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Bignamini, D. A.; Ferrari, S.
Potential analysis, 2023/1, Volume: 58, Issue: 1Journal Article
Let X be a separable Hilbert space with norm ⋅ and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on X , let F : X → X be a (smooth enough) function and let W ( t ) be a X -valued cylindrical Wiener process. For α ∈ 0, 1/2 we consider the operator A : = − ( 1 / 2 ) Q 2 α − 1 : Q 1 − 2 α ( X ) ⊆ X → X . We are interested in the mild solution X ( t , x ) of the semilinear stochastic partial differential equation d X ( t , x ) = A X ( t , x ) + F ( X ( t , x ) ) d t + Q α d W ( t ) , t ∈ ( 0 , T ; X ( 0 , x ) = x ∈ X , and in its associated transition semigroup P ( t ) φ ( x ) : = E φ ( X ( t , x ) ) , φ ∈ B b ( X ) , t ∈ 0 , T , x ∈ X ; where B b ( 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 X . More precisely there exists K := K ( F , T ) > 0 such that for every φ ∈ B b ( X ) , x ∈ X , t ∈ (0, T and h ∈ Q α ( X ) it holds | P ( t ) φ ( x + h ) − P ( t ) φ ( x ) | ≤ K t − 1 / 2 ∥ Q − α h ∥ .
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