The Dirichlet problem for stochastic partial differential equations with nonlocal operators in C1, open sets

This is a proceedings contribution for the presentation at XQCD 2025 based on our recent paper. We study the formulation of non-local operators on the lattice for analyzing the Higgs-confinement phase transition in a gauge-Higgs model. Since the Higgs and confined phase share the same symmetries, they cannot be distinguished by local order parameters, and one needs to use the expectation value of non-local operators. In this proceedings contribution, based on our recent work, we present lattice formulations and numerical results for three non-local operators: the Polyakov loop, the ’t Hooft loop, and the Aharonov-Bohm phase. As a testing ground, we employ the charge-2 Abelian Higgs model, whose phase diagram is well established. Our simulations reproduce the known phase structure and successfully demonstrate our lattice formulation.

The paper deals with homogenisation problems for high-contrast symmetric convolution-type operators with integrable kernels in media with a periodic microstructure. We adapt the two-scale convergence method to nonlocal convolution-type operators and obtain the homogenisation result both for problems stated in the whole space and in bounded domains with the homogeneous Dirichlet boundary condition. Our main focus is on spectral analysis. We describe the spectrum of the limit two-scale operator and characterise the limit behaviour of the spectrum of the original problem as the microstructure period tends to zero. It is shown that the spectrum of the limit operator is a subset of the limit of the spectrum of the original operator, and that they need not coincide.

Models of physical phenomena that use nonlocal operators are better suited for some applications than their classical counterparts that employ partial differential operators. However, the numerical solution of these nonlocal problems can be quite expensive. To address this issue, Local-to-Nonlocal couplings have emerged that combine partial differential operators with nonlocal operators. In this work, we make use of an energy-based Local-to-Nonlocal coupling that serves as a constraint for an interface identification problem.

We investigate a class of stochastic complex Ginzburg--Landau (CGL) equations driven by both interior and boundary Lévy noise on multidimensional positive orthants. The deterministic model incorporates a nonlocal diffusion operator given by a Caputo-type fractional Laplacian of order $\beta\in (\frac{3}{2},2) $, alongside a cubic-type of nonlinearity. Stochastic forcing acts independently on each boundary hyperplane through non-Gaussian jump processes, modeled via Poisson random measures, and within the interior through multiplicative Lévy-type perturbations. We construct mild solutions in weighted Sobolev spaces under minimal regularity assumptions, employing Laplace transform techniques, stochastic convolutions, and trace estimates to handle both nonlocality and boundary singularities. A priori estimates and second moment bounds are established, and local and global well-posedness results are proved for $L^{\infty}$ -integrable noise paths. Our analysis reveals how the interaction between Lévy jumps and anisotropic Dirichlet-type boundary conditions modifies the long-time asymptotic behavior of solutions. This work extends existing stochastic PDE theory to a new regime combining nonlocal diffusion, complex-valued nonlinearities, and discontinuous boundary noise, addressing significant analytical challenges associated with multidimensional geometry and jump-induced stochastic dynamics.

Using artificial dissipation to tame entanglement growth, we chart the emergence of diffusion in a generic interacting lattice model for varying U(1) charge densities. We follow the crossover from ballistic to diffusive transport above a scale set by the scattering length, finding the intuitive result that the diffusion constant scales as D ∝ 1 / ρ at low densities ρ . Our numerical approach generalizes the Dissipation-Assisted Operator Evolution algorithm: in the spirit of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, we effectively approximate nonlocal operators by their ensemble averages, rather than discarding them entirely. This greatly reduces the operator entanglement entropy, while still giving accurate predictions for diffusion constants across all density scales. We further construct a minimal model for the transport crossover, yielding charge correlation functions which agree well with our numerical data. Our results clarify the dominant contributions to hydrodynamic correlation functions of conserved densities, and serve as a guide for generalizations to low-temperature transport.

In this study, we examine the uniqueness conditions for solutions of fractal differential equations using the Krasnoselskii-Krein uniqueness theorem. The analysis establishes sufficient criteria that guarantee the existence of unique solutions. Additionally, we employ the successive midpoint method to numerically solve chaotic systems governed by both fractal and global derivatives. To evaluate the effectiveness of the proposed approach, graphical simulations are presented for various derivative orders. These results illustrate the method’s accuracy, stability, and reliability in capturing the intricate dynamics of the considered systems.

