ELLIPTIC GRADIENT ESTIMATES FOR DIFFUSION OPERATORS ON COMPLETE RIEMANNIAN MANIFOLDS

In this note, we obtain the elliptic estimate for diffusion operator L = △＋△Ф·△ on complete, noncompact Riemannian manifolds, under the curvature condition CD（K, m）, which generalizes B. L. Kotschwar＇s work [5]. As an application, we get estimate on the heat kernel. The Bernstein-type gradient estimate for SchrSdinger-type gradient is also derived.

Stability Analysis of Inverse Lax-Wendroff Procedure for a High order Compact Finite Difference Schemes

This paper considers the finite difference(FD)approximations of diffusion operators and the boundary treatments for different boundary conditions.The proposed schemes have the compact form and could achieve arbitrary even order of accuracy.The main idea is to make use of the lower order compact schemes recursively,so as to obtain the high order compact schemes formally.Moreover,the schemes can be implemented efficiently by solving a series of tridiagonal systems recursively or the fast Fourier transform(FFT).With mathematical induction,the eigenvalues of the proposed differencing operators are shown to be bounded away from zero,which indicates the positive definiteness of the operators.To obtain numerical boundary conditions for the high order schemes,the simplified inverse Lax-Wendroff(SILW)procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method.Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.

Construction of Generalized Diffusion Processes: the Resolvent Approach

In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusion among others.Via the standard resolvent approach,the associated generalized diffusion processes are constructed.

Criteria for Discrete Spectrum of 1D Operators

For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators.Interestingly,these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier,simply replacing the‘finite supremum’by‘vanishing at infinity’.Except a dual technique,the main tool used here is a transform in terms of the harmonic function,to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context.All of them are illustrated by examples.The main body of the paper is devoted to the hard part of the story,the easier part but powerful one is delayed to the end of the paper.

L-harmonic functions with polynomial growth of a fixed rate

Yau made the following conjecture:For a complete noncompact manifold with nonnegative Ricci curvature the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional.we extend the result on the Laplace operator to that on the symmetric diffusion operator,and prove the space of L-harmonic functions with polynomial growth of a fixed rate is finitedimensional,when m-dimensional Bakery-Emery Ricci curvature of the symmetric diffusion operator on the complete noncompact Riemannian manifold is nonnegative.

Study on a General Hopf Hierarchy

By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.