Long-time nonequilibrium fluctuations of the hard-sphere gas (finite-dimensional laws)

For a deterministic grand-canonical hard-sphere gas in $\mathbb R^3$ with smooth initial density having summable Gaussian bounds on it and its gradient, on every $[0,T]$ where the Boltzmann solution has uniform Gaussian velocity decay, the empirical measure centered at its exact microscopic expectation and scaled by $\sqrt{\mu_\varepsilon}$ has finite-dimensional distributions converging to those of the Gaussian solution of the linear fluctuating Boltzmann equation. It does not prove process-level convergence or tightness, does not treat periodic domains (the BGSS setting) or potentials, and assumes the regular kinetic solution exists.

For a deterministic grand-canonical hard-sphere gas in $\mathbb R^3$ with smooth initial density having summable Gaussian bounds on it and its gradient, on every $[0,T]$ where the Boltzmann solution has uniform Gaussian velocity decay, the empirical measure centered at its exact microscopic expectation and scaled by $\sqrt{\mu_\varepsilon}$ has finite-dimensional distributions converging to those of the Gaussian solution of the linear fluctuating Boltzmann equation. It does not prove process-level convergence or tightness, does not treat periodic domains (the BGSS setting) or potentials, and assumes the regular kinetic solution exists.