Convolution powers of complex functions on $\mathbb Z^d$

Abstract

The study of convolution powers of a finitely supported probability distribution φ on the d-dimensional square lattice is central to random walk theory. For instance, the nth convolution power φ (n) is the distribution of the nth step of the associated random walk and is described by the classical local limit theorem. Following previous work of P. Diaconis and the authors, we explore the more general setting in which φ takes on complex values. This problem, originally motivated by the problem of Erastus L. De Forest in data smoothing, has found applications to the theory of stability of numerical difference schemes in partial differential equations. For a complex valued function φ on Z d , we ask and address four basic and fundamental questions about the convolution powers φ (n) which concern sup-norm estimates, generalized local limit theorems, pointwise estimates, and stability. This work extends one-dimensional results of I. J. Schoenberg, T. N. E. Greville, P. Diaconis and the second author and, in the context of stability theory, results by V. Thomée and M. V. Fedoryuk.

Positive-Homogeneous Operators, Heat Kernel Estimates and the Legendre-Fenchel Transform

“…In fact, it is possible that multiple (distinct) operators can appear by looking at the Taylor expansions about distinct local extrema ofφ (when they exist) and, in such cases, the corresponding local limit theorems involve sums of of heat kernels-each corresponding to a distinct Λ. This study is carried out in the article [44] wherein local limit theorems involve the heat kernels of the positive-homoegeneous operators studied in the present article. We note that the theory presented in [44] is not complete, for there are cases in which the associated Taylor approximations yield symbols corresponding to operators Λ which fail to be positive-homogeneous (and hence fail to be positive-semi-elliptic) and further, the heat kernels of these (degenerate) operators appear as limits of oscillatory integrals which correspond to the presence of "odd" terms in Λ, e.g., the Airy function.…”

“…This study is carried out in the article [44] wherein local limit theorems involve the heat kernels of the positive-homoegeneous operators studied in the present article. We note that the theory presented in [44] is not complete, for there are cases in which the associated Taylor approximations yield symbols corresponding to operators Λ which fail to be positive-homogeneous (and hence fail to be positive-semi-elliptic) and further, the heat kernels of these (degenerate) operators appear as limits of oscillatory integrals which correspond to the presence of "odd" terms in Λ, e.g., the Airy function. In one dimension, a complete theory of local limit theorems is known for the class of finitely supported functions φ : Z → C. Beyond one dimension, a theory for local limit theorems of complex-valued functions, in which the results of [44] will fit, remains open.…”

“…By contrast, we fix no dilation structure on V and formulate homogeneity in terms of an operator Λ and the existence of a one-parameter group {δ E t } that "plays" well with Λ in sense defined above. As seen in the study of convolution powers on the square lattice (see [44]), it useful to have this freedom. Definition 2.2.…”

“…This a priori weaker condition is seen to be sufficient by an argument which makes use of the Jordan-Chevalley decomposition. In the positivehomogeneous case, this argument is carried out in [44] (specifically Proposition 2.2) wherein positivehomogeneous operators are first defined by this (a priori weaker) condition. For the nondegenerate case, the same argument pushes through with very little modification.…”

Davies’ method for heat-kernel estimates: An extension to the semi-elliptic setting

“…By contrast, we fix no dilation structure on V and formulate homogeneity in terms of an operator Λ and the existence of a oneparameter group {δ E t } that plays well with Λ in sense defined above. As seen in the study of convolution powers on the square lattice (see [22]), it useful to have this freedom. Definition 3.2.…”