Let \(\mathcal {X}\) be a separable Hilbert space with norm \(\left \|{\cdot }\right \|\) and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on \(\mathcal {X}\), let \(F:\mathcal {X}\rightarrow \mathcal {X}\) be a (smooth enough) function and let W(t) be a \(\mathcal {X}\)-valued cylindrical Wiener process. For α ∈ [0, 1/2] we consider the operator \(A:=-(1/2)Q^{2\alpha -1}:Q^{1-2\alpha }(\mathcal {X})\subseteq \mathcal {X}\rightarrow \mathcal {X}\). We are interested in the mild solution X(t, x) of the semilinear stochastic partial differential equation

and in its associated transition semigroup

where \(B_{b}(\mathcal {X})\) is the space of the bounded and Borel measurable functions. We will show that under suitable hypotheses on Q and F, P(t) enjoys regularizing properties, along a continuously embedded subspace of \(\mathcal {X}\). More precisely there exists K := K(F, T) > 0 such that for every \(\varphi \in B_{b}(\mathcal {X})\), \(x\in \mathcal {X}\), t ∈ (0, T] and \(h\in Q^{\alpha }(\mathcal {X})\) it holds

令\(\mathcal {X}\)为范数\(\left \|{\cdot }\right \|\)的可分希尔伯特空间，并令T > 0。令Q为线性的、自伴随的、正的，跟踪\(\mathcal {X}\)上的类运算符，让\(F:\mathcal {X}\rightarrow \mathcal {X}\)是一个（足够平滑的）函数，让W ( t ) 是一个\ (\mathcal {X}\) -值圆柱维纳过程。对于α ∈ [0, 1/2] 我们考虑算子\(A:=-(1/2)Q^{2\alpha -1}:Q^{1-2\alpha }(\mathcal {X} )\subseteq \mathcal {X}\rightarrow \mathcal {X}\)。我们对温和的溶液X (t , x ) 的半线性随机偏微分方程

及其相关的过渡半群

其中\(B_{b}(\mathcal {X})\)是有界函数和 Borel 可测函数的空间。我们将证明在Q和F 的合适假设下，P ( t ) 沿着\(\mathcal {X}\)的连续嵌入子空间具有正则化特性。更准确地说，存在K := K ( F , T ) > 0 使得对于每个\(\varphi \in B_{b}(\mathcal {X})\)，\(x\in \mathcal {X}\ ) , t ∈ (0, T ] and \(h\in Q^{\alpha }(\mathcal {X})\)成立