Asymptotic properties, stability and “near” stationary of parabolic partial differential equations with wide bandwidth inputs

Abstract

Let u^e(x, t) denote the weak solution to a parabolic PDE whose inputs and coefficients are wide bandwidth stochastic processes, and the bandwidth goes to as e-->0. Under known conditions, u^e(•,•) converges weakly to a certain parabolic PDE driven by a (finite dimensional or cylindrical) Wiener process. A key problem in applications concerns the behavior of u^e(x, t) for large time and small e . This is, perhaps, the main stability problem for such stochastic PDEs. We show (under appropriate conditions) that the tail of u^e is (in the sense of distributions) close to a stationary solution of the limit (Wiener process driven) system, for small e. The methods are purely probabilistic, and depend on interpretations of the solutions u^e(•,•) as functionals of certain diffusions with wide bandwidth coefficients. No PDE methods are used.

Keywords:

• Stochastic partial differential equations
• weak convergence
• stochastic stability for distributed systems
• approximations for stochastic distributed systems

^†This research was supported in part by NSF under grant No. ESC-82-11476, by AFOSR under grant AF-AFOSR-81-0116-C and by ONR under grant No. N00014-83-K-0542

^‡This research was supported in part by ARO under grant No. DAAG-29-84-K-0082.

^†This research was supported in part by NSF under grant No. ESC-82-11476, by AFOSR under grant AF-AFOSR-81-0116-C and by ONR under grant No. N00014-83-K-0542

^‡This research was supported in part by ARO under grant No. DAAG-29-84-K-0082.

Notes

^†This research was supported in part by NSF under grant No. ESC-82-11476, by AFOSR under grant AF-AFOSR-81-0116-C and by ONR under grant No. N00014-83-K-0542

^‡This research was supported in part by ARO under grant No. DAAG-29-84-K-0082.