On smooth approximations in the Wasserstein space

• Main Authors: Cosso, Andrea, Martini, Mattia
• Format: Journal Article
• Language: English
• Published: 11-08-2023
• Subjects: Mathematics - Analysis of PDEs; Mathematics - Probability
• DOI: 10.48550/arxiv.2303.15160

Electron. Commun. Probab. 28: 1-11 (2023) In this paper we investigate the approximation of continuous functions on the
Wasserstein space by smooth functions, with smoothness meant in the sense of
Lions differentiability. In particular, in the case of a Lipschitz function we
are able to construct a sequence of infinitely differentiable functions having
the same Lipschitz constant as the original function. This solves an open
problem raised in [11]. For (resp. twice) continuously differentiable function,
we show that our approximation also holds for the first-order derivative (resp.
second-order derivatives), therefore solving another open problem raised in
[11].