Nonlocal Operators Research Articles

Variational methods for nonpositive mixed local–nonlocal operators

We prove the existence of a weak solution for boundary value problems driven by a mixed local–nonlocal operator. The main novelty is that such an operator is allowed to be nonpositive definite.

Stability and convergence analysis of high-order numerical schemes with DtN-type absorbing boundary conditions for nonlocal wave equations

Abstract The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes to discretize the spatially nonlocal operator, and apply the explicit difference scheme to approximate the temporal derivative to achieve a fully discrete infinity system. After that, we construct the Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs), to reduce the infinite discrete system into a finite discrete system. To do so, we first adopt the idea in Du et al. (2018, Commun. Comput. Phys., 24, 1049–1072) and Du et al. (2018, SIAM J. Sci. Comp., 40, A1430–A1445) to derive the Dirichlet-to-Dirichlet (DtD)-type mappings for one- and two-dimensional cases, respectively. We then use the discrete nonlocal Green’s first identity to achieve the discrete DtN-type mappings from the DtD-type mappings. The resulting DtN-type mappings make it possible to perform the stability and convergence analysis of the reduced problem. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.

QCD factorization of quasi generalized gluon distributions

We study the factorization relations between quasi gluon GPDs and twist-2 GPDs. The perturbative coefficient functions are obtained at one-loop level. They are free from any collinear- or I.R. divergences. Unlike the case of the factorization of quasi quark GPDs at one-loop, we have to add ghost contributions for the factorization of quasi gluon GPDs in order to obtain gauge-invariant results. In general, operators will be mixed beyond tree-level. Our work shows that the mixing pattern of the nonlocal operators in quasi gluon GPDs is the same as those of local operators, i.e., the nonlocal operators considered are mixed with gauge-invariant operators, BRST-variation operators and operators involving EOM operator. The factorization relations are obtained for all quasi gluon GPDs. Taking the forward limit, we also obtain the relations between quasi gluon PDFs and twist-2 PDFs.

Heat kernels for time-dependent non-symmetric mixed Lévy-type operators

In this paper we establish the existence, uniqueness and regularity of heat kernels to a large class of time-inhomogeneous non-symmetric nonlocal operators with Dini's continuous kernels. Moreover, quantitative estimates including two-sided estimates, gradient and fractional derivative estimates of the heat kernels are obtained.

A Faber-Krahn inequality for mixed local and nonlocal operators

We consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the first eigenvalue among sets of given volume and we provide a stability result for sets that almost attain the minimum.

Smooth Solutions of Hyperbolic Equations with Translation by an Arbitrary Vector in the Free Term

We construct three-parameter families of solutions of hyperbolic differential-difference equations in a half-space with a general shift operator in the free term (or in a nonlocal operator potential). It is proved that the solutions obtained are classical if the real part of the symbol of the corresponding differential-difference operators is positive. Classes of equations for which the indicated condition is satisfied are given.

MFO-RIMS Tandem Workshop: Nonlocality in Analysis, Probability and Statistics

The central theme of the workshop were nonlocal operators which appear in various branches of mathematics (PDEs, fractional calculus, stochastic processes, statistics). Although the basic concepts are similar, both language and methods differ depending on one’s own community. The aim of the workshop was to bring together leading researchers from these disciplines, in order to alert the different communities about the problems, methods, and progress achieved separately, and to bridge the gap caused by different background and different mathematical terminology.

Heat- and Wave-Type Equations with Nonlocal Operators, I. Compact Lie Groups

Abstract We prove existence and uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a nonlocal (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p\leqslant 2\leqslant q<+\infty $. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave-type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat-type equation as well.

The concentration-compactness principle for the nonlocal anisotropic [formula omitted]-Laplacian of mixed order

In this paper, we study the existence of minimizers of the Sobolev quotient for a class of nonlocal operators with an orthotropic structure having different exponents of integrability and different orders of differentiability. Our method is based on the concentration-compactness principle which we extend to this class of operators. One consequence of our main result is the existence of a nontrivial nonnegative solution to the corresponding critical problem.

