Chaos expansion of heat equations with white noise potentials

Potential Analysis

Volumn 16, Issue 1, 2002, Pages 45-66

  Hu, Yaozhong ^a  

^a UNIVERSITY OF KANSAS   (United States)

Author keywords

Chaos expansion; Exact rate of convergence; Finite chaos approximation; Lyapunov exponent; Mittag Leffler functions; Space white noise potential; Space time white noise potential; Stochastic heat equation

Indexed keywords

EID: 26844440806     PISSN: 09262601     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1024878703232     Document Type: Article

References (28)

1. Abramowitz, M. and Stegun, I. A.: Handbook of Mathematical Functions, Dover Publications, INC, New York, 1992.
2. Benth, F. E.: 'On the positivity of the stochastic heat equation', Potential Anal. 6 (1997), 127-148.
3. Carmona, R.: Random Schrödinger operators, In: École d'été de probabilités de Saint-Flour, XIV(1984), Lecture Notes in Math. 1180, Springer, Berlin, 1986, pp. 1-124.
4. Carmona, R. and Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Probability and its Applications, Birkhäuser, Boston, 1990.
5. Carmona, R. A. and Molchanov, S. A.: 'Stationary parabolic Anderson model and intermittency', Probab. Theory Related Fields 102(4) (1995), 433-453.
6. Carmona, R. A. and Viens, F. G.: 'Almost-sure exponential behavior of a stochastic Anderson model with continuous space parameter', Stochastics Stochastics Rep. 62(3-4) (1998), 251-273.
7. Carmona, R., Viens, F. G. and Molchanov, S. A.: 'Sharp upper bound on the almost-sure exponential behavior of a stochastic parabolic partial differential equation', Random Oper. Stochastic Equations 4(1) (1996), 43-49.
8. Davies, E. B.: Spectral Theory and Differential Operators, Cambridge Stud. Adv. Math. 42, Cambridge Univ. Press, Cambridge, 1995.
9. Deck, Th. and Potthoff, J.: 'On a class of stochastic partial differential equations related to turbulent transport', Probab. Theory Related Fields 111 (1998), 101-122.
10. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.: High Transcendental Functions, Vol. 3, McGraw-Hill, 1955.
11. Glimm, J. and Jaffe, A.: Quantum Physics. A Functional Integral Point of View, 2nd edn, Springer-Verlag, New York, 1987.
12. Holden, H., Øksendal, B., Ubøe, J. and Zhang, T. S.: Stochastic Partial Differential Equations, Birkhäuser, 1996.
13. Hu, Y. Z. and Kallianpur, G.: Exponential integrability and application to stochastic quantization, Appl. Math. Optim. 37(3) (1998), 295-353.
14. Hu, Y. Z.: 'Itô-Wiener chaos expansion with exact residual and correlation, variance inequalities', J. Theoret. Probab. 10(4) (1997), 835-848.
15. Hu, Y Z.: 'On the self-intersection local time of Brownian motion-via chaos expansion', Publ. Mat 40(2) (1996), 337-350.
16. Hu, Y. Z.: Self-intersection local time of fractional Brownian motion-via chaos expansion, Preprint, 1998.
17. Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L.: White Noise. An Infinite-Dimensional Calculus, Math. Appl. 253, Kluwer Academic Publishers, Dordrecht, 1993.
18. d and renormalization', Stochastic Process. Appl. 56(1) (1995), 1-34.
19. Imkeller, P. and Yan, J. A.: 'Multiple intersection local time of planar Brownian motion as a particular Hida distribution', /. Funct. Anal. 140(1) (1996), 256-273.
20. Mueller, C.: 'Long-time existence for the heat equation with a noise term", Probab. Theory Related Fields 9 (1991), 505-517.
21. Nualart, D. and Rozovskii, B.: 'Weighted stochastic Sobolev spaces and bilinear SPDEs driven by space-time white noise', J. Funct. Anal. 149 (1997), 200-225.
22. Nualart, D. and Zakai, M.: 'Generalized Brownian functionals and the solution to a stochastic partial differential equation', J. Funct. Anal. 84(2) (1989), 279-296.
23. Potthoff, J., Våge, G. and Watanabe, H.: 'Generalized solutions of linear parabolic stochastic partial differential equations', Appl. Math. Optim. 38 (1998), 95-107.
24. Piatnitski, A. L., Zhao, H. Z. and Zheng, W. A.: Homogenization approximation to linear SPDEs in multi-dimensions, Preprint, 1998.
25. Rivasseau, V.: From Perturbative to Constructive Renormalization, Princeton Ser. Phys., Princeton Univ. Press, Princeton, NJ, 1991.
26. Simon, B.: The P(φ)2 Euclidean (Quantum) Field Theory, Princeton Ser. Phys., Princeton Univ. Press, Princeton, NJ, 1974.
27. Stroock, D. W.: 'Diffusion semigroups corresponding to uniformly elliptic divergence form operator', In: Lecture Notes in Math. 1321, Springer-Verlag, 1988, pp. 316-347.
28. Uemura, H.: 'Construction of the solution of 1-dimensional heat equation with white noise potential and its asymptotic behavior', Stochastic Anal. Appl. 14 (1996), 487-506.