Global well-posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain

In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the $d$-dimensional torus. This class includes the wave equation for $d=1$ and the beam equation for $d\le 3$. We show that the Gibbs measure of the equation without forcing and damping is the unique invariant measure for the flow of this system. Since the flow does not satisfy the Strong Feller property, we introduce a new technique for showing unique ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.Comment: The title has being changed from "Unique ergodicity for stochastic hyperbolic equations with additive space-time white noise" to "Unique ergodicity for a class of stochastic hyperbolic equations with additive space-time white noise", many typo corrections, some minor corrections in the proof

Development of a Multimodal MRI Study to Characterize Morpho-Functional Features in Rodent Models of Alzheimer's Disease

Alzheimer’s Disease (AD) is the most common form of dementia, recognized by the World Health Organization as a global public health priority. It is a complex pathology characterized by the accumulation of the Amyloid-β (Aβ) peptide as extracellular plaques and of the intracellullar neuro fibrillary tangles (NFT) alongside with different events, such as chronic neuroinflammation and astrogliosis. None of the existing biomarkers is simultaneously specific for the pathology and sensitive to its progression. Metabolic and functional alterations are the earliest events described in the AD pathological cascade but shared with other form of dementia, while specific structural alterations occurs in a later stage of the disease. The use of transgenic models could simplify the development of new imaging biomarkers that would enable early diagnosis and making new treatments more effective. The objective of this work was to develop a multi-modal panel of magnetic resonance imaging (MRI) techniques and automated analysis pipelines, characterized by a high translational impact, in order to investigate the metabolic, functional and structural alterations in the brains of AD transgenic models. Results obtained in the APP23 transgenic mouse show that chemical exchange saturation transfer (CEST) imaging can be used to detect alterations in the brain uptake of the glucose analogue 2-deoxy-d-glucose (2DG) with a better resolution than PET and without the need of radioactive tracers. Moreover, a longitudinal study highlighted that significant structural and metabolic alterations can be found only in a late stage of the pathology. Furthermore, an advanced pipeline for the analysis of the rodent brain functional connectivity has been developed. This thesis demonstrates that the advantage to the experimental design adopted is simplifying longitudinal studies of the same animal cohort. The translation of the analysis pipelines adopted in human studies enables more powerful results and reduces the number of animals involved in research

Phase transition for invariant measures of the focusing Schr\"odinger equation

In this paper, we consider the Gibbs measure for the focusing nonlinear Schr\"odinger equation on the one-dimensional torus, that was introduced in a seminal paper by Lebowitz, Rose and Speer (1988). We show that in the large torus limit, the measure exhibits a phase transition, depending on the size of the nonlinearity. This phase transition was originally conjectured on the basis of numerical simulation by Lebowitz, Rose and Speer (1988). Its existence is however striking in view of a series of negative results by McKean (1995) and Rider (2002).Comment: 50 page

The multifaced role of stat3 in cancer and its implication for anticancer therapy

Signal transducer and activator of transcription (STAT) 3 is one of the most complex regulators of transcription. Constitutive activation of STAT3 has been reported in many types of tumors and depends on mechanisms such as hyperactivation of receptors for pro-oncogenic cytokines and growth factors, loss of negative regulation, and excessive cytokine stimulation. In contrast, somatic STAT3 mutations are less frequent in cancer. Several oncogenic targets of STAT3 have been recently identified such as c-myc, c-Jun, PLK-1, Pim1/2, Bcl-2, VEGF, bFGF, and Cten, and inhibitors of STAT3 have been developed for cancer prevention and treatment. However, despite the oncogenic role of STAT3 having been widely demonstrated, an increasing amount of data indicate that STAT3 functions are multifaced and not easy to classify. In fact, the specific cellular role of STAT3 seems to be determined by the integration of multiple signals, by the oncogenic environment, and by the alternative splicing into two distinct isoforms, STAT3α and STAT3β. On the basis of these different conditions, STAT3 can act both as a potent tumor promoter or tumor suppressor factor. This implies that the therapies based on STAT3 modulators should be performed considering the pleiotropic functions of this transcription factor and tailored to the specific tumor type

