Wandering out to infinity of diffusion processes

Friedman, Avner

doi:10.1090/S0002-9947-1973-0341631-5

Wandering out to infinity of diffusion processes
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by Avner Friedman

Trans. Amer. Math. Soc. 184 (1973), 185-203

DOI: https://doi.org/10.1090/S0002-9947-1973-0341631-5

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Abstract:

Let $\xi (t)$ be a diffusion process in ${R^n}$, given by $d\xi = b(\xi )dt + \sigma (\xi )dw$. Conditions are given under which either $|\xi (t)| \to \infty$ as $t \to \infty$ with probability 1, or $\xi (t)$ visits any neighborhood at a sequence of times increasing to infinity, with probability 1. The results are obtained both in case (i) $\sigma (x)$ is nondegenerate, and (ii) $\sigma (x)$ is degenerate at a finite number of points and hypersurfaces.

References

A. Dvoretzky and P. Erdös, Some problems on random walk in space, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley-Los Angeles, Calif., 1951, pp. 353–367. MR 0047272
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Avner Friedman, Limit behavior of solutions of stochastic differential equations, Trans. Amer. Math. Soc. 170 (1972), 359–384. MR 378118, DOI 10.1090/S0002-9947-1972-0378118-9
Avner Friedman and Mark A. Pinsky, Asymptotic behavior of solutions of linear stochastic differential systems, Trans. Amer. Math. Soc. 181 (1973), 1–22. MR 319268, DOI 10.1090/S0002-9947-1973-0319268-3
Avner Friedman and Mark A. Pinsky, Asymptotic stability and spiraling properties for solutions of stochastic equations, Trans. Amer. Math. Soc. 186 (1973), 331–358 (1974). MR 329031, DOI 10.1090/S0002-9947-1973-0329031-5
Kiyoshi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-New York, 1965. MR 0199891
Norman Meyers and James Serrin, The exterior Dirichlet problem for second order elliptic partial differential equations, J. Math. Mech. 9 (1960), 513–538. MR 0117421, DOI 10.1512/iumj.1960.9.59029