TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 11, November 2006, Pages 5059–5082 S 0002-9947(06)03908-0 Article electronically published on June 13, 2006 RATIO LIMIT THEOREM FOR PARABOLIC HORN-SHAPED DOMAINS PIERRE COLLET, SERVET MARTINEZ, AND JAIME SAN MARTIN Abstract. We prove that for horn-shaped domains of parabolic type, the ratio of the heat kernel at different fixed points has a limit when the time tends to infinity. We also give an explicit formula for the limit in terms of the harmonic functions. 1. Introduction We consider the heat kernel in parabolic horn-shaped domains with Dirichlet boundary conditions. More precisely, we consider D the domain in R 3 obtained by revolving the parabolic region P = {(ρ, z):1+ z 2 <ρ} about the z−axis. Thus, more explicitly we have D =  (x 1 ,x 2 ,x 3 ): |x 3 | <a   (x 1 ) 2 +(x 2 ) 2  , where a(t)= √ t − 1. Although we will give a detailed proof for the case of the parabola, it is easy to verify that all our results can be extended to domains where the function a satisfies the conditions of [8] together with ∞ > lim s→∞ a(s)/s γ > 0, 0 <γ< 1. It is also easy to extend our results to higher dimensions. Throughout the paper p t will denote the heat kernel with Dirichlet boundary conditions in D. The main goal of the present paper is to derive the (normalized) limit of p t when t tends to infinity. The case of horn-shaped domains is very different from our previous works [4, 5] since the cone C of nonnegative harmonic functions vanishing at the boundary is a continuum. More precisely, the Martin boundary at infinity is a circle (see [11], [8] and [13]). The techniques are also rather different. The main point is that the associated stochastic process is at time t at a distance about t 2/3 away from the origin, very different from the usual t 1/2 typical of diffusion and occurring for example in cones. As we will see below (see Theorem 1.4 and formula (5)), an important consequence is that the integral appearing in a Feynman-Kac formula is convergent, ensuring the anisotropy of the process even in the infinite time limit. We will frequently use cylindrical coordinates defined by the axis of symmetry. The altitude will be denoted by z, the polar radius (for the projection in the plane z = 0) will be denoted by ρ and the angle (angles if we are in higher dimension) by ϑ. Received by the editors September 2, 2004 and, in revised form, November 19, 2004. 2000 Mathematics Subject Classification. Primary 60J65, 60J45, 35K05. Key words and phrases. Bessel process, Harnack inequality, heat kernel. c 2006 American Mathematical Society Reverts to public domain 28 years from publication 5059 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use