Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem
Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem
复制标题
可分离 Banach 空间中的随机卷积和随机线性柯西问题
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
J. Neerven
中科院分区：
文献类型：
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作者：
Z. Brzeźniak;J. Neerven
设H是可分的实Hilbert空间，E是可分的实Banach空间。本文在CameronMartin空间H上建立了关于柱面维纳过程{WHt}t−[0，T]的L(H，E)∈[0，T]的随机卷积的一般理论，并利用这个理论得到了随机抽象柯西问题(ACP)dXt=Axt dt+B dHt(t∈[0，T])，X0=0几乎必然存在弱解的充要条件，其中A是E和B上有界线性算子的C0−半群{S(T)}t≥0的生成元，B∈L(H，E)是有界线性算子。进一步证明了当弱解存在时，它是唯一的，且由随机卷积给出
Let H be a separable real Hilbert space and let E be a separable real Banach space. In this paper we develop a general theory of stochastic convolution of L(H,E)− valued functions with respect to a cylindrical Wiener process {WH t }t∈[0,T ] with CameronMartin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP ) dXt = AXt dt+B dW H t (t ∈ [0, T ]), X0 = 0 almost surely, where A is the generator of a C0−semigroup {S(t)}t≥0 of bounded linear operators on E and B ∈ L(H,E) is a bounded linear operator. We further show that whenever a weak solution exists, it is unique, and given by a stochastic convolution