We test the eigenstate thermalization hypothesis (ETH) in 1 + 1 -dimensional SU(2) lattice gauge theory (LGT) with one flavor of dynamical fermions. Using the loop-string-hadron framework of the LGT with a bosonic cutoff, we exactly diagonalize the Hamiltonian for finite size systems and calculate matrix elements (MEs) in the eigenbasis for both local and nonlocal operators. We analyze different indicators to identify the parameter space for quantum chaos at finite lattice sizes and investigate how the ETH behavior emerges in both the diagonal and off-diagonal MEs. Our investigations allow us to study various timescales of thermalization and the emergence of random matrix behavior, and highlight the interplays of the several diagnostics with each other. Furthermore, from the off-diagonal MEs, we extract a smooth function that is closely related to the spectral function for both local and nonlocal operators. We find numerical evidence of the spectral gap and the memory peak in the nonlocal operator case. Finally, we investigate aspects of subsystem ETH in the lattice gauge theory and identify certain features in the subsystem reduced density matrix that are unique to gauge theories.

The weak-field, quasi-static regime of gravity is commonly described by the Newton-Poisson equation as an effective response law. We construct this response within a cost-first discrete variational framework. The Recognition Composition Law (RCL) uniquely selects a reciprocal closure cost within the restricted quadratic symmetric composition class; together with the discrete ledger axioms AX1-AX5 (including conservation) and standard DEC refinement, the Newton-Poisson baseline is then recovered in the instantaneous-closure limit. Conditional on Assumption AS1 (scale-free latency) and Assumption AS2 (causal frequency-wavenumber ansatz), allowing finite equilibration introduces fractional memory into the response, yielding a scale-free modification of the source-potential relation characterized by a power-law kernel wker(k)=1+C(k0/k)α in Fourier space. The kernel exponent α=12(1-φ-1)≈0.191, where φ=(1+5)/2, is derived from self-similarity of the discrete ledger closure; the amplitude C=φ-2≈0.382 is identified as a hypothesis from a three-channel factorization argument. We evaluate this quasi-static kernel-motivated response against SPARC galaxy rotation curves under a strict global-only protocol (fixed M/L=1, no per-galaxy tuning, conservative σtot), using a controlled multiplicative surrogate for the full nonlocal disk operator implied by the kernel. In this deliberately over-constrained setting, the surrogate interface achieves median(χ2/N)=3.06 over 147 galaxies (2933 points), outperforming a strict global-only NFW benchmark and remaining less efficient than MOND under identical constraints. The analysis is restricted to the non-relativistic, quasi-static sector and should be read as a falsifier-oriented galactic-regime consistency check of the scaling window, not as a relativistic completion or a claim of Solar System viability without additional UV regularization/screening.

Recently, a great amount of attention has been focused on the study of fractional and nonlocal operators of the elliptic type both for pure mathematical research and in view of concrete real-world applications. We are interested in proving the existence and nonexistence of solutions of a minimizing problem involving a fractional Laplacian with weight. We consider the nonlocal minimizing problem on H0s(Ω)⊂Lqs(Ω), with qs:=2nn−2s, s∈(0,1), and n≥3infu∈H0s(Ω)||u||Lqs(Ω)=1∫Rnp(x)|(−Δ)s2u(x)|2dx−λ∫Ω|u(x)|2dx, where Ω is a bounded domain in Rn, p:Rn→R is a given positive weight presenting a global positive minimum p0>0 at a∈Ω, and λ is a real constant. The objective of this paper is to show that minimizers do not exist for some k,s,λ, and n. After that, we show some nonexistence results thanks to a fractional Pohozaev identity and fractional Hardy inequality.

This paper investigates the existence and regularity theory of steady fractional diffusion equations with first-order convection terms in the whole spaceRn. Specifically, within the framework of the Bessel potential spaceLαp(Rn), we analyze the interaction between the nonlocal operator(-Δ)sand the divergence-type drift termdiv(b(x)m). The main challenges of this study lie in the regularity competition between the fractional diffusion operator and the first-order derivative drift term, and the analytical challenges arising from the lack of compact embedding properties in unbounded regions. The fractional Fokker-Planck equation is an important generalization of the classical Fokker-Planck equation combined with fractional calculus and is a core mathematical model for describing anomalous diffusion and non-Markovian stochastic processes. The classical Fokker-Planck equation mainly characterizes normal diffusion behaviors such as Brownian motion and is suitable for transport processes that are local, memoryless, and obey Gaussian distributions. However, a large number of practical systems (such as diffusion in complex media, movement of biological cells, financial price fluctuations, relaxation in amorphous materials, etc.) exhibit long-range memory, non-local interactions, heavy-tailed distributions, and anomalous diffusion characteristics that deviate from Fick's law, which are difficult to accurately describe using integer-order differential models.

