Stability of Solutions of Parabolic PDEs with Random Drift and Viscosity Limit

Stability of Solutions of Parabolic PDEs with Random Drift and Viscosity Limit Let u α be the solution of the Itô stochastic parabolic Cauchy problem $\partial u/\partial t-L_\a u = \xi \cdot \nabla u, \ u\v_{t=0} = f$ , where ξ is a space—time noise. We prove that u α depends continuously on α , when the coefficients in L α converge to those in L 0 . This result is used to study the diffusion limit for the Cauchy problem in the Stratonovich sense: when the coefficients of L α tend to 0 the corresponding solutions u α converge to the solution u 0 of the degenerate Cauchy problem $\partial u_0/ \partial t =\xi\circ \nabla u_0, \ u_0\v_{t=0} = f$ . These results are based on a criterion for the existence of strong limits in the space of Hida distributions (S) * . As a by-product it is proved that weak solutions of the above Cauchy problem are in fact strong solutions. Applied Mathematics and Optimization Springer Journals

Stability of Solutions of Parabolic PDEs with Random Drift and Viscosity Limit

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Copyright © Inc. by 1999 Springer-Verlag New York
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
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