The stationary AKPZ equation: logarithmic superdiffusivity

We study the two-dimensional Anisotropic KPZ equation (AKPZ) formally given by \begin{equation*} \partial_t H=\frac12\Delta H+\lambda((\partial_1 H)^2-(\partial_2 H)^2)+\xi\,, \end{equation*} where $\xi$ is a space-time white noise and $\lambda$ is a strictly positive constant. While the classical two-dimensional KPZ equation, whose nonlinearity is $|\nabla H|^2=(\partial_1 H)^2+(\partial_2 H)^2$, can be linearised via the Cole-Hopf transformation, this is not the case for AKPZ. We prove that the stationary solution to AKPZ (whose invariant measure is the Gaussian Free Field) is superdiffusive: its diffusion coefficient diverges for large times as $(\log t)^\delta$ for some $\delta\in (0,1)$, in a Tauberian sense. Morally, this says that the correlation length grows with time like $t^{1/2}\times (\log t)^{\delta/2}$. Moreover, we show that if the process is rescaled diffusively ($t\to t/\varepsilon^2, x\to x/\varepsilon$), then it evolves non-trivially already on time-scales of order $1/|\log\varepsilon|^\delta$. Both claims hold as soon as the coefficient $\lambda$ of the nonlinearity is non-zero, and the constant $\delta$ is uniformly bounded away from zero for $\lambda$ small. Based on the mode-coupling approximation (see e.g. [Spohn, H., J. Stat. Phys., '14]), we conjecture that the optimal value is $\delta=1/2$. These results are in contrast with the belief, based on one-loop renormalization group calculations (see [Wolf D., Phys. Rev. Lett., '91] and [Barab\`asi A.-L., Stanley H.-E., Cambridge University Press, '95]) and numerical simulations [Halpin-Healy T.,Assdah A., Phys. Rev. A, '92], that the AKPZ equation has the same large-scale behaviour as the two-dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e. the case $\lambda=0$).Comment: Added a conjecture for the value of $\delta$ and a heuristics supporting i

Gaussian Fluctuations for the Stochastic Burgers Equation in Dimension d ≥ 2

The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a d-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in van Beijeren et al. (Phys Rev Lett 54(18):2026–2029, 1985. https://doi.org/10.1103/PhysRevLett.54.2026). In both the critical d=2 and super-critical d≥3 cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For d≥3 the scaling adopted is the classical diffusive one, while in d=2 it is the so-called weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way

Gaussian Fluctuations for the stochastic Burgers equation in dimension d≥2d\geq 2

The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a $d$-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in [H. van Beijeren, R. Kutner and H. Spohn, Excess noise for driven diffusive systems, PRL, 1985]. In both the critical $d=2$ and super-critical $d\geq 3$ cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For $d\ge3$ the scaling adopted is the classical diffusive one, while in $d=2$ it is the weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way.Comment: 68 pages, 4 figures. A mistake in an earlier version has been correcte

Gaussian fluctuations for the Stochastic Burgers equation in dimension d ≥ 2

The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a d-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in van Beijeren et al. (Phys Rev Lett 54(18):2026–2029, 1985. https://doi.org/10.1103/ PhysRevLett.54.2026). In both the critical d = 2 and super-critical d ≥ 3 cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For d ≥ 3 the scaling adopted is the classical diffusive one, while in d = 2 it is the so-called weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way

Weak coupling limit of the Anisotropic KPZ equation

We study the 2-dimensional anisotropic KPZ equation (AKPZ), which is formally given by ∂th=12Δh+λ((∂1h)2−(∂2h)2)+ξ,where ξ denotes a space-time white noise and λ>0 is the so-called coupling constant. The AKPZ equation is a critical SPDE, meaning that not only is it analytically ill posed but also the breakthrough pathwise techniques for singular SPDEs in earlier works by Hairer, Gubinelli, Imkeller, and Perkowski are not applicable. As shown in recent work by the authors, the equation regularized at scale N has a diffusion coefficient that diverges logarithmically as the regularization is removed in the limit N→∞. Here, we study the weak coupling limit where λ=λN=ˆλ/√logN: this is the correct scaling that guarantees that the nonlinearity has a still nontrivial but nondivergent effect. In fact, as N→∞ the sequence of equations converges to the linear stochastic heat equation ∂th=νeff2Δh+√νeffξ, where νeff>1 is explicit and depends nontrivially on ˆλ. This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearized via Cole–Hopf or any other transformation

