Abstract
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 [12]. 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 [12].
Acknowledgments
M. Martini conducted this research during his PhD at Università degli Studi di Milano. Currently, M. Martini is supported by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (ELISA project, Grant agreement nr. 101054746).
Citation
Andrea Cosso. Mattia Martini. "On smooth approximations in the Wasserstein space." Electron. Commun. Probab. 28 1 - 11, 2023. https://doi.org/10.1214/23-ECP538
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