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Escape rate of the Brownian motions on hyperbolic spaces.

1. Introduction. Let [H.sup.d] be the d-dimensional hyperbolic space and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the Brownian motion on [H.sup.d] generated by the half of the Laplace-Beltrami operator. For a fixed point o [member of] [H.sup.d], define P = [P.sub.o] and [R.sub.t] = d (o, [X.sub.t]), where d is the distance function of [H.sup.d]. In this note, we show

Theorem 1.1. Let g(t) be a positive function on (0, [infinity]) such that for some [t.sub.0] > 0, [square root of t]g(t) is nondecreasing and g(t)/[square root of t] is bounded for all t [greater than or equal to] [t.sub.0].

(i) For the function [r.sub.1](t):= (d-1)t/2 + [square root of t]g(t),

(1.1) P (there exists T > 0 such that [R.sub.t] < [r.sub.1] (t) for all t [greater than or equal to] T) = 1 or 0

according as

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

(ii) For the function [r.sub.2](t) := (d - 1)t/2 -[square root of t]g(t),

(1.3) P (there exists T > 0 such that [R.sub.t] > [r.sub.2] (t) for all t [greater than or equal to] T) = 1 or 0

according as (1.2) holds.

The function [r.sub.1](t) is called an upper rate function for M if the probability in (1.1) is 1. By the same way, the function [r.sub.2](t) is called a lower rate function for M if the probability in (1.3) is 1. According to Theorem 1.1, we have for c > 0,

* the function r(t) := (d - 1)t/2 + [square root of (ct log log t)] is an upper rate function for M if and only if c > 2;

* the function r(t) := (d - 1)t/2 + [square root of (ct log log t)] is a lower rate function for M if and only if c > 2.

For the Brownian motions on Riemannian manifolds, more generally symmetric diffusion processes generated by regular Dirichlet forms, upper and lower rate functions are given in terms of volume growth rate ([1-4,6,11]). As for the upper rate functions, the results in [2-4,6,11] are applicable to the Brownian motions on Riemannian manifolds with exponential volume growth rate, as to M; however, as for the lower rate functions, the results in [1-3] are not applicable to M because the doubling condition is imposed on the volume growth. Grigor'yan and Hsu [4] also discussed the sharpness of the upper rate functions for M or for the Brownian motion on a model manifold, that is, a spherically symmetric Riemannian manifold with a pole. Using the fact that

(1.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

(which follows from (2.2) below), they remarked that the function r(t) = ct is an upper rate function for M if c > (d - 1)/2, and not if 0 < c < (d - 1)/2. This observation is still valid for the lower rate functions. See also [7] for the result of the law of the iterated logarithms-type to the Brownian motions on model manifolds.

For the proof of Theorem 1.1, we make use of the Brownian expression of the radial part [R.sub.t] ((2.2) below) as in [4,7], together with a generalized version of Kolmogorov's test for the one dimensional Brownian motion ([9,10]). In fact, the integral in (1.2) is the same with that in this test. The assumption on g(t)/[square root of t] will be needed in (2.7) and (2.8) below.

2. Proof of Theorem 1.1. Let B = ([{[B.sub.t]}.sub.t[greater than or equal to]0], P) be the one dimensional Brownian motion starting from the origin. Then a generalized Kolmogorov's test holds:

Theorem 2.1 ([9, Theorem 3.1 and Lemma 3.3] and [10, Theorem 2.1]). Under the full conditions of Theorem 1.1,

(2:1) P (there exists T > 0 such that [absolute value of [B.sub.t]] < [square root of t]g(t) for all t [greater than or equal to] T) = 1 or 0

according as (1.2) holds. This assertion is valid even if [absolute value of [B.sub.t]] in the equality above is replaced by [B.sub.t] or - [B.sub.t].

By comparison with Kolmogorov's test (see, e.g., [8, 4.12]), we do not need to assume that g(t)[??] [infinity] as t [right arrow] [infinity] in Theorem 2.1.

Proof of Theorem 1.1. Recall that M = ([{[X.sub.t]}.sub.t[greater than or equal to]0], P) is the Brownian motion on [H.sup.d] starting from a fixed point o [member of] [H.sup.d] and [R.sub.t] = d(o, [X.sub.t]) is the radial part of [X.sub.t]. Then by [5, Example 3.3.3],

(2.2) [R.sub.t] = [B.sub.t] + [[d-1]/2] [[integral].sup.t.sub.0] coth [R.sub.s] ds.

Assume the full conditions of Theorem 1.1. We first discuss the lower bound of [R.sub.t]. Since coth x [greater than or equal to] 1 for any x > 0, we obtain by (2.2),

(2.3) [R.sub.t] [greater than or equal to] [B.sub.t] + [[d-1]/2]t for any t [greater than or equal to] 0.

Hence if the integral in (1.2) is convergent, then the probability in (1.3) is 1 by Kolmogorov's test. By the same way, if the integral in (1.2) is divergent, then the probability in (1.1) is 0.

We next discuss the upper bound of [R.sub.t]. Since [B.sub.t] = o(t) as t [right arrow] [infinity], we see by (2.3) that there exists c > 0 such that P(A) = 1 for

(2.4) A := {there exists [T.sub.1] > 0 such that [R.sub.t] [greater than or equal to] ct for all t [greater than or equal to] [T.sub.1]}.

