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We consider the stochastic differential equation $$ \left\{ \begin{array}{lc} dX(t)=[AX(t)+F(X(t))]dt+C^{1/2}dW(t), & t>0;\\ X(0)=x \in \mathcal{X}; \end{array}\right. $$ where $\mathcal{X}$ is a Hilbert space, $\{W(t)\}_{t\geq 0}$ is a $\mathcal{X}$-valued cylindrical Wiener process, $A, C$ are suitable operators on $\mathcal{X}$ and $F:{\rm Dom}\,(F)\subseteq \mathcal{X}\to \mathcal{X}$ is a smooth enough function. We establish a logarithmic Harnack inequality for the transition semigroup $\{P(t)\}_{t\geq 0}$ associated with the stochastic problem above, under less restrictive conditions than those considered in the literature. Some applications to these inequalities are also shown.

We consider generalised Mehler semigroups and, assuming the existence of an associated invariant measure $\sigma$, we prove functional integral inequalities with respect to $\sigma$, such as logarithmic Sobolev and Poincar\'{e} type. Consequently, some integrability properties of exponential functions with respect to $\sigma$ are deduced.

In this work we aim to characterise the dust motion in the inner coma of comet 67P/Churyumov- Gerasimenko to provide constraints for theoretical 3D coma models. The OSIRIS camera onboard the Rosetta mission was able for the first time to acquire images of single dust particles from inside the cometary coma, very close to the nucleus. We analyse a large number of particles, performing a significant statistic of their behaviour during the post perihelion period, when the spacecraft covered distances from the nucleus ranging between 80 and 400 km. We describe the particle trajectories, investigating their orientation and finding highly radial motion with respect to the nucleus. Then, from the particle brightness profiles, we derive a particle rotational frequency of v < 3.6 Hz, revealing that they are slow rotators and do not undergo fragmentation. We use scattering models to compare the observed spectral radiance of the particles with the simulated ones in order to estimate their size, finding values that range from millimetres up to centimetres. The statistics performed in this paper provide useful parameters to constrain the cometary coma dynamical models.

Let $X$ be a separable Banach space and let $Q:X^*\rightarrow X$ be a linear, bounded, non-negative and symmetric operator and let $A:D(A)\subseteq X\rightarrow X$ be the infinitesimal generator of a strongly continuous semigroup of contractions on $X$. We consider the abstract Wiener space $(X,\mu_\infty,H_\infty)$ where $\mu_\infty$ is a centred non-degenerate Gaussian measure on $X$ with covariance operator defined, at least formally, as \begin{align*} Q_\infty=\int_0^{+\infty} e^{sA}Qe^{sA^*}ds, \end{align*} and $H_\infty$ is the Cameron--Martin space associated to $\mu_\infty$. Let $H$ be the reproducing kernel Hilbert space associated with $Q$ with inner product $[\cdot,\cdot]_H$. We assume that the operator $Q_\infty A^*:D(A^*)\subseteq X^*\rightarrow X$ extends to a bounded linear operator $B\in \mathcal L(H)$ which satisfies $B+B^*=-{\rm Id}_H$, where ${\rm Id}_H$ denotes the identity operator on $H$. Let $D$ and $D^2$ be the first and second order Fr\'echet derivative operators, we denote by $D_H$ and $D^2_H$ the closure in $L^2(X,\mu_\infty)$ of the operators $QD$ and $QD^2$ and by $W^{1,2}_H(X,\mu_\infty)$ and and $W^{2,2}_H(X,\mu_\infty)$ their domains in $L^2(X,\mu_\infty)$, respectively,. Furthermore, we denote by $D_{A_\infty}$ the closure of the operator $Q_\infty A^*D$ and by $W^{1,2}_{A_\infty}(X,\mu_\infty)$ its domain in $L^2(X,\mu_\infty)$. We characterize the domain of the operator $L$, associated to the bilinear form \begin{align*} (u,v)\mapsto-\int_{X}[BD_Hu,D_Hv]_Hd\mu_\infty, \qquad u,v\in W^{1,2}_H(X,\mu_\infty), \end{align*} in $L^2(X,\mu_\infty)$. More precisely, we prove that $D(L)$ coincides, up to an equivalent remorming, with a subspace of $W^{2,2}_H(X,\mu_\infty)\cap W^{1,2}_{A_\infty}(X,\mu_\infty)$. We stress that we are able to treat the case when $L$ is degenerate and non-symmetric.

