Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise

We consider stochastic reaction-diffusion equations with colored noise and prove Schauder type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup, defined on the Banach space of bounded and uniformly continuous functions

Log-Sobolev inequalities and hypercontractivity for Ornstein-Uhlenbeck evolution operators in infinite dimensions

In an infinite dimensional separable Hilbert space $X$, we study the realizations of Ornstein-Uhlenbeck evolution operators \pst in the spaces L^p(X,\g_t), \{\g_t\}_{t\in\R} being the unique evolution system of measures for \pst in $\R$. We prove hyperconctractivity results, relying on suitable Log-Sobolev estimates. Among the examples we consider the transition evolution operator of a non autonomous stochastic parabolic PDE

Differentiability in infinite dimension and the Malliavin calculus

In this paper we study two notions of differentiability introduced by P. Cannarsa and G. Da Prato (see [28]) and L. Gross (see [56]) in both the framework of infinite dimensional analysis and the framework of Malliavin calculus

Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces

Let $\mathcal{X}$ be a separable Hilbert space with norm $\|\cdot\|$ 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 $\alpha\in [0,1/2]$ 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 \begin{gather*} \left\{\begin{array}{ll} dX(t,x)=\big(AX(t,x)+F(X(t,x))\big)dt+ Q^{\alpha}dW(t), & t\in(0,T];\\ X(0,x)=x\in \mathcal{X}, \end{array}\right. \end{gather*} and in its associated transition semigroup \begin{align*} P(t)\varphi(x):=\mathbb{E}[\varphi(X(t,x))], \qquad \varphi\in B_b(\mathcal{X}),\ t\in[0,T],\ x\in \mathcal{X}; \end{align*} 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 $\varphi\in B_b(\mathcal{X})$, $x\in \mathcal{X}$, $t\in(0,T]$ and $h\in Q^\alpha(\mathcal{X})$ it holds \[|P(t)\varphi(x+h)-P(t)\varphi(x)|\leq Kt^{-1/2}\|Q^{-\alpha}h\|.\

Novel decay dynamics revealed for virus-mediated drug activation in cytomegalovirus infection

Human cytomegalovirus (CMV) infection is a substantial cause of morbidity and mortality in immunocompromised hosts and globally is one of the most important congenital infections. The nucleoside analogue ganciclovir (GCV), which requires initial phosphorylation by the viral UL97 kinase, is the mainstay for treatment. To date, CMV decay kinetics during GCV therapy have not been extensively investigated and its clinical implications not fully appreciated. We measured CMV DNA levels in the blood of 92 solid organ transplant recipients with CMV disease over the initial 21 days of ganciclovir therapy and identified four distinct decay patterns, including a new pattern exhibiting a transient viral rebound (Hump) following initial decline. Since current viral dynamics models were unable to account for this Hump profile, we developed a novel multi-level model, which includes the intracellular role of UL97 in the continued activation of ganciclovir, that successfully described all the decline patterns observed. Fitting the data allowed us to estimate ganciclovir effectiveness in vivo (mean 92%), infected cell half-life (mean 0.7 days), and other viral dynamics parameters that determine which of the four kinetic patterns will ensue. An important clinical implication of our results is that the virological efficacy of GCV operates over a broad dose range. The model also raises the possibility that GCV can drive replication to a new lower steady state but ultimately cannot fully eradicate it. This model is likely to be generalizable to other anti-CMV nucleoside analogs that require activation by viral enzymes such as UL97 or its homologues

Simulating the WFXT sky

We investigate the scientific impact of the Wide Field X-ray Telescope mission. We present simulated images and spectra of X-ray sources as observed from the three surveys planned for the nominal 5-year WFXT lifetime. The goal of these simulations is to provide WFXT images of the extragalactic sky in different energy bands based on accurate description of AGN populations, normal and star forming galaxies, groups and clusters of galaxies. The images are realized using a detailed PSF model, instrumental and physical backgrounds/foregrounds, accurate model of the effective area and the related vignetting effect. Thanks to this comprehensive modelization of the WFXT properties, the simulated images can be used to evaluate the flux limits for detection of point and extended sources, the effect of source confusion at very faint fluxes, and in general the efficiency of detection algorithms. We also simulate the spectra of the detected sources, in order to address specific science topics which are unique to WFXT. Among them, we focus on the characterization of the Intra Cluster Medium (ICM) of high-z clusters, and in particular on the measurement of the redshift from the ICM spectrum in order to build a cosmological sample of galaxy clusters. The end-to-end simulation procedure presented here, is a valuable tool in optimizing the mission design. Therefore, these simulations can be used to reliably characterize the WFXT discovery space and to verify the connection between mission requirements and scientific goals. Thanks to this effort, we can conclude on firm basis that an X-ray mission optimized for surveys like WFXT is necessary to bring X-ray astronomy at the level of the optical, IR, submm and radio wavebands as foreseen in the coming decade.Comment: "Proceedings of "The Wide Field X-ray Telescope Workshop", held in Bologna, Italy, Nov. 25-26 2009. To appear in Memorie della Societa Astronomica Italiana 2010 (arXiv:1010.5889)

On generators of transition semigroups associated to semilinear stochastic partial differential equations

Let $\mathcal{X}$ be a real separable Hilbert space. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $\mathcal{X}$, let $F:\mathcal{X}\rightarrow\mathcal{X}$ be a (smooth enough) function and let $\{W(t)\}_{t\geq 0}$ be a $\mathcal{X}$-valued cylindrical Wiener process. For $\alpha\in [0,1/2]$ 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 \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)=x\in \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 \varphi\in B_b(\mathcal{X}),\ t\geq 0,\ x\in \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 $$\lambda u-N_2 u=f,\qquad \lambda>0,\ f\in L^2(\mathcal{X},\nu);$$ where $N_2$ is the infinitesimal generator of $P(t)$ in $L^2(\mathcal{X},\nu)$