arXiv subjects

Tyler Pierce

Publications and source records attributed to Tyler Pierce.

2 recordsLinked to original sources

arXiv2024

Brownian Motion in the $p$-Adic Integers is a Limit of Discrete Time Random Walks

Vladimirov defined an operator on balls in $\mathbb Q_p$, the $p$-adic numbers, that is analogous to the Laplace operator in the real setting. Kochubei later provided a probabilistic interpretation of the operator. This Vladimirov-Kochubei operator generates a real-time diffusion process in the ring of $p$-adic integers, a Brownian motion in $\mathbb Z_p$. The current work shows that this process is a limit of discrete time random walks. It motivates the construction of the Vladimirov-Kochubei operator, provides further intuition about the properties of ultrametric diffusion, and gives an example of the weak convergence of stochastic processes in a profinite group.

math.PR↗

arXiv2024

Brownian Motion in a Vector Space over a Local Field is a Scaling Limit

For any natural number $d$, the Vladimirov-Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a $d$-dimensional vector space $V$ over a local field $K$. Just as the Laplace operator on $L^2(\mathbb R^d)$ is the infinitesimal generator of Brownian motion with state space $\mathbb R^d$, the Vladimirov-Taibleson operator on $L^2(V)$ is the infinitesimal generator of real-time Brownian motion with state space $V$. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted $V$ to be the $p$-adic numbers.

math.PR↗