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$p$ -Adic Diffusion and Random Walks on $[0, 1]$

This paper constructs strong Markov processes on the real unit interval by transporting $p$ -adic diffusion operators via the Monna map, analyzes their spectral properties and heat equation solutions, and approximates these dynamics using continuous-time random walks on finite hierarchical partitions to provide a new visualization method for $p$ -adic diffusion on real domains.

Original authors: Patrick Erik Bradley, Paulina Halwas, Ell Ari Nitsche, Ángel Morán Ledezma

Published 2026-08-04

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Patrick Erik Bradley, Paulina Halwas, Ell Ari Nitsche, Ángel Morán Ledezma

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand how a drop of ink spreads through a glass of water. In the real world, this is a smooth, continuous process; the ink drifts slowly, blending into every corner. But what if the universe wasn't a smooth glass of water, but a giant, infinite tree where you can only jump from one branch to another? This is the strange, fascinating world of p-adic numbers. Unlike the real numbers we use for measuring distances on a ruler, p-adic numbers measure "closeness" based on how many times a number can be divided by a specific prime number (like 2, 3, or 5). In this world, two numbers are close if they share a long history of being divisible by that prime, creating a landscape that looks more like a fractal tree than a flat line.

For a long time, mathematicians treated the smooth, real world and this bumpy, tree-like p-adic world as completely separate neighborhoods. They spoke different languages and lived by different rules. However, a clever tool called the Monna map acts like a universal translator or a magical bridge between these two worlds. It takes the jagged, tree-like structure of p-adic numbers and lays them out flat onto the familiar interval from 0 to 1, like unrolling a scroll. This paper is about what happens when we take the rules of "diffusion" (how things spread or random-walk) from the p-adic tree, translate them across this bridge, and watch how they behave on our real, flat line. Why does this matter? Because understanding how things move in these strange, tree-like spaces helps us model complex systems in physics, biology, and even the way information flows in networks, and this paper shows us how to visualize and calculate those movements in a way that feels much more familiar.

The Paper's Big Idea: Translating the Untranslatable

This paper, written by Patrick Erik Bradley and colleagues, is essentially a guidebook for turning a very abstract, hard-to-visualize mathematical process into something you can actually see and calculate. The authors are tackling a problem where they want to study random walks (think of a drunk person stumbling around) and diffusion (like heat spreading) on the real number line between 0 and 1, but they want to do it using the rules of p-adic math.

Here is the core trick they use: They start with a process that happens on a p-adic unit disc (a specific type of p-adic space). In this p-adic world, the "distance" between points is measured differently than in our normal world. The authors use the Monna map to transport this entire process across the bridge to the real interval $[0, 1]$ .

The result is a new kind of random process on the real line. If you were to watch a particle moving according to these rules, it wouldn't drift smoothly like ink in water. Instead, it would stay still for a while and then suddenly jump to a new location. The paper proves that these paths are "right-continuous," meaning they don't have weird, infinite wiggles; they just sit there and then jump. This gives us a way to model "jumping" behavior on the real line using the clean, structured math of the p-adic world.

The Magic of the Bridge: Spectra and Eigenvalues

One of the paper's most exciting findings is about the spectra of these diffusion operators. In simple terms, the "spectrum" is like the set of natural frequencies or "notes" a system can play. When the authors translate the p-adic diffusion to the real line, they find that the "notes" (eigenvalues) of the system can be calculated very quickly.

They use a special formula discovered by a mathematician named Kozyrev. Because the p-adic distance is so structured (it's like a perfect tree), the math for finding these frequencies becomes incredibly efficient. The paper shows that for these specific types of operators, you can calculate the eigenvalues with linear complexity. This means if you double the size of your problem, you only double the work, rather than making it exponentially harder. This is a huge win for computers trying to simulate these processes.

From Theory to Pixels: The Algorithm

The authors don't just stop at theory; they build a practical algorithm to visualize these processes. They break the interval $[0, 1]$ into smaller and smaller chunks, creating a hierarchy that mimics the p-adic tree. They then use a method called "breadthwise decomposition" to break a function down into layers, much like peeling an onion or looking at a tree from the trunk out to the leaves.

They tested this with two specific examples:

A Gaussian Kernel: This acts like a bell curve, where the "jumps" are more likely to happen to nearby points (in the p-adic sense) and less likely for far-away points.
A Power Kernel: This uses an inverse power law, which changes how the "jumping" probability decays with distance.

In their simulations, they started with a smooth "hill" of probability (a Gaussian bump) and watched it evolve over time. As time passed, the smooth hill didn't just flatten out; it broke apart into blocky, step-like shapes. The finer details of the shape vanished first, leaving behind a coarse, step-like structure that looked like a digital image with low resolution. This happens because the p-adic rules favor large-scale jumps over tiny, smooth drifts.

What They Don't Claim

It is important to note what this paper does not do. The authors are not claiming to have discovered a new law of physics or to have solved the mystery of how heat moves in the real world. They are not saying that real-world diffusion is p-adic. Instead, they are constructing a specific mathematical model where a p-adic process is transported to the real world to create a new type of random process.

The results regarding the visualization and the convergence of the approximations are demonstrated through simulations and mathematical proofs within the specific context of their defined operators. They show that their method works to approximate the solution to the "Cauchy problem" (a standard way of describing how a system changes from a starting point) for these specific operators. They do not claim that this method works for every possible diffusion process in nature, only for those that can be constructed via this specific Monna map translation.

