For second-order Langevin particle systems, the associated Dean-Kawasaki SPDE is well posed exactly for atomic, empirical-measure initial data and provably ill posed for smooth initial data.
Multilevel Monte Carlo methods for the Dean-Kawasaki equation from fluctuating hydrodynamics.SIAM J
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Well-Posedness for Dean-Kawasaki Models of Vlasov-Fokker-Planck Type
For second-order Langevin particle systems, the associated Dean-Kawasaki SPDE is well posed exactly for atomic, empirical-measure initial data and provably ill posed for smooth initial data.