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Well-Posedness of the Dean–Kawasaki and the Nonlinear Dawson–Watanabe Equation with Correlated Noise

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Abstract

In this paper we prove the well-posedness of the generalized Dean–Kawasaki equation driven by noise that is white in time and colored in space. The results treat diffusion coefficients that are only locally \({1}/{2}\)-Hölder continuous, including the square root. This solves several open problems, including the well-posedness of the Dean–Kawasaki equation and the nonlinear Dawson–Watanabe equation with correlated noise.

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Notes

  1. In the introduction, we formally write \(\partial _t\rho \) and \(\dot{\xi }^F\) to denote the time-derivative of the stochastic processes \(\rho \) and \(\xi ^F\). In the remainder of the paper, we will use the probabilistic notation \(\, \textrm{d} \rho \) and \(\, \textrm{d} \xi ^F\).

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Acknowledgements

The first author acknowledges financial support from the EPSRC through the EPSRC Early Career Fellowship EP/V027824/1. The second author is co-funded by the European Union (ERC, FluCo, 101088488). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

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  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70802, USA

    Benjamin Fehrman

  2. Max Planck Institute for Mathematics in the Sciences, Leipzig, 04103, Germany

    Benjamin Gess

  3. Fakultät für Mathematik, Universität Bielefeld, Bielefeld, 33615, Germany

    Benjamin Gess

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  1. Benjamin Fehrman
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Fehrman, B., Gess, B. Well-Posedness of the Dean–Kawasaki and the Nonlinear Dawson–Watanabe Equation with Correlated Noise. Arch Rational Mech Anal 248, 20 (2024). https://doi.org/10.1007/s00205-024-01963-3

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