论文标题
具有线性漂移的Fokker-Planck方程的繁殖标准和尖锐的衰减估计值
Propagator norm and sharp decay estimates for Fokker-Planck equations with linear drift
论文作者
论文摘要
我们关注的是，$ l^2 $ - propagator的短期行为具有线性漂移的fokker-planck方程，即$ \ partial_t f = \ mathrm {div} _ {x} _ {x}} {（d \ nabla_x f+cxf）} $。通过坐标转换，这些方程式可以标准化，以便将扩散和漂移矩阵链接为$ d = c_s $，$ c $的对称部分。本文的主要结果（定理3.4）是归一化的fokker-planck方程与它们的漂移 - dot $ \ dot x = -cx $之间的连接：他们的$ l^2 $ propagator norms实际上是一致的。这意味着对漂移盘的最佳衰减估计值（W.R.T.最大指数衰减率和最小乘法常数）将fokker-Planck溶液的急剧指数衰减估计值延伸到稳态。该定理的第二次应用是解决方案的短时间行为：短时间正规化（在某些加权Sobolev空间中）由其低调指数确定，最近已针对Fokker-Planck方程式和ODES引入了该指数（请参见[5，1，2]）。在证明中，我们意识到，每个不变光谱子空间中的演变可以表示为相应漂移对应的明确给出的，张开的版本。实际上，Fokker-Planck方程甚至可以视为$ \ dot x = -cx $的第二个量化。
We are concerned with the short- and large-time behavior of the $L^2$-propagator norm of Fokker-Planck equations with linear drift, i.e. $\partial_t f=\mathrm{div}_{x}{(D \nabla_x f+Cxf)}$. With a coordinate transformation these equations can be normalized such that the diffusion and drift matrices are linked as $D=C_S$, the symmetric part of $C$. The main result of this paper (Theorem 3.4) is the connection between normalized Fokker-Planck equations and their drift-ODE $\dot x=-Cx$: Their $L^2$-propagator norms actually coincide. This implies that optimal decay estimates on the drift-ODE (w.r.t. both the maximum exponential decay rate and the minimum multiplicative constant) carry over to sharp exponential decay estimates of the Fokker-Planck solution towards the steady state. A second application of the theorem regards the short time behaviour of the solution: The short time regularization (in some weighted Sobolev space) is determined by its hypocoercivity index, which has recently been introduced for Fokker-Planck equations and ODEs (see [5, 1, 2]). In the proof we realize that the evolution in each invariant spectral subspace can be represented as an explicitly given, tensored version of the corresponding drift-ODE. In fact, the Fokker-Planck equation can even be considered as the second quantization of $\dot x=-Cx$.