Under the event A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

for any s [greater than or equal to] [T.sub.1], which implies that for all t [greater than or equal to] [T.sub.1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

Since there exists an integer valued random variable N such that

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

we obtain for such N,

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

for all t [greater than or equal to] [T.sub.1].

Assume first that the integral in (1.2) is convergent. Then there exists a positive constant [c.sub.n] for each n [greater than or equal to] 1 such that the function [h.sup.(n).sub.1](t) :=g(t) - n/[square root of t] satisfies

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

Hence Theorem 2.1 implies that for each n [greater than or equal to] 1,

P(there exists T > 0 such that [absolute value of [B.sub.t]] < [r.sup.(n).sub.1](t) for all t [greater than or equal to] T) = 1

for [r.sup.(n).sub.1](t): [square root of t][h.sup.(n).sub.1](t)(= [square root of t]g(t) - n). In particular, we get P([B.sub.1]) = 1 for

[B.sub.1] := {for each n [greater than or equal to] 1, there exists [S.sub.n] > 0 such that [absolute value of [B.sub.t]] < [r.sup.(n).sub.1](t) for all t [greater than or equal to] [S.sub.n]}.

Under the event A [intersection] [B.sub.1], since there exists [T.sub.2] > 0 for N [greater than or equal to] 1 in (2.5) such that

[B.sub.t] < [r.sup.(N).sub.1](t) = [square root of t]g(t) - N for all t [greater than or equal to] [T.sub.2],

we have by (2.6),

[R.sub.t] < [[d - 1]/2] t + [square root of t]g(t) for all t [greater than or equal to] [T.sub.1] [disjunction] [T.sub.2],

Therefore, the probability in (1.1) is 1.

Assume next that the integral in (1.2) is divergent. Then by the same way as in (2.7), the function [h.sup.(n).sub.2](t) := g(t) + n/[square root of t] satisfies for each n [greater than or equal to] 1,

(2.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

Hence Theorem 2.1 yields that for each n [greater than or equal to] 1,

P(for any t > 0, there exists T [greater than or equal to] t such that [B.sub.T] [less than or equal to] -[r.sup.(n).sub.2](T)) = 1 for [r.sup.(n).sub.2](t) := [square root of t][h.sup.(n).sub.2](t)(=[square root of t]g(t) + n). In particular, P([B.sub.2]) = 1 for

[B.sub.2] := {for each n [greater than or equal to] 1, there exists [U.sub.n] [greater than or equal to] t for any t > 0 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

Under the event A [intersection] [B.sub.2], since there exists [T.sub.3] [greater than or equal to] t [disjunction] [T.sub.1] for any t > 0 and N [greater than or equal to] 1 in (2.5) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

we have for such [T.sub.3],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCI]

by (2.6). Therefore, the probability in (1.3) is 0.

doi: 10.3792/pjaa.93.27

2010 Mathematics Subject Classification. Primary 60G17; Secondary 58J65, 60F20.

By Yuichi SHIOZAWA

Department of Mathematics, Graduate School of Science, Osaka University,

Toyonaka, Osaka 560-0043, Japan

(Communicated by Masaki KASHIWARA, M.J.A., March 13, 2017)

Acknowledgements. The author would like to thank Prof. Masayoshi Takeda for his valuable discussion motivating this work and his comment on the draft of this paper. This work was supported in part by the Grant-in-Aid for Scientific Research (C) 26400135.

References

[1] A. Bendikov and L. Saloff-Coste, On the regularity of sample paths of sub-elliptic diffusions on manifolds, Osaka J. Math. 42 (2005), no. 3, 677-722.

[2] A. Grigor'yan, Escape rate of Brownian motion on Riemannian manifolds, Appl. Anal. 71 (1999), no. 1-4, 63-89.

[3] A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135-249.

[4] A. Grigor'yan and E. Hsu, Volume growth and escape rate of Brownian motion on a Cartan-Hadamard manifold, in Sobolev spaces in mathematics. II, Int. Math. Ser. (N. Y.), 9, Springer, New York, 2009, pp. 209-225.

[5] E. P. Hsu, Stochastic analysis on manifolds, Graduate Studies in Mathematics, 38, Amer. Math. Soc., Providence, RI, 2002.

[6] E. P. Hsu and G. Qin, Volume growth and escape rate of Brownian motion on a complete Riemannian manifold, Ann. Probab. 38 (2010), no. 4, 1570-1582.

[7] K. Ichihara, Comparison theorems for Brownian motions on Riemannian manifolds and their applications, J. Multivariate Anal. 24 (1988), no. 2, 177-188.

[8] K. Ito and H. P. McKean, Jr., Diffusion processes and their sample paths, Springer, Berlin, 1974.

[9] S. Keprta, Integral tests for Brownian motion and some related processes, 1997. (Ph.D. Thesis, Carleton University, Canada). http://www.nlc-bnc. ca/obj/s4/f2/dsk2/ftp03/NQ26856.pdf

[10] S. Keprta, Integral tests for some processes related to Brownian motion, in Asymptotic methods in probability and statistics (Ottawa, ON, 1997), 253-279, North-Holland, Amsterdam, 1998.

[11] S. Ouyang, Volume growth and escape rate of symmetric diffusion processes, Stochastics 88 (2016), no. 3, 353-372.
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Author:Shiozawa, Yuichi
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
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Date:Apr 1, 2017
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