论文标题
强$ l^2 H^2 $ fokker-Planck方程的JKO方案的收敛
Strong $L^2 H^2$ convergence of the JKO scheme for the Fokker-Planck equation
论文作者
论文摘要
Following a celebrated paper by Jordan, Kinderleherer and Otto it is possible to discretize in time the Fokker-Planck equation $\partial_t\varrho=Δ\varrho+\nabla\cdot(ρ\nabla V)$ by solving a sequence of iterated variational problems in the Wasserstein space, and the sequence of piecewise constant curves obtained from已知该方案会收敛到连续PDE的溶液。这种融合在瓦斯坦斯坦空间中价值的时间均匀，并且在时空的$ l^1 $中也很强。在本文中，我们证明了在域（一个有界和平稳的凸域）以及最初的数据（应该从零和无穷大的范围内，并且属于$ w^{1，p} $的指标$ p $比维度大于$ l^2_th^2_t^2_2_2_2_x $ s的强大的序列，属于$ w^{1，p} $，属于$ w^{1，p} $，属于$ w^{1，p $，在本文中，我们证明了这一点。 空间。该技术基于一些使用最佳运输技术获得的一些不平等，可以根据近似解决方案的离散序列证明，并模仿相应的连续计算。
Following a celebrated paper by Jordan, Kinderleherer and Otto it is possible to discretize in time the Fokker-Planck equation $\partial_t\varrho=Δ\varrho+\nabla\cdot(ρ\nabla V)$ by solving a sequence of iterated variational problems in the Wasserstein space, and the sequence of piecewise constant curves obtained from the scheme is known to converge to the solution of the continuous PDE. This convergence is uniform in time valued in the Wasserstein space and also strong in $L^1$ in space-time. We prove in this paper, under some assumptions on the domain (a bounded and smooth convex domain) and on the initial datum (which is supposed to be bounded away from zero and infinity and belong to $W^{1,p}$ for an exponent $p$ larger than the dimension), that the convergence is actually strong in $L^2_tH^2_x$, hence strongly improving the previously known results in terms of the order of derivation in space. The technique is based on some inequalities, obtained with optimal transport techniques, that can be proven on the discrete sequence of approximate solutions, and that mimic the corresponding continuous computations.