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 $L^2(\mathcal{X},\nu)$.

We study functions of bounded variation (and sets of finite perimeter) on a convex open set $\Omega\subseteq X$, $X$ being an infinite dimensional real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein--Uhlenbeck operator.

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 \[|P(t)\varphi(x+h)-P(t)\varphi(x)|\leq Kt^{-1/2}\|Q^{-\alpha}h\|.\]

Let $Bo(T,\tau)$ be the Borel $\sigma$-algebra generated by the topology $\tau$ on $T$. In this paper we show that if $K$ is a Hausdorff compact space, then every subset of $K$ is a Borel set if, and only if, $$Bo(C^*(K),w^*)=Bo(C^*(K),\|\cdot\|);$$ where $w^*$ denotes the weak-star topology and $\|\cdot\|$ is the dual norm with respect to the sup-norm on the space of real-valued continuous functions $C(K)$. Furthermore we study the topological properties of the Hausdorff compact spaces $K$ such that every subset is a Borel set. In particular we show that, if the axiom of choice holds true, then $K$ is scattered.

The southern hemisphere of comet 67P/Churyumov-Gerasimenko (67P) became observable by the Rosetta mission in March 2015, a few months before cometary southern vernal equinox. The Anhur region in the southern part of the comet's larger lobe was found to be highly eroded, enriched in volatiles, and highly active. We analyze high-resolution images of the Anhur region pre- and post-perihelion acquired by the OSIRIS imaging system on board the Rosetta mission. The Narrow Angle Camera is particularly useful for studying the evolution in Anhur in terms of morphological changes and color variations.}{Radiance factor images processed by the OSIRIS pipeline were coregistered, reprojected onto the 3D shape model of the comet, and corrected for the illumination conditions. We find a number of morphological changes in the Anhur region that are related to formation of new scarps; removal of dust coatings; localized resurfacing in some areas, including boulders displacements; and vanishing structures, which implies localized mass loss that we estimate to be higher than 50 million kg. The strongest changes took place in and nearby the Anhur canyon-like structure, where significant dust cover was removed, an entire structure vanished, and many boulders were rearranged. All such changes are potentially associated with one of the most intense outbursts registered by Rosetta during its observations, which occurred one day before perihelion passage. Moreover, in the niche at the foot of a new observed scarp, we also see evidence of water ice exposure that persisted for at least six months. The abundance of water ice, evaluated from a linear mixing model, is relatively high (> 20%). Our results confirm that the Anhur region is volatile-rich and probably is the area on 67P with the most pristine exposures near perihelion., Comment: 19 pages, 16 figures; accepted for publication in Astronomy and Astrophysics for the Rosetta 2 special number

Significant progress has been made in the development of superconducting quantum circuits, however new quantum devices that have longer decoherence times at higher temperatures are urgently required for quantum technologies. Superconducting nanowires with quantum phase slips are promising candidates for use in novel devices that operate on quantum principles. Here, we demonstrate ultra-thin YBa2Cu3O7-x nanowires with phase-slip dynamics and study their switching-current statistics at temperatures below 20 K. We apply theoretical models that were developed for Josephson junctions and show that our results provide strong evidence for energy-level quantization in the nanowires. The crossover temperature to the quantum regime is 12-13 K, while the lifetime in the excited state exceeds 20 ms at 5.4 K. Both values are at least one order of magnitude higher than those in conventional Josephson junctions based on low-temperature superconductors. We also show how the absorption of a single photon changes the phase-slip and quantum state of a nanowire, which is important for the development of single-photon detectors with high operating temperature and superior temporal resolution. Our findings pave the way for a new class of superconducting nanowire devices for quantum sensing and computing.

