Abstract
We have examined the influence of parametric noise on the solution behavioru(t, x) of a nonlinear initial value(ϕ) problem arising in cell kinetics. In terms of ensemble statistics, the eventual limiting solution mean\(\mathop \xi \limits^ - _u\) and variance\(\mathop {\sigma _u^2 }\limits^ -\) are well-characterized functions of the noise statistics, and\(\mathop \xi \limits^ - _u\) and\(\mathop {\sigma _u^2 }\limits^ -\) depend on ϕ. When noise is continuously present along the trajectory,\(\mathop \xi \limits^ - _u\) and\(\mathop {\sigma _u^2 }\limits^ -\) are independent of the noise statistics and ϕ. However, in their evolution toward\(\mathop \xi \limits^ - _u\) and\(\mathop {\sigma _u^2 }\limits^ -\), bothξ u (t, x) andσ 2 u (t, x) depend on the noise andϕ.
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Mackey, M.C., Schwegler, H. Ensemble and trajectory statistics in a nonlinear partial differential equation. J Stat Phys 70, 281–295 (1993). https://doi.org/10.1007/BF01053968
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DOI: https://doi.org/10.1007/BF01053968