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Two Contrasting Properties of Solutions for One-Dimensional Stochastic Partial Differential Equations

Published online by Cambridge University Press:  20 November 2018

Tokuzo Shiga*
Affiliation:
Department of Applied Physics Tokyo Institute of Technology Oh-okayama, Meguro Tokyo 152 Japan
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Abstract

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The paper is concerned with the comparison of two solutions for a one-dimensional stochastic partial differential equation. Noting that support compactness of solutions propagates with passage of time, we define the SCP property and show that the SCP property and the strong positivity are two contrasting properties of solutions for one-dimensional SPDEs, which are due to degeneracy of the noise-term coefficient

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Dawson, D. A., Stochastic evolution equations and related measure processes, J. Multi. Anal. 5(1975), 152.Google Scholar
2. Dawson, D. A. and Hochberg, K. J., Wandering random measures in the Fleming-Viot model, Ann. Probab. 10(1982), 554580.Google Scholar
3. Fans, W. G. and G. Jona-Lasinio, Large fluctuations for a non-linear heat equation with white noise, J. Phys. A, Math. Gen. 15(1982), 30253055.Google Scholar
4. Funaki, T., Derivation of the hydrodynamical equation for one-dimensional Ginzburg-Landau model, Probab. Theor. Relat. Fields 82(1989), 3993.Google Scholar
5. Ikeda, N. and Watanbe, S., Stochastic differential equations and diffusion processes, 2nd edition, North- Holland, Kodansha, 1989.Google Scholar
6. Iscoe, I., On the supports of measure-valued critical branching Brownian motion, Ann. Probab. 16(1988), 200221.Google Scholar
7. Iwata, K., An infinite dimensional stochastic differential equation with state space C(R), Probab. Theor. Relat. Fields 74(1987), 141159.Google Scholar
8. Konno, N. and Shiga, T., Stochastic partial differential equations for some measure-value diffusions, Probab. Theor. Relat. Fields 79(1988), 201255.Google Scholar
9. Mueller, C., On the support of solutions to the heat equation with noise, Stochastics and Stoch. Rep. 37(1991), 225245.Google Scholar
10. Reimers, M., One dimensional stochastic partial differential equations and the branching measure diffusion, Probab. Theor. Relat. Fields 81(1989), 319340.Google Scholar
11. Shiga, T., Multi-allelic Gillespie-Sato diffusion models and their extension to infinite allelic ones, Lecture Notes in Biomath. 70, Stochastic Method in Biology, 1985, 8799.Google Scholar
12 Shiga, T. Stepping stone models in population genetics and population dynamics, Stochastic Process in Physics and Engineering, (eds. Albeverio, S. et al.), 1988, 345355.Google Scholar
13. Totoki, H., A method of construction of measures on functional spaces and its applications to stochastic processes, Mem. Fac. Kyushu Univ. (A) 15(1961), 178190.Google Scholar
14. Walsh, J. B., An introduction to stochastic partial differential equations, Lecture Notes in Math. 1180(1986), 265439.Google Scholar