The "Janus" Role of C/EBPs Family Members in Cancer Progression

CCAAT/enhancer-binding proteins (C/EBPs) constitute a family of transcription factors composed of six members that are critical for normal cellular differentiation in a variety of tissues. They promote the expression of genes through interaction with their promoters. Moreover, they have a key role in regulating cellular proliferation through interaction with cell cycle proteins. C/EBPs are considered to be tumor suppressor factors due to their ability to arrest cell growth (contributing to the terminal differentiation of several cell types) and for their role in cellular response to DNA damage, nutrient deprivation, hypoxia, and genotoxic agents. However, C/EBPs can elicit completely opposite effects on cell proliferation and cancer development and they have been described as both tumor promoters and tumor suppressors. This "Janus" role of C/EBPs depends on different factors, such as the type of tumor, the isoform/s expressed in cells, the type of dimerization (homo- or heterodimerization), the presence of inhibitory elements, and the ability to inhibit the expression of other tumor suppressors. In this review, we discuss the implication of the C/EBPs family in cancer, focusing on the molecular aspects that make these transcription factors tumor promoters or tumor suppressors

Focusing Φ34\Phi^4_3-model with a Hartree-type nonlinearity

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study a focusing $\Phi^4_3$-model with a Hartree-type nonlinearity, where the potential for the Hartree nonlinearity is given by the Bessel potential of order $\beta$. We first construct the focusing Gibbs measure for $\beta > 2$. We also show that the threshold $\beta = 2$ is sharp in the sense that the associated Gibbs measure is not normalizable for $\beta < 2$. Furthermore, we prove the following dichotomy at the critical value $\beta = 2$: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime. We then study the three-dimensional stochastic damped nonlinear wave equation (SdNLW) with a cubic Hartree nonlinearity, forced by an additive space-time white noise. Using ideas from paracontrolled calculus, we rewrite the equation into a system of three unknowns and prove its local well-posedness. We then establish almost sure global well-posedness and invariance of the focusing Gibbs measure via Bourgain's invariant measure argument. In view of the non-normalizability result, our almost sure global well-posedness result is sharp. In Appendix, we also consider the (parabolic) stochastic quantization for the focusing Hartree $\Phi^4_3$-measure and construct global-in-time invariant dynamics for the same range of $\beta$. We also consider the defocusing case. By introducing further renormalizations at $\beta = 1$ and $\beta = \frac 12$, we extend the construction of the defocusing Hartree $\Phi^4_3$-measure for $\beta > 0$, where the resulting measure is shown to be singular with respect to the reference Gaussian free field for $0 < \beta \le \frac 12$. The dynamical problem is studied only for $\beta > 1$ in the defocusing case.Comment: 112 page

Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus

We find the optimal exponent of normalizability for certain Gibbs-type measures based on variants of Brownian motion which have appeared in the PDE literature, starting with an influential paper of Lebowitz, Rose and Speer (1988). We give a proof of a result stated in that paper. The proof also applies to the 2D radial measures introduced by Tzvetkov, which were later also studied by Bourgain and Bulut. In this case, we answer a question of the latter two authors.Comment: 10 page

Convolution kernels versus spectral multipliers for sub-Laplacians on groups of polynomial growth

Let $\mathcal{L}$ be a sub-Laplacian on a connected Lie group $G$ of polynomial growth. It is well known that, if $F : \mathbb{R} \to \mathbb{C}$ is in the Schwartz class $\mathcal{S}(\mathbb{R})$, then the convolution kernel $\mathcal{K}_{F(\mathcal{L})}$ of the operator $F(\mathcal{L})$ is in the Schwartz class $\mathcal{S}(G)$. Here we prove a sort of converse implication for a class of groups $G$ including all solvable noncompact groups of polynomial growth. We also discuss the problem whether integrability of $\mathcal{K}_{F(\mathcal{L})}$ implies continuity of $F$.Comment: 27 page