Nonlocal operators in divergence form and existence theory for integrable data

We analyze the Gray–Scott reaction–diffusion system on Ω ⊂ R n ( n ≥ 1 ) with mixed diffusion combining local and nonlocal operators. Using semigroup methods and duality estimates, we prove global existence of component-wise nonnegative solutions and establish uniform bounds. Numerical simulations illustrate pattern formation and highlight qualitative differences between the purely local and mixed-diffusion models.

This paper is devoted to the study of a class of nonlocal singular elliptic problems involving the fractional ψ-Hilfer p ( x ) -Laplace operator H L p ( x ) α , β , ψ u , with positive weight functions. The presence of both the singular nonlinearity and the generalized nonlocal operator introduces significant analytical challenges. To address these, we employ the Nehari manifold technique, which enables us to investigate the variational structure of the problem and establish the existence of at least two distinct solutions.

We consider static linear elastic composite materials (CMs) with periodic structure. The core of the proposed methodology is the generation of a novel dataset using specially designed body force fields with compact support (BFCS), enabling a new RVE concept that reduces the infinite periodic medium to a finite domain without boundary artifacts. This functionally reduced RVE is used for translated averaging of DNS results, efficiently computed via a newly developed fast Fourier transform (FFT)-based solver for BFCS loading. The resulting dataset captures localized field responses and is used to train machine learning (ML) and neural networks (NN) models to learn effective nonlocal surrogate operators. These operators accurately predict macroscopic responses while reflecting microstructural features and nonlocal interactions. By accounting for field localization while simultaneously eliminating influences from finite sample size and boundary effects, it provides a physically grounded and data-driven framework for constructing accurate surrogate models for the homogenization of complex materials.

We address the problem given by the following partial differential equation: some semi-Linear parabolic equations with uniformly elliptic non-local operators in Half-Space. Initially, we establish a generalized weighted average inequality and a maximum principle in unbounded domains, which are crucial for the sliding method. Then, we employ sliding to demonstrate the monotonicity of bounded positive solutions. In this paper, we will remove the monotonicity assumption of the kernel function $a(x)$ by using the sliding method. The techniques employed in the process of this method have applications to other problems related to uniformly elliptic operators.

The Lefschetz number of an endomorphism of an elliptic complex is expressed in terms of regularized traces of the operators defining the endomorphism. This result is obtained under certain conditions on the wavefront sets of the operators in question. In the particular case of geometric endomorphisms of the complex, we obtain the classical Atiyah–Bott formula. As an application, we compute the Lefschetz numbers of nonlocal elliptic operators associated with an action of a finite group on a closed smooth manifold. For the de Rham complex, this gives a formula for the Lefschetz number in the cohomology of the orbit space in terms of fixed points.

Physics-informed neural network approach to unsteady fractional flow in a vertical coaxial annulus with thermal effects and magneto-hall interaction

ABSTRACT In this paper, we study a nonlocal nonlinear Schrödinger equation (MMT model). We investigate the effect of the nonlocal operator appearing in the nonlinearity on the long‐term behavior of solutions, and we identify the conditions under which the solutions of the Cauchy problem associated with this equation are bounded globally in time in the energy space. We also explore the dynamical behavior of standing wave solutions. Therefore, we first numerically generate standing wave solutions of nonlocal nonlinear Schrödinger equation by using the Petviashvili's iteration method and their stability is investigated by the split‐step Fourier method. This equation also has a two‐parameter family of standing wave solutions. In a second step, we meticulously concern with the construction and stability of a two‐parameter family of standing wave solutions numerically. Finally, we investigate the semiclassical limit of the nonlocal nonlinear Schrödinger equation in both focusing and defocusing cases.

We consider a class of multi-population pedestrian models consisting in a system of nonlocal conservation laws coupled in the nonlocal components and describing several groups of pedestrians moving toward their respective targets while trying to avoid each other and the obstacles limiting the walking domain. Specifically, the nonlocal operators account for interactions occurring at the microscopic level as a reaction to the presence of other individuals or obstacles along the preferred path. In particular, the presence of obstacles is implemented in the nonlocal terms of the equations and not as classical boundary conditions. This allows to rewrite domain shape optimization problems as PDE-constrained problems. In this paper, we investigate the well-posedness of such optimization problems by proving the stability of solutions with respect to the positions and shapes of the obstacles. A differentiability result in the linear case is also provided. These properties are illustrated with a numerical example.