6,911 research outputs found
The Brownian traveller on manifolds
We study the influence of the intrinsic curvature on the large time behaviour
of the heat equation in a tubular neighbourhood of an unbounded geodesic in a
two-dimensional Riemannian manifold. Since we consider killing boundary
conditions, there is always an exponential-type decay for the heat semigroup.
We show that this exponential-type decay is slower for positively curved
manifolds comparing to the flat case. As the main result, we establish a sharp
extra polynomial-type decay for the heat semigroup on negatively curved
manifolds comparing to the flat case. The proof employs the existence of
Hardy-type inequalities for the Dirichlet Laplacian in the tubular
neighbourhoods on negatively curved manifolds and the method of self-similar
variables and weighted Sobolev spaces for the heat equation.Comment: 42 pages, 1 figur
Theoretical properties of quasi-stationary Monte Carlo methods
This paper gives foundational results for the application of
quasi-stationarity to Monte Carlo inference problems. We prove natural
sufficient conditions for the quasi-limiting distribution of a killed diffusion
to coincide with a target density of interest. We also quantify the rate of
convergence to quasi-stationarity by relating the killed diffusion to an
appropriate Langevin diffusion. As an example, we consider in detail a killed
Ornstein--Uhlenbeck process with Gaussian quasi-stationary distribution.Comment: 27 pages, 1 figure. Final version of accepted paper. Minor typos
correcte
Recommended from our members
The Brownian traveller on manifolds
We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the at case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the at case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation
- …