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We study the Hermite operator $H=-\Delta+|x|^2$ in $\mathbb{R}^d$ and itsfractional powers $H^\beta$, $\beta>0$ in phase space. Namely, we representfunctions $f$ via the so-called short-time Fourier, alias Fourier-Wigner orBargmann transform $V_g f$ ($g$ being a fixed window function), and we measuretheir regularity and decay by means of mixed Lebesgue norms in phase space of$V_g f$, that is in terms of membership to modulation spaces $M^{p,q}$, $0<p,q\leq \infty$. We prove the complete range of fixed-time estimates for thesemigroup $e^{-tH^\beta}$ when acting on $M^{p,q}$, for every $0< p,q\leq\infty$, exhibiting the optimal global-in-time decay as well as phase-spacesmoothing. As an application, we establish global well-posedness for thenonlinear heat equation for $H^{\beta}$ with power-type nonlinearity (focusingor defocusing), with small initial data in modulation spaces or in Wieneramalgam spaces. We show that such a global solution exhibits the same optimaldecay $e^{-c t}$ as the solution of the corresponding linear equation, where$c=d^\beta$ is the bottom of the spectrum of $H^\beta$. This is in sharpcontrast to what happens for the nonlinear focusing heat equation withoutpotential, where blow-up in finite time always occurs for (even small) constantinitial data - hence in $M^{\infty,1}$.

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness