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Spectral Equivalence of Solvable Lie Algebras with Nilradical I: General and Heisenberg Cases
Gary Hu
https://arxiv.org/abs/2509.21418 https://arxiv.org/pd…

Spectral Equivalence of Solvable Lie Algebras with Nilradical I: General and Heisenberg Cases
Building on the work of Yang, Key, and Muller, we investigate the spectral theory of solvable Lie algebras through their characteristic polynomials and associated spectral invariants. We begin by studying the general case: we compare two notions of spectral equivalence, establish the first known lower bound for the spectral invariant $k(L)$ via the weight decomposition of the nilradical, and introduce a new notion of spectral rigidity to study spectral equivalence within parameterized families.…

Dynamical interface above a hard wall and reflected SPDE on the half-line
Pierre Faug\`ere, Cyril Labb\'e
https://arxiv.org/abs/2509.03328 https://arxi…

Dynamical interface above a hard wall and reflected SPDE on the half-line
We consider a dynamical random interface on the infinite lattice $\mathbb{N}$ evolving according to a "corner flip" dynamic above a hard wall, with an additional pinning at the origin. We study the stationary fluctuations under a diffusive scaling and prove convergence in law towards the solution of an SPDE of Nualart-Pardoux's type, namely the Reflected Stochastic Heat Equation on the half-line. We also obtain that the law of the 3-dimensional Bessel process is an invariant measure for this SP…

Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types
Mikhail Ignatev, Alexey Petukhov
https://arxiv.org/abs/2507.20332 https://

Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types
Let $\mathfrak g$ be a classical simple Lie algebra over an algebraically closed field $\mathbb F$ of characteristic zero or large enough, and let $\mathfrak n$ be a maximal nilpotent subalgebra of $\mathfrak g$. The main tool in representation theory of $\mathfrak n$ is the orbit method, which classifies primitive ideals in the universal enveloping algebra ${\rm U}(\mathfrak n)$ and unitary representations of the unipotent group $N=\exp(\mathfrak n)$ in terms of coadjoint orbits on the dual sp…

On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term
Mohammud Foondun, Davar Khoshnevisan, Eulalia Nualart
https://arxiv.org/abs/2510.00214

On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term
We consider a parabolic stochastic partial differential equation (SPDE) on $[0\,,1]$ that is forced with multiplicative space-time white noise with a bounded and Lipschitz diffusion coefficient and a drift coefficient that is locally Lipschitz and satisfies an $L\log L$ growth condition. We prove that the SPDE is well posed when the initial data is in $L^2[0\,,1]$. This solves a strong form of an open problem.

Biderivations of some classes of solvable Leibniz algebras
Bakhtiyor Yusupov, Doston Jumaniyozov, Majidkhon Azizov
https://arxiv.org/abs/2507.18134 https://

Biderivations of some classes of solvable Leibniz algebras
In this work, we investigate anti-derivations and biderivation of Leibniz algebras. We describe general form of anti-derivations and biderivations on null-filiform and filiform Leibniz algebras. Moreover, we show how to construct Leibniz algebras, while using these biderivations. We describe general form of anti-derivations and biderivations on solvable Leibniz algebras with null-filiform and filiform nilradicals. Interesting fact that, any biderivations of solvable Leibniz algebras with null-f…

Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line
Robert C. Dalang, David Nualart, Fei Pu
https://arxiv.org/abs/2508.11859

Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line
For Gaussian random fields with values in $\mathbb{R}^d$, sharp upper and lower bounds on the probability of hitting a fixed set have been available for many years. These apply in particular to the solutions of systems of linear SPDEs. For non-Gaussian random fields, the available bounds are less sharp. For nonlinear systems of stochastic heat equations, a sharp lower bound was obtained in a previous paper by two of the authors. Here, we obtain the corresponding sharp upper bound. The proof req…