Stochastic and Integral Geometry by R. V. Ambartzumian (auth.), R. V. Ambartzumian (eds.)

By R. V. Ambartzumian (auth.), R. V. Ambartzumian (eds.)

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6) dsX= SdsB, where B is BM(V). Calculations concerning X generally begin with an Ito analysis of the semimartingale f(X) for some suitable smooth real-valued function f. This is closely related to the martingale characterization (Stroock and Varadhan, 1979). It is actually more efficient first to analyze g(S) for g a smooth real-valued function on O(M), and then to specialise by setting g = f 0 71". 6). Choosing an orthonormal basis {Wi: i = 1, ... 9) the horizontal tangent vector field on the orthonormal frame bundle corresponding to the vector Wi.

The lower bound on curvature ensures that X does not explode. Com- parison arguments show that as t tends to 00 so lim inf r(Xt )! 2. Thus, X diverges to infinity. But the negative curvature means that the surface measure of the geodesic ball {x : r(x) = p} increases exponentially fast with p and the effect of this is that the diffusive component of e decreases rapidly. The drift of e can be controlle(j by techniques related to the existence of the lim sup bound and so e can be shown to 'freeze to a halt'.

The above theorem illustrates the basic strategy as propounded in Kendall (1981): one can deduce geometric implications in harmonic map theory by contrasting properties of BM(M) with properties of families of r -martingales in N. Equally, a property of Brownian motion or of r-martingales that leads to such implications is thereby interesting and worthy of further study. 2. LIMITING DIRECTIONS Theorem 9 can be mimicked for r -martingales of bounded dilatation. The next theorem follows by a lifting argument and by arguing as in Theorem 11 above.

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