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On the rate at which a homogeneous diffusion approaches a limit, an application of large deviation theory to certain stochastic integrals. (English) Zbl 0604.60076
This paper deals with the asymptotics of P(\(| X(T,x)| \in Borel\) in \(R^ 1)\) as \(T\to \infty\) for a diffusion X(T,x) that solves the following equation \[ X(T,x)=x+\int^{T}_{0}V_ 0(X(t,x))dt+\sum^{d}_{k=1}\int^{T}_{0}V_ k(X(t,x))\circ d\beta_ k(t) \] where \(V_ k(x)=| x| V_ k(x/| x|)\) (i.e. \(V_ k\) is homogeneous of degree 1) and \(\beta_ k\) are independent Brownian motions in \(R^ 1\). The main ingredient of the analysis is the state of the art machinery, known as the large deviation principle. Due to homogeneity of \(V_ k's\) the problem reduces to the study of the diffusion \(\theta (T,x)=X(T,x)/| X(T,x)|\) on the unit sphere in \(R^ N.\)
By identifying the rate function corresponding to \(\theta\) (T,x) the author obtains extremely detailed information about the exponential decay of P(\(| X(T,x)| /| x| \geq R)\) as \(T\to \infty\), which happens to be uniform in \(x\in R^ N\). The analysis relies, in part, on certain ”nice things” available on the sphere (e.g. ergodic measure, discrete spectrum of the corresponding generator, etc.). In addition, very enlightening comments are made in numerous remarks some of which deserve to be separate theorems.
Reviewer: A.Korzeniowski

MSC:
60J60 Diffusion processes
60F10 Large deviations
60H05 Stochastic integrals
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