In April 2016, the Rosetta spacecraft performed a low-altitude low-phase-angle flyby over the Imhotep-Khepry transition of 67P/Churyumov-Gerasimenko's nucleus. The OSIRIS/Narrow-Angle-Camera (NAC) acquired 112 images with mainly 3 broadband filters in the visible at a resolution of up to 0.53 m/px and for phase angles between 0.095{\deg} and 62{\deg}. Using those images, we have investigated the morphological and spectrophotometrical properties of this area. We assembled the images into coregistered color cubes. Using a 3D shape model, we produced the illumination conditions and georeference for each image. We projected the observations on a map to investigate its geomorphology. Observations were photometrically corrected using the Lommel-Seeliger disk law. Spectrophotometric analyses were performed on the coregistered color cubes. These data were used to estimate the local phase reddening. This region of the nucleus hosts numerous and varied types of terrains and features. We observe an association between a feature's nature, its reflectance, and its spectral slope. Fine material deposits exhibit an average reflectance and spectral slope, while terrains with diamictons, consolidated material, degraded outcrops, or features such as somber boulders, present a lower-than-average reflectance and higher-than-average spectral slope. Bright surfaces present here a spectral behavior consistent with terrains enriched in water-ice. We find a phase-reddening slope of 0.064{\pm}0.001{\%}/100nm/{\deg} at 2.7 au outbound, similarly to the one obtained at 2.3 au inbound during the February 2015 flyby. Identified as the source region of multiple jets and a host of water-ice material, the Imhotep-Khepry transition appeared in April 2016, close to the frost line, to further harbor several potential locations with exposed water-ice material among its numerous different morphological terrain units., Comment: 23 pages, 14 figures, 5 tables

The Rosetta space probe accompanied comet 67P/Churyumov-Gerasimenko for more than two years, obtaining an unprecedented amount of unique data of the comet nucleus and inner coma. This work focuses identifying the source regions of faint jets and outbursts and on studying the spectrophotometric properties of some outbursts. We use observations acquired with the OSIRIS/NAC camera during July-October 2015, that is, close to perihelion. More than 200 jets of different intensities were identified directly on the nucleus. Some of the more intense outbursts appear spectrally bluer than the comet dark terrain in the vivible-to-near-infrared region. We attribute this spectral behavior to icy grains mixed with the ejected dust. Some of the jets have an extremely short lifetime. They appear on the cometary surface during the color sequence observations, and vanish in less than some few minutes after reaching their peak. We also report a resolved dust plume observed in May 2016 at a resolution of 55 cm/pixel, which allowed us to estimate an optical depth of $\sim$0.65 and an ejected mass of $\sim$ 2200 kg. We present the results on the location, duration, and colors of active sources on the nucleus of 67P from the medium-resolution (i.e., 6-10 m/pixel) images acquired close to perihelion passage. The observed jets are mainly located close to boundaries between different morphological regions. Jets depart not only from cliffs, but also from smooth and dust-covered areas, from fractures, pits, or cavities that cast shadows and favor the recondensation of volatiles. This study shows that faint jets or outbursts continuously contribute to the cometary activity close to perihelion passage, and that these events are triggered by illumination conditions. Faint jets or outbursts are not associated with a particular terrain type or morphology., Comment: Accepted for publication on Astronomy and Astrophysics on 27 August 2018. 27 pages, 18 figures, 2 tables

The Southern hemisphere of the 67P/Churyumov-Gerasimenko comet has become visible from Rosetta only since March 2015. It was illuminated during the perihelion passage and therefore it contains the regions that experienced the strongest heating and erosion rate, thus exposing the subsurface most pristine material. In this work we investigate, thanks to the OSIRIS images, the geomorphology, the spectrophotometry and some transient events of two Southern hemisphere regions: Anhur and part of Bes. Bes is dominated by outcropping consolidated terrain covered with fine particle deposits, while Anhur appears strongly eroded with elongated canyon-like structures, scarp retreats, different kinds of deposits, and degraded sequences of strata indicating a pervasive layering. We discovered a new 140 m long and 10 m high scarp formed in the Anhur/Bes boundary during/after the perihelion passage, close to the area where exposed CO$_2$ and H$_2$O ices were previously detected. Several jets have been observed originating from these regions, including the strong perihelion outburst, an active pit, and a faint optically thick dust plume. We identify several areas with a relatively bluer slope (i.e. a lower spectral slope value) than their surroundings, indicating a surface composition enriched with some water ice. These spectrally bluer areas are observed especially in talus and gravitational accumulation deposits where freshly exposed material had fallen from nearby scarps and cliffs. The investigated regions become spectrally redder beyond 2 au outbound when the dust mantle became thicker, masking the underlying ice-rich layers., Comment: 16 pages, 15 figures, published online on 24 May 2017 on Mon. Not. R. Astron. Soc. stx1275, https://doi.org/10.1093/mnras/stx1275