The Takeaway

In the end, this paper is a beautiful demonstration of how two seemingly disconnected worlds of mathematics can talk to each other. By using the Monna map as a translator, the authors have taken the rigid, tree-like structure of p-adic diffusion and turned it into a set of jumping paths on the real number line. They have provided a fast, efficient way to calculate how these paths behave and a clear way to visualize them. The result is a new tool for mathematicians and scientists to model complex, jump-heavy systems, showing that sometimes, to understand the smooth flow of the real world, you have to look at it through the lens of a fractal tree.

Technical Summary: p-Adic Diffusion and Random Walks on [0, 1]

Problem Statement
The paper addresses the challenge of bridging the gap between $p$ -adic analysis and real analysis within the context of stochastic processes. While $p$ -adic diffusion on the unit disc $\mathbb{Z}_p$ is well-studied, its direct application and visualization on the real unit interval $[0, 1]$ have been limited. The authors aim to construct strong Markov processes on $[0, 1]$ that are "transported" from $p$ -adic counterparts, solve the associated Cauchy problems for heat equations, and provide a method for visualizing these processes. A specific focus is placed on overcoming the computational inefficiency of solving master equations for jump processes when jump rates depend on $p$ -adic distances, leveraging the hierarchical structure of ultrametric spaces.

Methodology
The core methodology relies on the Monna map ( $\rho$ ), a function that maps $p$ -adic numbers to non-negative real numbers by interpreting $p$ -adic expansions as real decimal expansions. The authors utilize this map to establish an isometric isomorphism between the Hilbert spaces $L^2(\mathbb{Z}_p, \mu)$ and $L^2([0, 1], \lambda)$ , where $\mu$ is the Haar measure and $\lambda$ is the Lebesgue measure.

Operator Transport: Integral operators on $\mathbb{Z}_p$ with radial kernels (dependent on the $p$ -adic norm $|\cdot|_p$ ) are transported to the real interval via the Monna map. The resulting operator $D_f$ on $[0, 1]$ is defined by a kernel $k(x, y) = f(|\rho^{-1}(x) - \rho^{-1}(y)|_p)$ .
Spectral Analysis: The authors employ Kozyrev wavelets (generalized to $p > 2$ ) as an orthonormal basis. Because the kernel is radial with respect to the $p$ -adic distance, these wavelets serve as eigenfunctions for the integral operators. This allows for the explicit calculation of eigenvalues using a formula derived from the $p$ -adic theory, avoiding the need for complex numerical diagonalization.
Finite Approximation: To solve the Cauchy problem for the heat equation $\partial_t u = D_f u$ , the authors discretize the unit interval using regular partitions. They define transition rate matrices based on averaging the kernel over partition elements. Theorem 3.3 establishes that the solutions to these discrete Cauchy problems converge uniformly to the solution of the continuous problem as the partition size increases.
Algorithmic Visualization: An algorithm is proposed to visualize the diffusion process. Instead of a full wavelet decomposition, the method utilizes a breadthwise decomposition. This exploits the fact that for radially symmetric kernels, all wavelets with supports of the same size share the same eigenvalue. The algorithm computes local averages on a hierarchical tree structure (representing $\mathbb{Z}_p/p^d\mathbb{Z}_p$ ) to decompose the initial function, applies the time evolution via the heat kernel (exponential of eigenvalues), and recombines the components.

Key Contributions and Results

Theoretical Equivalence: The paper proves that the semigroup generated by the transported operator $D_f$ on $[0, 1]$ is unitarily equivalent to the semigroup generated by the $p$ -adic operator $P_f$ on $\mathbb{Z}_p$ . Consequently, the pull-back of the real Markov process $X_t$ via the Monna map yields a strong Markov process on $\mathbb{Z}_p$ .
Eigenvalue Formula: An explicit formula for the eigenvalues $\lambda_r$ of the diffusion operator is provided, derived from the $p$ -adic wavelet theory. This formula depends on the kernel function $f$ and the $p$ -adic scale $p^{-r}$ .
Convergence of Discretization: The authors demonstrate that solutions to the master equation on finite sets (derived from the hierarchical partition of $[0, 1]$ ) converge uniformly to the solution of the continuous heat equation. This validates the use of discrete random walks to approximate $p$ -adic diffusion on the real line.
Computational Efficiency: By utilizing the breadthwise decomposition and the specific eigenvalue structure of ultrametric kernels, the proposed algorithm achieves linear complexity $O(N)$ in both space and time for computing the evolution of the process, where $N$ is the number of discretization points.
Visualization Examples: The paper provides visualizations of diffusion processes on $[0, 1]$ for two specific kernels: a Gaussian-like kernel and an inverse power-law kernel. These examples illustrate how perturbations on small scales vanish faster than those on larger scales, leading to a step-wise smoothing of the initial function.

Significance and Claims
The paper claims that the Monna map serves as a "translator" between $p$ -adic and real-domain stochastic theories, enabling the transfer of powerful analytical tools (such as wavelet-based spectral decomposition) from the $p$ -adic domain to the real domain.

The authors emphasize that this approach provides a simple visualization method for $p$ -adic diffusion processes, which are otherwise abstract. By transporting the process to $[0, 1]$ , one can observe the dynamics of ultrametric diffusion using standard real-variable intuition. Furthermore, the work highlights the utility of the Monna map in establishing an isometric isomorphism between $L^2$ spaces, a connection that facilitates the application of $p$ -adic spectral methods to real-valued problems, such as the approximation of diffusion on graphs and the analysis of Deep Neural Networks (referencing recent work [28]).

The paper modestly frames its contribution as updating the understanding of the Monna map's role, moving beyond sporadic interest to a systematic application in constructing Markov processes and solving Cauchy problems. It does not propose new physical applications but rather establishes the mathematical machinery to transport existing $p$ -adic diffusion models to the real unit interval for analysis and visualization.