D ec 2 00 4 NARROW ESCAPE , part III : Riemann surfaces and non - smooth domains
نویسندگان
چکیده
We consider Brownian motion in a bounded domain Ω on a twodimensional Riemannian manifold (Σ, g). We assume that the boundary ∂Ω is smooth and reflects the trajectories, except for a small absorbing arc ∂Ωa ⊂ ∂Ω. As ∂Ωa is shrunk to zero the expected time to absorption in ∂Ωa becomes infinite. The narrow escape problem consists in constructing an asymptotic expansion of the expected lifetime, denoted Eτ , as ε = |∂Ωa|g/|∂Ω|g → 0. We derive a leading order asymptotic approximation Eτ = |Ω|g Dπ [
منابع مشابه
D ec 2 00 4 NARROW ESCAPE , part II : The circular disk
We consider Brownian motion in a circular disk Ω, whose boundary ∂Ω is reflecting, except for a small arc, ∂Ωa, which is absorbing. As ε = |∂Ωa|/|∂Ω| decreases to zero the mean time to absorption in ∂Ωa, denoted Eτ , becomes infinite. The narrow escape problem is to find an asymptotic expansion of Eτ for ε ≪ 1. We find the first two terms in the expansion and an estimate of the error. The resul...
متن کاملNarrow Escape, Part III: Non-Smooth Domains and Riemann Surfaces
We consider the narrow escape problem in two-dimensional Riemannian manifolds (with a metric g) with corners and cusps, in an annulus, and on a sphere. Specifically, we calculate the mean time it takes a Brownian particle diffusing in a domain to reach an absorbing window when the ratio ε = |∂ a |g |∂ |g between the absorbing window and the otherwise reflecting boundary is small. If the boundar...
متن کامل0 D ec 2 00 2 REPRESENTATIONS OF FINITE GROUPS ON RIEMANN - ROCH SPACES
We study the action of a finite group on the Riemann-Roch space of certain divisors on a curve. If G is a finite subgroup of the automorphism group of a projective curve X and D is a divisor on X left stable by G then we show the natural representation of G on the Riemann-Roch space L(D) = LX (D) is a direct sum of irreducible representations of dimension ≤ d, where d is the size of the smalles...
متن کاملec 2 00 4 Boundary Regularity for the ∂̄ b - Neumann Problem , Part 1
We establish sharp regularity and Fredholm theorems for the ∂̄b-Neumann problem on domains satisfying some non-generic geometric conditions. We use these domains to construct explicit examples of bad behaviour of the Kohn Laplacian: it is not always hypoelliptic up to the boundary, its partial inverse is not compact and it is not globally subelliptic.
متن کاملD ec 2 00 7 Character sums to smooth moduli are small
(where χ (mod q) is a Dirichlet character) arise naturally in many classical problems of analytic number theory, from estimating the least quadratic nonresidue (mod p) to bounding L-functions. Recall that for any character χ (mod q) , |Sχ(x)| is trivially bounded above by φ(q). A folklore conjecture (which is a consequence of the Generalized Riemann Hypothesis) predicts that for nonprincipal ch...
متن کامل