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $\nu=e^{-U}\mu$ and $\Omega=G^{-1}(-\infty,0]$. In this paper we are interested in the domain of the the self-adjoint operator associated with the quadratic form \begin{gather} (\psi,\varphi)\mapsto \int_\Omega\langle\nabla_H\psi,\nabla_H\varphi\rangle_Hd\nu\qquad\psi,\varphi\in W^{1,2}(\Omega,\nu).\qquad\qquad (\star) \end{gather} In particular we obtain a complete characterization of the Ornstein-Uhlenbeck operator on half-spaces, namely if $U\equiv 0$ and $G$ is an affine function, then the domain of the operator defined via $(\star)$ is the space \[\{u\in W^{2,2}(\Omega,\mu)\,|\, \langle\nabla_H u(x),\nabla_H G(x)\rangle_H=0\text{ for }\rho\text{-a.e. }x\in G^{-1}(0)\},\] where $\rho$ is the Feyel-de La Pradelle Hausdorff-Gauss surface measure., Comment: arXiv admin note: text overlap with arXiv:1609.07337

The detection of individual photons is an inherently binary mechanism, revealing either their absence or presence while concealing their spectral information. For multi-color imaging techniques, such as single photon spectroscopy, fluorescence resonance energy transfer microscopy and fluorescence correlation spectroscopy, wavelength discrimination is essential and mandates spectral separation prior to detection. Here, we adopt an approach borrowed from quantum photonic integration to realize a compact and scalable waveguide-integrated single-photon spectrometer capable of parallel detection on multiple wavelength channels, with temporal resolution below 50 ps and dark count rates below 10 Hz. We demonstrate multi-detector devices for telecommunication and visible wavelengths and showcase their performance by imaging silicon vacancy color centers in diamond nanoclusters. The fully integrated hybrid superconducting-nanophotonic circuits enable simultaneous spectroscopy and lifetime mapping for correlative imaging and provide the ingredients for quantum wavelength division multiplexing on a chip.

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron--Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $\nu=e^{-U}\mu$ and $\Omega=G^{-1}(-\infty,0]$. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \begin{align}\label{Probelma in abstract} \lambda u-L_{\nu,\Omega} u=f, \end{align} where $\lambda>0$, $f\in L^2(\Omega,\nu)$ and $L_{\nu,\Omega}$ is the self-adjoint operator associated with the quadratic form \[(\psi,\phi)\mapsto \int_\Omega\langle\nabla_H\psi,\nabla_H\phi\rangle_Hd\nu\qquad\psi,\phi\in W^{1,2}(\Omega,\nu).\] In addition we will show that if $u$ is a weak solution of problem $\lambda u-L_{\nu,\Omega} u=f$, with $\lambda>0$ and $f\in L^2(\Omega,\nu)$, then it satisfies a Neumann type condition at the boundary, namely for $\rho$-a.e. $x\in G^{-1}(0)$ \[\left\langle\,\text{Tr}\,(\nabla_Hu)(x),\,\text{Tr}\,(\nabla_H G)(x)\right\rangle_H=0,\] where $\rho$ is the Feyel--de La Pradelle Hausdorff--Gauss surface measure and $\text{Tr}$ is the trace operator.

We study some generalized metric properties of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and their relationships with other geometrical properties of norms. In case of dual Banach space $X^*$, we prove that there exists a dual norm such that its unit sphere is a Moore space for the weak$^*$-topology (has a G$_\delta$-diagonal for the weak$^*$-topology, respectively) if, and only if, $X^*$ admits an equivalent weak$^*$-LUR dual norm (rotund dual norm, respectively).

We have fabricated ultra-thin YBa2Cu3O7-x nanowires with a high critical current density and studied their voltage switching behavior in the 4.2 - 90 K temperature range. A comparison of our experimental data with theoretical models indicates that, depending on the temperature and nanowire cross section, voltage switching originates from two different mechanisms: hotspot-assisted suppression of the edge barrier by the transport current and the appearance of phase-slip lines in the nanowire. Our observation of hotspot-assisted voltage switching is in good quantitative agreement with predictions based on the Aslamazov-Larkin model for an edge barrier in a wide superconducting bridge.

This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio statistics and Barndorff-Nielsen's adjusted signed likelihood ratio statistics and we conduct a simulation study. The simulations suggest that the proposed tests display superior finite sample behavior as compared to the standard tests. Two applications are presented in order to illustrate the methods., Comment: 20 pages, 3 figures