In this short note, we prove the convexity of the first eigenfunction of the drifting Laplacian operator with zero Dirichlet boundary value provided a suitable assumption to the drifting term is added. After giving a gradient estimate, we then use the convexity of the first eigenfunction to get a lower bound of the difference of the first and second eigenvalues of the drifting Laplacian.

We propose an efficient method to evaluate callable and putable bonds under a wide class of interest rate models, including the popular short rate diffusion models, as well as their time changed versions with jumps. The method is based on the eigenfunction expansion of the pricing operator. Given the set of call and put dates, the callable and putable bond pricing function is the value function...

In earlier work, we developed a Monte Carlo method to compute the principal eigenvalue of linear operators, which was based on the simulation of exit times. In this paper, we generalize this approach by showing how to use a branching method to improve the efficacy of simulating large exit times for the purpose of computing eigenvalues. Furthermore, we show that this new method provides a natura...

Abstract This paper studies the J -method of [ E. Jarlebring, S. Kvaal, W. Michiels. SIAM J. Sci. Comput. 36-4:A1978-A2001, 2014 ] for nonlinear eigenvector problems in a general Hilbert space framework. is basis variational discretization techniques and mesh-independent numerical analysis. A simple modification method mimics an energy-decreasing discrete gradient flow. In case Gross–Pitaevskii...

This paper proposes a novel computationally efficient dynamic bi-orthogonality based approach for calibration of a computer simulator with high dimensional parametric and model structure uncertainty. The proposed method is based on a decomposition of the solution into mean and a random field using a generic Karhunnen-Loeve expansion. The random field is represented as a convolution of separable...

We present a simplified theory of generalized eigenfunction expansions for a commuting family of bounded operators and with finitely many unbounded operators. We also study the convergence of these expansions, giving an abstract type of uniform convergence result, and illustrate the theory by giving two examples: The Fourier transform on Hecke operators, and the Laplacian operators in hyperboli...

Generalized eigenfunctions may be regarded as vectors of a basis in a particular direct integral of Hilbert spaces or as elements of the antidual space Φ× in a convenient Gelfand triplet Φ ⊆ H ⊆ Φ×. This work presents a fit treatment for computational purposes of transformations formulas relating different generalized bases of eigenfunctions in both frameworks direct integrals and Gelfand tripl...