Solutions of a stochastic differential equation forced onto a manifold by a large drift. (English) Zbl 0749.60053

It is considered a sequence of \(R^ d\)-valued semimartingales \(X_ n\) satisfying \[ X_ n(t)=X_ n(0)+\int_ 0^ t\sigma_ n(X_ n(s- ))dZ_ n(s)+\int_ 0^ t F(X_ n(s-))dA_ n(s), \] where \(Z_ n\) is a sequence of \(R^ k\)-valued semimartingales, \(\sigma_ n\) is a continuous \(d\times k\) matrix-valued function, \(F\) is a vector field whose deterministic flow has an asymptotically stable manifold of fixed points \(\Gamma\) and \(A_ n\) is a nondecreasing process. The author investigates stability of this system. Stability means, roughly speaking, if \(X_ n(0)\) is close to \(\Gamma\) (or is only in the domain of attraction of \(\Gamma\) under the flow of \(F\)) and \(X_ n\) is close to \(\Gamma\), then any limiting process is close to \(\Gamma\). There are given conditions under which a sequence \((X_ n)\) is relatively compact in the Skorokhod topology and which ensure the existence of a stochastic integral equation which any limiting process has to satisfy.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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