The Fading Memory Formalism with Mittag-Leffler-Type Kernels as A Generator of Non-Local Operators

Transient heat conduction problems are systematically applied to the fading memory formalism with different Mittag-Leffler-type memory kernels. With such an approach, using various memories naturally results in definitions of various fractional operators. Six examples are given and interpreted from a common perspective, covering the most well-liked versions of the Mittag-Leffler function. The fading memory approach was used as a template and demonstrated that, if the constitutive equations are correctly built, it is also possible to directly determine where the hereditary terms are located in the models.

Functional-Differential Operators on Geometrical Graphs with Global Delay and Inverse Spectral Problems

We suggest a new concept of functional-differential operators with constant delay on geometrical graphs that involves global delay parameter. Differential operators on graphs model various processes in many areas of science and technology. Although a vast majority of studies in this direction concern purely differential operators on graphs (often referred to as quantum graphs), recently there also appeared some considerations of nonlocal operators on star-type graphs. In particular, there belong functional-differential operators with constant delays but in a locally nonlocal version. The latter means that each edge of the graph has its own delay parameter, which does not affect any other edge. In this paper, we introduce globally nonlocal operators that are expected to be more natural for modelling nonlocal processes on graphs. We also extend this idea to arbitrary trees, which opens a wide area for further research. Another goal of the paper is to study inverse spectral problems for operators with global delay in one illustrative case by addressing a wide range of questions including uniqueness, characterization of the spectral data as well as the uniform stability.

Scaling laws of out-of-time-order correlators in a non-Hermitian kicked rotor model

We investigate the dynamics of the out-of-time-order correlators (OTOCs) via a non-Hermitian extension of the quantum kicked rotor model, where the kicking potential satisfies PT-symmetry. The spontaneous PT-symmetry breaking emerges when the strength of the imaginary part of the kicking potential exceeds a threshold value. We find, both analytically and numerically, that in the broken phase of PT symmetry, the OTOCs rapidly saturate with time evolution. Interestingly, the late-time saturation value scales as a pow-law in the system size. The mechanism of such scaling law results from the interplay between the effects of the nonlocal operator in OTOCs and the time reversal induced by non-Hermitian-driven potential.

Horizontally quasiconvex envelope in the Heisenberg group

This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex, for short) envelope of a given continuous function in the Heisenberg group.We provide a characterization for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to a first-order nonlocal Hamilton– Jacobi equation. We also construct the corresponding envelope of a continuous function by iterating the nonlocal operator. One important step in our arguments is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problem for the nonlocal Hamilton–Jacobi equation. Applications of our approach to the h-convex hull of a given set in the Heisenberg group are discussed as well.

The nonlocal inverse problem of Donsker and Varadhan

In this paper we prove a version of the celebrated Inverse Problem of Donsker and Varadhan [13] in the setting of nonlocal elliptic operators of the formLu=LKu+BK(h,u), where LK is a uniformly elliptic nonlocal operator with smooth coefficients, and, for h:RN→R smooth and bounded, BK(h,u) is the associated bilinear form, which can be regarded as a nonlocal transport term.

Eliminating Artificial Boundary Conditions in Time-Dependent Density Functional Theory Using Fourier Contour Deformation.

We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain, FCD yields a high-order accurate numerical solution of the time-dependent Schrödinger equation directly in free space, without the need for artificial boundary conditions. Of the many existing artificial boundary condition schemes, FCD is most similar to an exact nonlocal transparent boundary condition, but it works directly on Cartesian grids in any dimension, and runs on top of the fast Fourier transform rather than fast algorithms for the application of nonlocal history integral operators. We adapt FCD to time-dependent density functional theory (TDDFT), and describe a simple algorithm to smoothly and automatically truncate long-range Coulomb-like potentials to a time-dependent constant outside of a bounded domain of interest, so that FCD can be used. This approach eliminates errors originating from the use of artificial boundary conditions, leaving only the error of the potential truncation, which is controlled and can be systematically reduced. The method enables accurate simulations of ultrastrong nonlinear electronic processes in molecular complexes in which the interference between bound and continuum states is of paramount importance. We demonstrate results for many-electron TDDFT calculations of absorption and strong field photoelectron spectra for one and two-dimensional models, and observe a significant reduction in the size of the computational domain required to achieve high quality results, as compared with the popular method of complex absorbing potentials.

