解析Ornstein-Uhlenbeck算子的L^p-Poincaré不等式
The L^p-Poincaré inequality for analytic Ornstein-Uhlenbeck operators
摘要：
考虑线性随机发展方程dU（t）= AU（t）+ dW_H（t），t \ge 0，其中A在巴拿赫空间E上生成C_0-半群，W_H是E的连续嵌入Hilbert子空间H中的柱布朗运动.在此方程的解有不变测度μ_ \infty的假设下，我们证明了：若Ornstein-Uhlenbeck半群是解析的且有紧预解式，则Poincaré不等式\overline f \n_{L^p(E,μ_\infty)} \le \n D_H f \n_{L^p(E,μ_\infty)}对所有1<p< \infty成立.这里，\overline f表示f相对于μ_ \infty的平均值，D_H表示H方向上的Fréchet导数。
英文摘要：
Consider the linear stochastic evolution equation dU(t) = AU(t) + dW_H(t), t\ge 0, where A generates a C_0-semigroup on a Banach space E and W_H is a cylindrical Brownian motion in a continuously embedded Hilbert subspace H of E. Under the assumption that the solutions to this equation admit an invariant measure μ_\infty we prove that if the associated Ornstein-Uhlenbeck semigroup is analytic and has compact resolvent, then the Poincaré inequality \n f - \overline f\n_{L^p(E,μ_\infty)} \le \n D_H f\n_{L^p(E,μ_\infty)} holds for all 1<p<\infty. Here \overline f denotes the average of f with respect to μ_\infty and D_H the Fréchet derivative in the direction of H.
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