tech. J.41, 463–501 (1962)
Walsh, J.B.: A stochastic model of neural response. Adv. Appl. Probab.13, 231–281 (1981)
Walsh, J.B.: Regularity properties of a stochastic partial differential equation. In: Cinlar, E., Chung, K.L., Getoor, R.K. (eds.) Proc. Seminar on Stochastic Processes. Boston: Birkhäuser 1984
Author information
Authors and Affiliations
Additional information
Work partially supported by an NSERC Canada Operating Grant at Carleton University
Work supported by an NSERC Canada International Scientific Exchange Award at Carleton University and by National Natural Science Foundation of China
Rights and permissions
About this article
Cite this article
Csörgő, M., Lin, Z. Path properties of kernel generated two-time parameter Gaussian processes. Probab. Th. Rel. Fields 89, 423–445 (1991). https://doi.org/10.1007/BF01199787
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01199787
Keywords
- Stochastic Process
- Probability Theory
- Kernel Function
- Mathematical Biology
- Gaussian Process