Large deviations for the radial processes of the Brownian motions on hyperbolic spaces

On the Asymptotic Behavior of the Hyperbolic Brownian Motion

“…Firstly we should have Λ * (x) = I 1 (x) for all x ≥ 0 if we consider t 2 in place of t because the exponential part of function h (k,m) F in eq. (4) in Hirao [20] with (k, m) = (0, n−1) would coincide with the one of h n (η, t 2 ) in this paper. Furthermore the proof of Theorem 1.1 in Hirao [20] shows the existence of the limit for all x ∈ R. We remark that, since the random variables d F Z F z (t), z : t > 0 should be nonnegative, there is a slight inexactness in this proof.…”

“…(4) in Hirao [20] with (k, m) = (0, n−1) would coincide with the one of h n (η, t 2 ) in this paper. Furthermore the proof of Theorem 1.1 in Hirao [20] shows the existence of the limit for all x ∈ R. We remark that, since the random variables d F Z F z (t), z : t > 0 should be nonnegative, there is a slight inexactness in this proof. Actually, if a family of nonnegative random variables {Z(t) : t > 0} satisfies the LDP with a rate function I, the lower bound for the open set G = (−∞, 0) in the definition of LDP would yield I(x) = ∞ for x ∈ (−∞, 0); moreover, for the same reason, the limit (5.1) fails because the function Λ should be nondecreasing.…”

“…In Section 4 we investigate the asymptotic behavior of the hitting probabilities of an hyperbolic ball centered at the origin as the distance of the starting point of the process goes to infinity. Finally, in Section 5, we discuss some connections with the results in Hirao [20].…”

“…instead of (4.2). Finally the cases n ∈ {3, 5, 7}: 5 On a recent LDP in the literature Theorem 1.1 in Hirao [20] provides the LDP for…”

“…instead of (4.2). Finally the cases n ∈ {3, 5, 7}: 5 On a recent LDP in the literature Theorem 1.1 in Hirao [20] provides the LDP for d F( Z F z (t),z) t : t > 0 , where Z F z (t) : t ≥ 0 is a Brownian motion on a hyperbolic space H n (F) with distance function d F (several choices of F are allowed), and Z F z (0) = z. Here we want to illustrate the relationship between Theorem 1.1 in Hirao [20] when F concerns the parameters (k, m) = (0, n − 1) in eq.…”