Theory | Non-Archimedean Analysis and Applications

Date & Time: Tuesday, September 29, 2026 | 11:00 - 13:30
Place: tba
Chairs: Brian Luna Zambrano, Patrick Erik Bradley

Abstract: 
The mini symposium aims at bringing together researchers in non-archimedean analysis and applications, in order to present the state of the art of new developments in non-archimedean analysis and applications in the Sciences, Engineering and Economics. Having initially diffusion processes over the p-adic number field, now non-linear dynamics (e.g. an analogue of Navier-Stokes or the porous medium equation), as well as other p-adic domains like p-adic analytic manifolds are now included. First applied and further developped in mathematical physics, the methods are now part of computer science, biology, earth sciences and other domains. Bringing together researchers on applications in Science and Engineering aims at fostering new insights and ideas towards a deeper understanding of general hierarchical structures in these domains outside mathematics. 

Confirmed Speakers:

Speaker: Christoph Kriegler: U Clermont
Title: Functional calculus and semilinear evolution equations for the Taibleson operator on non-Archimedean local fields
Abstract: For any non-Archimedean local field \(\mathbb{K}\) and any integer \( n \geq 1 \) , we show that the Taibleson operator admits a bounded \( H^{\infty} \) \( (\Sigma_{\theta} \)) functional calculus on the Bochner space \( L^p ( \mathbb{K}^n, Y ) \) for any  U M D Banach function space Y and any angle \( \theta \gt 0 \) , where \( \Sigma_{\theta} = {z~\in \mathbb{C}^*: \lvert (\mathrm{arg}~z) \rvert \lt \theta} \) and \( 1 \lt p \lt \infty \). Moreover, we prove that it even admits a bounded H\"ormander functional calculus of order \( \frac{3}{2} \). In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups and establish the R-boundedness of a family of convolution operators. Our results contribute to the theory of functional calculi for operators acting on vector-valued \( L^p \)-spaces over totally disconnected spaces. As an application, we obtain maximal regularity results and well-posedness for a class of evolution equations driven by the Taibleson operator.

Speaker: Oleksandra Antoniouk: American University Kyiv
Title: Properties of the solutions to the linear and nonlinear equations in the non-Archimedean case
Abstract: This talk presents recent results on linear and nonlinear pseudo-differential equations over non-Archimedean fields. We discuss the well-posedness and qualitative properties of solutions to several classes of evolution and boundary value problems. In the linear setting, we consider the non-Archimedean Neumann problem and describe the existence, uniqueness, and regularity of weak and strong solutions. In the nonlinear setting, we focus on p-adic counterparts of the porous medium equation and more general multidimensional nonlinear pseudo-differential evolution equations. The methods combine functional-analytic techniques with the theory of non-Archimedean pseudo-differential operators, providing a unified framework for the analysis of linear and nonlinear models. Connections with diffusion processes on ultrametric spaces and directions for further research are also outlined.

Speaker: Mariia V. Serdiuk: Taras Shevchenko National University of Kyiv, Ukraine
Title: Radial Solutions of Pseudo-differential Equations on the Field of p-adic Numbers
Abstract:   The theory of p-adic mathematical physics concerned with the mappings from \(t\in\mathbb{Q}_p\) to \(t\in\mathbb{C}\) is well developed [Albeverio et.al.]. A typical linear operator is the Vladimirov-Taibleson operator \(D^{\alpha,n}\), which can be seen as a kind of an elliptic operator.

In a paper by A. N. Kochubei [Kochubei et.al.] a right inverse to the operator \(D^{\alpha},\;\alpha>0\) was found. It turns out that this permits to reduce the \(p\)-adic Cauchy problem for radial functions to an integral equation whose properties resemble those of classical Volterra equations.

