## 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.

### MSC:

 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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