Brownian motion in Minkowski normed spaces
Abstract
A Minkowski normed space is the Euclidean space equipped with a (possibly asymmetric) uniformly convex and smooth norm, forming a particular class of Finsler manifolds. We construct a stochastic process with one-dimensional time marginal densities given by the fundamental solution to the nonlinear Finsler heat equation in Minkowski normed spaces. This process is constructed as a solution to a singular McKean--Vlasov stochastic differential equation and constitutes a nonlinear Markov process in the sense of McKean. We show that solutions to this stochastic differential equation are pathwise unique, and, thus, probabilistically strong solutions, though the equation has singular coefficients beyond the subcritical regime. Since our construction is a natural extension of the construction of standard Brownian motion from the standard heat kernel, we call this process \emph{Brownian motion in Minkowski normed spaces.} Furthermore, we obtain a universality property, namely that whichever Minkowski norm is chosen, the radial part of this process is the standard Bessel process. To the best of our knowledge, this is the first construction of stochastic processes associated with nonlinear heat equation in Finslerian spaces.
Related
- Geometric functionals of Brownian motion on Hermitian symmetric spaces of non-compact type
- Lipschitz clustering in normed spaces and barycentric spaces
- The Strong Mankiewicz Property for Closed Convex Body in Normed Spaces
- Soft normed linear spaces via soft elements: decomposition, convergence, and operators
- Lpμ-Normed Spaces of Bounded Multilinear Operator
BibTeX
@misc{ohta2026brownian,
title = {{Brownian motion in Minkowski normed spaces}},
author = {Shin-ichi Ohta and Marco Rehmeier and Kohei Suzuki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.00800},
eprint = {2607.00800},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.00800},
}
RIS
TY - UNPB
TI - Brownian motion in Minkowski normed spaces
AU - Shin-ichi Ohta
AU - Marco Rehmeier
AU - Kohei Suzuki
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2607.00800
UR - https://arxiv.org/abs/2607.00800
AB - A Minkowski normed space is the Euclidean space equipped with a (possibly asymmetric) uniformly convex and smooth norm, forming a particular class of Finsler manifolds. We construct a stochastic process with one-dimensional time marginal densities given by the fundamental solution to the nonlinear Finsler heat equation in Minkowski normed spaces. This process is constructed as a solution to a singular McKean--Vlasov stochastic differential equation and constitutes a nonlinear Markov process in the sense of McKean. We show that solutions to this stochastic differential equation are pathwise unique, and, thus, probabilistically strong solutions, though the equation has singular coefficients beyond the subcritical regime. Since our construction is a natural extension of the construction of standard Brownian motion from the standard heat kernel, we call this process \emph{Brownian motion in Minkowski normed spaces.} Furthermore, we obtain a universality property, namely that whichever Minkowski norm is chosen, the radial part of this process is the standard Bessel process. To the best of our knowledge, this is the first construction of stochastic processes associated with nonlinear heat equation in Finslerian spaces.
ER -