Let \( \alpha \gt 0 \), \(\gamma \gt 0\). We consider the problem

\begin{equation}\label{equation1}\tag{1}\
|t|_p^{\gamma}(D^{\alpha}u)(|t|_p)=f(|t|_p,u(|t|_p)),\;0\neq t\in\mathbb{Q}_p,\;u(0)=u_0.
\end{equation}

We suppose that the function \(f:p^{\mathbb{Z}}\times\mathbb{R}\to\mathbb{R}\) satisfies the conditions

\begin{equation}\\
|f(|t|_p,x)|\leq M, \label{equation2}\tag{2}
\end{equation}

\begin{equation}\\
|f(|t|_p,x)-f(|t|_p,y)|\leq F|x-y|, \label{equation3}\tag{3}
\end{equation}

for all \(t\in\mathbb{Q}_p,\;x,y\in\mathbb{R}\) and some constants \(M \) ,\(F \) independent on \(t \), \(x \), \(y \).

With the problem \ref{equation1} we associate the integral equation

\begin{equation}\\
u(|t|_p)=u_0+I^{\alpha}\left[|\cdot|_p^{-\gamma}f(|\cdot|_p,u(|\cdot|_p))\right](|t|_p). \label{equation4}\tag{4}
\end{equation}

We call a solution \(u\) to the equations \ref{equation4}, if it exists, a mild solution to the Cauchy problem \ref{equation1}.

Theorem 1.
Suppose that \(\gamma<\min(1,\alpha)\) and the conditions \ref{equation2}, \ref{equation3} hold. Then the problem \ref{equation1} has a unique local mild solution, that is the integral equation \ref{equation4} has a solution \(u(|t|_p)\) defined for \(|t|_p\leq q^N\), where \(N\in\mathbb{Z}\) is sufficiently negative, and any other solution \(\overline{u}(|t|_p)\), if it exists, coincides with \(u\) for \(|t|_p\leq p^K\), where \(K\leq N\).

Speaker: Trond Digernes (NTNU, The Norwegian University of Science and Technology), Mads Jakobsen
Title: Non-Archimedean Test Function Spaces
Abstract: We will comment on the non-existence of Schwartz type test function spaces in the non-Archimedean context and present a construction of test function spaces based on the multiplicative structure of a local field.

Speaker: Angel Alfredo Moran Ledezma  (Karlsruhe Institute of Technology)
Title: Ultrametric Graphons As Limits of Hierarchical Community Networks: Spectral Theory and Applications
Abstract: We develop a theory of ultrametric graphons as limiting objects for random networks with nested hierarchical community structure. A graphon W: [0,1]^2 --> [0,1] is called ultrametric if W (x,y) = w (d(x,y)), where d is an ultrametric on [0,1] induced by a family of nested partitions and w is a positive kernel. The resulting random graphs exhibit a hierarchical community structure in which the density of connections is governed by the ultrametric distance between vertices.

The Laplacian L_d^k of the deterministic graph sampled from an ultrametric graphon is itself an ultrametric Laplacian, whose eigenvalues and spectral projectors admit completely explicit closed-form expressions in terms of the community sizes and inter-community connection densities. We show that the normalized eigenvalues and spectral projectors of the random Laplacian L_r^k are arbitrarily close to those of L_d^k with high probability as k--> \infty; the explicit formulas for L_d^k therefore provide closed-form analytical approximations for the spectrum and spectral projectors of L_r^k. We then develop the following applications. A sign structure theorem for the empirical spectral projectors provides a rigorous generalization of the Fiedler vector criterion to hierarchical networks with arbitrarily many communities. We further establish a detectability threshold in spectral community detection for one-level hierarchical graphons, governed by a threshold p^* = min_i \rho_i, where \rho_i is the normalized spectral gap of the -th sub-community. For random walks, we construct a limiting pseudo-inverse Laplacian operator L_w^(+) and establish its almost sure convergence from the pseudo-inverse of the random Laplacian L_r^2 in (L^2) norm.

Since the hitting and commute times of the continuous-time Markov chain are expressed in terms of the pseudo-inverse Laplacian, this convergence implies that both collapse almost surely, in the large-graph limit, to quantities depending only on the expected degrees of the endpoints, losing all information about the hierarchical community structure. Finally, we apply the framework to the SIS epidemic model on hierarchical community networks, deriving explicit closed-form stability conditions for the disease-free equilibrium in terms of the cluster sizes and inter-community connectivities. These reveal a fundamental tension between homogeneous and heterogeneous community structures: global cooperation is optimal in the homogeneous case, whereas targeted intervention on the most connected sub-community is substantially more effective in the heterogeneous case, as confirmed by numerical experiments.