A maximal $$L_p$$-regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes

Let $$Z=(Z_t)_{t\ge 0}$$ be an additive process with a bounded triplet $$(0,0,\varLambda _t)_{t\ge 0}$$ . Then the infinitesimal generators of Z is given by time dependent nonlocal operators as follows: $$\begin{aligned} {\mathcal {A}}_Z(t)u(t,x)&=\lim _{h \downarrow 0}\frac{{\mathbb {E}}[u(t,x+Z_{t+h}-Z_t)-u(t,x)]}{h} \\&=\int _{{\mathbb {R}}^d}(u(t,x+y)-u(t,x)-y\cdot \nabla _x u(t,x)1_{|y|\le 1})\varLambda _t(dy). \end{aligned}$$ Suppose that for any Schwartz function $$\varphi $$ on $${\mathbb {R}}^d$$ whose Fourier transform is in $$C_c^{\infty }(B_{c_s} {\setminus } B_{c_s^{-1}} )$$ , there exist positive constants $$N_0$$ , $$N_1$$ , and $$N_2$$ such that $$\begin{aligned} \int _{{\mathbb {R}}^d}|{\mathbb {E}}[\varphi (x+r^{-1}Z_t)]|dx\le N_0 e^{- \frac{ N_1 t}{s(r)}},\quad \forall (r,t)\in (0,1)\times [0,T], \end{aligned}$$ and $$\begin{aligned} \Vert \psi ^{\mu }(r^{-1}D)\varphi \Vert _{L_1({\mathbb {R}}^d)}\le \frac{N_2}{s(r)},\quad \forall r\in (0,1). \end{aligned}$$ where s is a scaling function (Definition 2.4), $$c_s$$ is a positive constant related to s, $$\mu $$ is a symmetric Lévy measure on $${\mathbb {R}}^d$$ , $$\psi ^{\mu }(r^{-1}D)\varphi (x)= {\mathcal {F}}^{-1} \left[ \psi ^{\mu }(r^{-1}\xi ) {\mathcal {F}}[\varphi ]\right] (x)$$ and $$\psi ^{\mu }(\xi ){:=}\int _{{\mathbb {R}}^d}(e^{iy\cdot \xi }-1-iy\cdot \xi 1_{|y|\le 1})\mu (dy)$$ . In particular, above assumptions hold for Lévy measures $$\varLambda _t$$ having a nice lower bound and $$\mu $$ satisfying a weak-scaling property (Propositions 3.3, 3.5, and 3.6). We emphasize that there is no regularity condition on Lévy measures $$\varLambda _t$$ and they do not have to be symmetric. In this paper, we establish the $$L_p$$ -solvability to the initial value problem 0.2 $$\begin{aligned} \frac{\partial u}{\partial t}(t,x)={\mathcal {A}}_Z(t)u(t,x),\quad u(0,\cdot )=u_0,\quad (t,x)\in (0,T)\times {\mathbb {R}}^d, \end{aligned}$$ where $$u_0$$ is contained in a scaled Besov space $$B_{p,q}^{s;\gamma -\frac{2}{q}}({\mathbb {R}}^d)$$ (see Definition 2.8) with a scaling function s, exponent $$p \in (1,\infty )$$ , $$q\in [1,\infty )$$ , and order $$\gamma \in [0,\infty )$$ . We show that equation (0.2) is uniquely solvable and the solution u obtains full-regularity gain from the diffusion generated by a stochastic process Z. In other words, there exists a unique solution u to equation (0.2) in $$L_q((0,T);H_p^{\mu ;\gamma }({\mathbb {R}}^d))$$ , where $$H_p^{\mu ;\gamma }({\mathbb {R}}^d)$$ is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution u satisfies $$\begin{aligned} \Vert u\Vert _{L_q((0,T);H_p^{\mu ;\gamma }({\mathbb {R}}^d))}\le N\Vert u_0\Vert _{B_{p,q}^{s;\gamma -\frac{2}{q}}({\mathbb {R}}^d)}, \end{aligned}$$ where N is independent of u and $$u_0$$ . We finally remark that our operators $${\mathcal {A}}_{Z}(t)$$ include logarithmic operators such as $$-a(t)\log (1-\varDelta )$$ (Corollary 3.2) and operators whose symbols are non-smooth such as $$-\sum _{j=1}^dc_j(t)(-\varDelta )^{\alpha /2}_{x^j}$$ (Corollary 3.9).

