Abstract

We study the relationship between random processes with values in Euclidian space \(\mathbb{R}^{d}\) and operator valued functions \(\mathbb{R}_{+}\to B(H)\), where \(H=L_{2}(\mathbb{R}^{d})\) and \(B(H)\) is the Banach space of bounded linear operators in the space \(H\). A random process with values in the space \(\mathbb{R}^{d}\) can be presented by the pseudomeasures on the algebra \(\mathcal{A}_{\mathcal{C}}\) of cylindrical subsets of the space of mappings \(\mathbb{R}_{+}\to\mathbb{R}^{d}\). The mapping of the locally convex space of complex valued pseudomeasures on algebra \(\mathcal{A}_{\mathcal{C}}\) into the locally convex space \(M(\mathbb{R}_{+},B(H))\) of operator valued functions and its invertable restrictions on some subspaces are investigated. The iterations of the random processes (or the iterations of pseudomeasures) are defined by the preimage of the iteration of operator valued functions of the above inverse mapping. The semigroup property for operator valued functions and the Markovian property for random processes are obtained by taking the limit for iterations in the framework of Chernoff theorem for operator valued functions and its analogue for pseudomeasures.

中文翻译：

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圆柱伪测度的切尔诺夫定理的类比

摘要

我们研究了具有欧几里得空间\（\ mathbb {R} ^ {d} \）中的值的随机过程与运算符值函数\（\ mathbb {R} _ {+} \ to B（H）\）之间的关系。\（H = L_ {2}（\ mathbb {R} ^ {d}）\）和\（B（H）\）是空间\（H \）中有界线性算子的Banach空间。可以通过在圆柱子集的代数\（\ mathcal {A} _ {\ mathcal {C}}} \上的伪度量来表示在值空间为\（\ mathbb {R} ^ {d} \）中的随机过程映射空间\（\ mathbb {R} _ {+} \至\ mathbb {R} ^ {d} \）的空间。复数值伪测度的局部凸空间在代数\（\ mathcal {A} _ {\ mathcal {C}}} \）上的映射研究了算子值函数的局部凸空间\（M（\ mathbb {R} _ {+}，B（H））\）及其在某些子空间上的可逆约束。随机过程的迭代（或伪测量的迭代）由上述逆映射的算子值函数的迭代的原像定义。算子值函数的半群性质和随机过程的马尔可夫性质是通过对算子值函数的切尔诺夫定理及其伪度量的模拟取极限来获得的。