We study measures on a real separable Hilbert space ##IMG## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn1.gif] {$E$} that are invariant under translations by arbitrary vectors in ##IMG## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn1.gif] {$E$} . We define the Hilbert space ##IMG## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn2.gif] {$\mathcal H$} of complex-valued functions on ##IMG## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn1.gif] {$E$} square-integrable with respect to some translation-invariant measure ##IMG## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn3.gif] {$\lambda$} . We determine the expectations of the operators of shift by random vectors whose distributions are given by semigroups (with respect to convolution) of Gaussian measures on ##IMG## [http://...] {$E$}

中文翻译：

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Hilbert空间上具有平移不变测度和半群近似的函数的Sobolev空间

我们研究了在真实可分希尔伯特空间## IMG ## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn1.gif] {$ E $}下不变的度量## IMG ## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn1.gif] {$ E $}中任意向量的翻译。我们定义了希尔伯特空间## IMG ## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn2.gif] {$ \ mathcal H $}的复数值函数## IMG ## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn1.gif] {$ E $}关于某些平移不变量度是可平方的## IMG ## [http://ej.iop.org/images/1064-5632/84/4/694/IZV_84_4_694ieqn3.gif] {$ \ lambda $}。我们通过随机矢量确定移位算子的期望，该随机矢量的分布由高斯测度在## IMG ## [http：//上的半群（相对于卷积）给出。