Variational formulation of the Fokker-Planck equation with decay: a particle approach (Q2859242)

The authors consider iterative variational schemes defined on a space \( \mathcal{X}\) as follows: given \(\rho ^{k-1}\), find \(\rho ^{k}\in \arg \min_{\rho \in \mathcal{X}}\mathcal{K}^{h}(\rho \mid \rho ^{k-1})\) where \( \mathcal{K}^{h}\) is some functional. For example if \(\mathcal{X}\) is a Hilbert space and \(\mathcal{K}^{h}\) is defined through \(\mathcal{K}^{h}(\rho \mid \overline{\rho })=\mathcal{F}(\rho )+\frac{1}{2h}\left\| \rho - \overline{\rho }\right\| ^{2}\) then \(\rho ^{k}\) satisfies \(\frac{\rho ^{k}-\rho ^{k-1}}{h}=-\mathrm{grad}\mathcal{F}(\rho ^{k})\), which is the backward Euler approximation of the evolution equation \(\partial _{t}\rho =- \mathrm{grad}\mathcal{F}(\rho )\). Now taking \(\mathcal{X}=\mathcal{P}_{2}( \mathbb{R}^{d})\) the space of probability measures on \(\mathbb{R}^{d}\) with finite second moment and \(\mathcal{K}_{FP}^{h}(\rho \mid \overline{\rho })= \frac{1}{2}\mathcal{F}(\rho )-\frac{1}{2}\mathcal{F}(\overline{\rho })+\frac{ 1}{4h}d^{2}(\overline{\rho },\rho )\) where \(\mathcal{F}\) is the Helmholtz free energy (sum of the negative Gibbs-Boltzmann entropy and of an energy arising from a potential \(\Psi \)) and \(d\) is Wasserstein's metric, the above minimization problem leads to an approximation of the Fokker-Planck equation \(\partial _{t}u=\Delta u+\mathrm{div}(u\nabla \Psi )\), in \(\mathbb{R} ^{d}\times (0,\infty )\), see the paper by \textit{R. Jordan} et al. [SIAM J. Math. Anal. 29, No. 1, 1--17 (1998; Zbl 0915.35120)].NEWLINENEWLINEIn the present paper, the authors first derive a large-deviation principle for a system of Brownian particles with drift but without decay, which models the above Fokker-Planck equation. They also derive a large-deviation principle for a particle system which models the evolution equation \(\partial _{t}u=\Delta u-\lambda u\) on \(\mathbb{R} ^{d}\times (0,\infty )\). The authors start gathering the tools which are necessary for their study: Wasserstein's distance between measures in \( \mathcal{P}_{2}(\mathbb{R}^{d})\), large-deviations for random variables and Mosco convergence. For the description of the asymptotic behaviours, the authors assume a conjecture on the Mosco convergence of a quantity involving the decay of convergence and the Wasserstein distance.