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- W3139313963 abstract "Let $mathcal{X}$ be a real separable Hilbert space. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $mathcal{X}$, let $F:mathcal{X}rightarrowmathcal{X}$ be a (smooth enough) function and let ${W(t)}_{tgeq 0}$ be a $mathcal{X}$-valued cylindrical Wiener process. For $alphain [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2alpha-1}:Q^{1-2alpha}(mathcal{X})subseteqmathcal{X}rightarrowmathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation begin{gather} left{begin{array}{ll} dX(t,x)=big(AX(t,x)+F(X(t,x))big)dt+ Q^{alpha}dW(t), & t>0; X(0,x)=xin mathcal{X}, end{array} right. end{gather} and in its associated transition semigroup begin{align} P(t)varphi(x):=E[varphi(X(t,x))], qquad varphiin B_b(mathcal{X}), tgeq 0, xin mathcal{X}; end{align} where $B_b(mathcal{X})$ is the space of the real-valued, bounded and Borel measurable functions on $mathcal{X}$. In this paper we study the behavior of the semigroup $P(t)$ in the space $L^2(mathcal{X},nu)$, where $nu$ is the unique invariant probability measure of eqref{Tropical}, when $F$ is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincar'e inequalities and we study the maximal Sobolev regularity for the stationary equation [lambda u-N_2 u=f,qquad lambda>0, fin L^2(mathcal{X},nu);] where $N_2$ is the infinitesimal generator of $P(t)$ in $L^2(mathcal{X},nu)$." @default.
- W3139313963 created "2021-03-29" @default.
- W3139313963 creator A5013570193 @default.
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- W3139313963 date "2020-10-08" @default.
- W3139313963 modified "2023-10-16" @default.
- W3139313963 title "On generators of transition semigroups associated to semilinear stochastic partial differential equations" @default.
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- W3139313963 doi "https://doi.org/10.48550/arxiv.2010.03908" @default.
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