Related papers: A kernel formula for regularized Wasserstein proxi…
We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schr\"odinger operator with negative Hardy potential $$\Delta^{\alpha/2} -\lambda |x|^{-\alpha}$$ on $\RR^d$, where…
We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in \cite{CS}. Since the order of differentiability of…
In this note we establish the large time non-negativity of the heat kernel for a class of elliptic differential operators on closed, Riemannian manifolds, and apply this result to a problem from conformal differential geometry.
We study the spatially homogeneous granular medium equation \[\partial_t\mu=\rm{div}(\mu\nabla V)+\rm{div}(\mu(\nabla W \ast \mu))+\Delta\mu\,,\] within a large and natural class of the confinement potentials $V$ and interaction potentials…
This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we study the inverse problem of determining the…
We propose a kernel compression method for solving Distributed-Order (DO) Fractional Partial Differential Equations (DOFPDEs) at the cost of solving corresponding local-in-time PDEs. The key concepts are (1) discretization of the integral…
We analyze the spectra of general non-minimal second-order operators. To do this, we derive the local part of the trace of the second Seeley-DeWitt heat kernel coefficient for such operators in a completely model-independent way.…
We construct mollification operators in strongly Lipschitz domains that do not invoke non-trivial extensions, are $L^p$ stable for any real number $p\in[1,\infty]$, and commute with the differential operators $\nabla$, $\nabla{\times}$, and…
This paper investigates the eigenvalue problem of integral operators whose kernels can be expressed as a finite sum of pairwise products of single-variable functions, making them separable. By consdiering the matrix form of the separable…
Using our recently proposed covariant algebraic approach the heat kernel for a Laplace-like differential operator in low-energy approximation is studied. Neglecting all the covariant derivatives of the gauge field strength (Yang-Mills…
The fundamental solution and the heat semigroup of the Vladimirov-Taibleson operator on constant-order noncommutative Vilenkin groups are obtained, together with some estimates on the associated heat kernel. We also show the existence of a…
Polterovich proved a remarkable closed formula for heat kernel coefficients of the Laplace operator on compact Riemannian manifolds involving powers of Laplacians acting on the distance function. In the case of K\"ahler manifolds, we prove…
We find an explicit integral formula for the eigenfunctions of a fourth order differential operator against the kernel involving two Bessel functions. Our formula establishes the relation between K-types in two different realizations of the…
This work introduces novel numerical algorithms for computational quantum mechanics, grounded in a representation of the Laplace operator -- frequently used to model kinetic energy in quantum systems -- via the heat semigroup. The key…
In this paper we deal with a scale of reproducing kernel Hilbert spaces $H^{(n)}_2$, $n\ge 0$, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane $\mathbb{C}^+$. They are obtained as ranges of…
This work is the third part of a program initiated in arXiv:2111.13258, arXiv:2302.06571 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in…
For a given pseudo-Hermitian Hamiltonian of the standard form: H=p^2/2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator \eta satisfying H^\dagger=\eta H \eta^{-1} to the solution of a differential equation.…
Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDE) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations…
The past two decades have seen a revolution in ab initio calculations of nuclear properties. One key element has been the development of a rigorous effective interaction theory, applying unitary transformations to soften the nuclear…
We construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic partial differential equation (PDE) in two variables from input-output training pairs. The primary challenge of…