Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.06214

EXISTENCE AND UPPER BOUNDS OF FUNDAMENTAL SOLUTIONS TO KINETIC EQUATIONS WITH ROUGH NON-LOCAL DIFFUSION

Pascal Auscher
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the fundamental solution operators associated to Kolmogorov equations with symmetric non-local diffusion with rough kernels. We prove that these operators are induced by measurable kernels and prove sharp upper bounds in terms of a kinetic profile governed by the interplay of transport and non-local diffusion. Our proof combines the small-jump with loss decomposition of P.-A. Meyer with the ideas of Nash. The proof is purely analytic.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Pascal Auscher. 2026-10-05. EXISTENCE AND UPPER BOUNDS OF FUNDAMENTAL SOLUTIONS TO KINETIC EQUATIONS WITH ROUGH NON-LOCAL DIFFUSION. https://arxiv.org/abs/2610.06214

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A nonlocal-to-local transfer principle for parabolic Calderón problems

We prove a nonlocal-to-local transfer principle for fractional powers of variable coefficient parabolic operators. Let $H_σ=\partial_t-\nabla\cdotσ\nabla$, where $σ$ is a uniformly elliptic $C^2$ matrix-valued coefficient. For any $s\in(0,1)$ and any nonempty bounded Lipschitz exterior observation set $W$ with $\overline W\cap\overlineΩ=\emptyset$, we show that equality of the partial exterior DN maps for $H_{σ_j}^s$, $j=1,2$, together with the exterior agreement $σ_1=σ_2$ in $Ω_e$, implies equality of the full lateral local Cauchy data sets for $H_{σ_j}$ on $(\partialΩ)_T$. The proof uses the parabolic Caffarelli--Silvestre extension and the averaging transform \[ v(t,x)=\int_0^\infty y^{1-2s}\widetilde u(t,x,y)\,dy. \] This transform sends fractional parabolic solutions to local parabolic energy solutions. The main analytic ingredient is a boundary density theorem: the lateral traces generated by these averaged exterior solutions are dense in the corresponding closed local energy trace space. Unlike the elliptic reduction, no exterior normalization to a constant is imposed; the common exterior coefficient may be variable. The result holds for all spatial dimensions $n\geq2$. Consequently, local uniqueness results for parabolic Calderón problems transfer to the corresponding nonlocal problems. In particular, scalar leading coefficients are uniquely determined in $Ω$ from partial exterior nonlocal measurements, relative to their exterior values.

math.AP↗
arXiv2026

A proof of Onsager's Conjecture for the SQG equation

We construct solutions to the SQG equation that fail to conserve the Hamiltonian while having the maximal allowable regularity for this property to hold. This result solves the generalized Onsager conjecture on the threshold regularity for Hamiltonian conservation for SQG. We also prove nonuniqueness for arbitrary smooth data and an h-principle result.

math.AP↗
arXiv2026

An inverse problem for the Monge-Ampère equation

We prove uniqueness for an inverse source problem for the Monge--Ampère equation $\det D^2u=F$ on a bounded uniformly convex planar domain. The source is smooth and strictly positive, and its boundary derivatives through order two are prescribed. We show that the Dirichlet-to-Neumann map on the convex solution branch, restricted to a neighborhood of zero boundary data, determines the source throughout the domain. The first linearization gives an anisotropic equation whose principal coefficient is the inverse Hessian of an unknown background solution. We determine its conformal class in the original Euclidean coordinates and use the second linearized equation to determine the remaining scalar factor. We construct complex geometric optics solutions and prove arbitrary order expansions of their correction terms in $W^{2,p}$ for phases without critical points. A termwise stationary phase analysis of the second linearized identity gives a differential equation with a Cauchy integral term. We prove the required unique continuation property by incorporating that integral as an additional unknown in a first order system and applying a Carleman estimate.

math.AP↗