On the Bakry-Emery criterion for linear diffusions and weighted porous media equations
On the Bakry-Emery criterion for linear diffusions and weighted porous media equations
复制标题
关于线性扩散和加权多孔介质方程的 Bakry-Emery 准则
DOI：
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发表时间：
2007
影响因子：
1
通讯作者：
Giuseppe Savaré
中科院分区：
文献类型：
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作者：
Jean Dolbeault;B. Nazaret;Giuseppe Savaré
给出了广义Poincare不等式的一个非局部充分条件，推广了著名的Bakry-Emery条件. W. Beckner在高斯情形下的结果，并沿着Ornstein-Uhlenbeck流，给出了某些广义熵的指数衰减，这些广义熵在L^2范数和通常熵之间插值。我们的标准改进的结果，例如，可以推导出从Bakry-Emery准则和Holley-Stroock型扰动结果。在第二步中，我们将相同的策略应用于多孔介质类型的非线性方程。这为发展问题的解提供了新的插值不等式和衰减估计。该标准再次是一个非局部条件的基础上的积极性的最低本征值的薛定谔算子。在这两种情况下，我们将Fisher信息与其时间导数联系起来。由于所得到的标准是非本地的，它是更好地适应的潜力，例如，在无穷大的非二次增长，或无界扰动的潜力。
The goal of this paper is to give a non-local sufficient condition for generalized Poincare inequalities, which extends the well-known Bakry-Emery condition. Such generalized Poincare inequalities have been introduced by W. Beckner in the gaussian case and provide, along the Ornstein-Uhlenbeck flow, the exponential decay of some generalized entropies which interpolate between the $L^2$ norm and the usual entropy. Our criterion improves on results which, for instance, can be deduced from the Bakry-Emery criterion and Holley-Stroock type perturbation results. In a second step, we apply the same strategy to non-linear equations of porous media type. This provides new interpolation inequalities and decay estimates for the solutions of the evolution problem. The criterion is again a non-local condition based on the positivity of the lowest eigenvalue of a Schrodinger operator. In both cases, we relate the Fisher information with its time derivative. Since the resulting criterion is non-local, it is better adapted to potentials with, for instance, a non-quadratic growth at infinity, or to unbounded perturbations of the potential.
影响因子：
1
作者：
J. Dolbeault;I. Gentil;A. Jungel
通讯作者：
J. Dolbeault;I. Gentil;A. Jungel
影响因子：
1.7
作者：
A. Jüngel;D. Matthes
通讯作者：
A. Jüngel;D. Matthes