PreprintCC BY-SA 4.0arXiv
The Ornstein−Uhlenbeck process on 𝒫2 with a volatility operator
Abstract
We analyze diffusion processes (μt)t≥ 0 on the $2$-Wasserstein space 𝒫2 over ℝd, which can be classified as Ornstein--Uhlenbeck processes by means of their Fukushima decomposition. Characteristically for this class, $ | μ_t|_2^2-|μ_0|_2^2-2ct+2\int_0 ^t|μ_s|_2^2\,d s,\, t\geq 0,$ is a martingale, where c∈(0,∞) is a constant and |μt|2:=(∫ℝdxT xμt(d x ))1/2 denotes the second moment. Making a particular choice from this class, we obtain a process (μt)t≥ 0 such that all moments exist and μt(θ)−μ0(θ)−∫0 tμs(Δθ−⟨ idℝd,∇ θ⟩)d s, t≥ 0, is a martingale for each multivariate polynomial θ:ℝd→ℝ.
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BibTeX
@misc{grothaus2026ornsteinuhlenbeck,
title = {{The Ornstein$-$Uhlenbeck process on $\mathscr P_2$ with a volatility operator}},
author = {Martin Grothaus and Simon Wittmann},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.14917},
eprint = {2606.14917},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.14917},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - The Ornstein−Uhlenbeck process on 𝒫_2 with a volatility operator
AU - Martin Grothaus
AU - Simon Wittmann
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2606.14917
UR - https://arxiv.org/abs/2606.14917
AB - We analyze diffusion processes (μ_t)_t≥ 0 on the $2$-Wasserstein space 𝒫_2 over ℝ^d, which can be classified as Ornstein--Uhlenbeck processes by means of their Fukushima decomposition. Characteristically for this class, $ | μ_t|_2^2-|μ_0|_2^2-2ct+2\int_0 ^t|μ_s|_2^2\,d s,\, t\geq 0,$ is a martingale, where c∈(0,∞) is a constant and |μ_t|_2:=(∫_ℝ^dx^T xμ_t(d x ))^1/2 denotes the second moment. Making a particular choice from this class, we obtain a process (μ_t)_t≥ 0 such that all moments exist and μ_t(θ)−μ_0(θ)−∫_0 ^tμ_s(Δθ−⟨ id_ℝ^d,∇ θ⟩)d s, t≥ 0, is a martingale for each multivariate polynomial θ:ℝ^d→ℝ.
ER -