Sobolev estimates for fractional parabolic equations with space-time non-local operators

We obtain $$L_p$$ estimates for fractional parabolic equations with space-time non-local operators $$\begin{aligned} \partial _t^\alpha u - Lu + \lambda u= f \quad \text {in } (0,T) \times {\mathbb {R}}^d , \end{aligned}$$ where $$\partial _t^\alpha u$$ is the Caputo fractional derivative of order $$\alpha \in (0,1]$$ , $$T\in (0,\infty )$$ , and $$\begin{aligned} Lu(t,x) {:}{=} \int _{{\mathbb {R}}^d} \bigg ( u(t,x+y)-u(t,x) - y\cdot \nabla _xu(t,x)\chi ^{(\sigma )}(y)\bigg )K(t,x,y)\,dy \end{aligned}$$ is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel K with respect to t and y. We also derive a weighted mixed-norm estimate for the equations with operators that are local in time, i.e., $$\alpha = 1$$ , which extend the previous results in Mikulevičius and Pragarauskas (J Differ Equ 256(4):1581–1626, 2014) and Zhang (Annales l’IHPAnalyse Nonlinéaire 30:573–614, 2013) by using a quite different method.

Dynamical study of a novel 4D hyperchaotic system: An integer and fractional order analysis

In this article, a new nonlinear four-dimensional hyperchaotic model is presented. The dynamical aspects of the complex system are analyzed covering equilibrium points, linear stability, dissipation, bifurcations, Lyapunov exponent, phase portraits, Poincaré mapping, attractor projection, sensitivity and time series analysis. To analyze hidden attractors, the proposed system is investigated through nonlocal operator in Caputo sense. The existence of solution of the system in fractional sense is studied by fixed point theory. The stability of fractional order system is demonstrated via Matignon stability criteria. The fractional order system is numerically studied via newly developed numerical method which is based on Newton polynomial interpolation. The evolution of the attractors are depicted with different fractional orders. For few fractional orders, some hidden strange chaotic attractors are observed through graphs. Theoretical and numerical studies demonstrate that this model has complex dynamics with some stimulating physical characteristics. To verify and validate the results, we implement Field Programmable Analog Arrays (FPAA).

On operator estimates in homogenization of nonlocal operators of convolution type

The paper studies a bounded symmetric operator Aε in L2(Rd) with(Aεu)(x)=ε−d−2∫Rda((x−y)/ε)μ(x/ε,y/ε)(u(x)−u(y))dy; here ε is a small positive parameter. It is assumed that a(x) is a non-negative L1(Rd) function such that a(−x)=a(x) and the moments Mk=∫Rd|x|ka(x)dx, k=1,2,3, are finite. It is also assumed that μ(x,y) is Zd-periodic both in x and y function such that μ(x,y)=μ(y,x) and 0<μ−⩽μ(x,y)⩽μ+<∞. Our goal is to study the limit behaviour of the resolvent (Aε+I)−1, as ε→0. We show that, as ε→0, the operator (Aε+I)−1 converges in the operator norm in L2(Rd) to the resolvent (A0+I)−1 of the effective operator A0 being a second order elliptic differential operator with constant coefficients of the form A0=−divg0∇. We then obtain sharp in order estimates of the rate of convergence.

Upper heat kernel estimates for nonlocal operators via Aronson’s method

In his celebrated article, Aronson established Gaussian bounds for the fundamental solution to the Cauchy problem governed by a second order divergence form operator with uniformly elliptic coefficients. We extend Aronson’s proof of upper heat kernel estimates to nonlocal operators whose jumping kernel satisfies a pointwise upper bound and whose energy form is coercive. A detailed proof is given in the Euclidean space and extensions to doubling metric measure spaces are discussed.