The stationary AKPZ equation : logarithmic superdiffusivity

We study the two‐dimensional Anisotropic KPZ equation (AKPZ) formally given by ∂ t H = 1 2 Δ H + λ ( ( ∂ 1 H ) 2 − ( ∂ 2 H ) 2 ) + ξ , $$\begin{equation*} \hspace*{3.4pc}\partial _t H=\frac{1}{2}\Delta H+\lambda ((\partial _1 H)^2-(\partial _2 H)^2)+\xi , \end{equation*}$$ where ξ is a space‐time white noise and λ is a strictly positive constant. While the classical two‐dimensional KPZ equation, whose nonlinearity is | ∇ H | 2 = ( ∂ 1 H ) 2 + ( ∂ 2 H ) 2 $|\nabla H|^2=(\partial _1 H)^2+(\partial _2 H)^2$ , can be linearised via the Cole‐Hopf transformation, this is not the case for AKPZ. We prove that the stationary solution to AKPZ (whose invariant measure is the Gaussian Free Field (GFF)) is superdiffusive: its diffusion coefficient diverges for large times as log t $\sqrt {\mathop {\mathrm{log}}\nolimits t}$ up to log log t $\mathop {\mathrm{log}}\nolimits \mathop {\mathrm{log}}\nolimits t$ corrections, in a Tauberian sense. Morally, this says that the correlation length grows with time like t 1 / 2 × ( log t ) 1 / 4 $t^{1/2}\times (\mathop {\mathrm{log}}\nolimits t)^{1/4}$ . Moreover, we show that if the process is rescaled diffusively ( t → t / ε 2 , x → x / ε , ε → 0 $t\rightarrow t/\varepsilon ^2, x\rightarrow x/\varepsilon , \varepsilon \rightarrow 0$ ), then it evolves non‐trivially already on time‐scales of order approximately 1 / | log ε | ≪ 1 $1/\sqrt {|\mathop {\mathrm{log}}\nolimits \varepsilon |}\ll 1$ . Both claims hold as soon as the coefficient λ of the nonlinearity is non‐zero. These results are in contrast with the belief, common in the mathematics community, that the AKPZ equation is diffusive at large scales and, under simple diffusive scaling, converges to the two‐dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e., the case λ = 0 $\lambda =0$ )

Giant hepatocellular adenoma as cause of severe abdominal pain: a case report

The authors describe the case of a large hepatocellular adenoma diagnosed in a 30-year old woman who came to us complaining of acute pain in the upper abdominal quadrants. The patient had been taking an oral contraceptive pill for the last ten years. We present the clinical features, the diagnostic work-up and the treatment prescribed

Chronic constipation diagnosis and treatment evaluation: The "CHRO.CO.DI.T.E." study

Background: According to Rome criteria, chronic constipation (CC) includes functional constipation (FC) and irritable bowel syndrome with constipation (IBS-C). Some patients do not meet these criteria (No Rome Constipation, NRC). The aim of the study was is to evaluate the various clinical presentation and management of FC, IBS-C and NRC in Italy. Methods: During a 2-month period, 52 Italian gastroenterologists recorded clinical data of FC, IBS-C and NRC patients, using Bristol scale, PAC-SYM and PAC-QoL questionnaires. In addition, gastroenterologists were also asked to record whether the patients were clinically assessed for CC for the first time or were in follow up. Diagnostic tests and prescribed therapies were also recorded. Results: Eight hundred seventy-eight consecutive CC patients (706 F) were enrolled (FC 62.5%, IBS-C 31.3%, NRC 6.2%). PAC-SYM and PAC-QoL scores were higher in IBS-C than in FC and NRC. 49.5% were at their first gastroenterological evaluation for CC. In 48.5% CC duration was longer than 10 years. A specialist consultation was requested in 31.6%, more frequently in IBS-C than in NRC. Digital rectal examination was performed in only 56.4%. Diagnostic tests were prescribed to 80.0%. Faecal calprotectin, thyroid tests, celiac serology, breath tests were more frequently suggested in IBS-C and anorectal manometry in FC. More than 90% had at least one treatment suggested on chronic constipation, most frequently dietary changes, macrogol and fibers. Antispasmodics and psychotherapy were more frequently prescribed in IBS-C, prucalopride and pelvic floor rehabilitation in FC. Conclusions: Patients with IBS-C reported more severe symptoms and worse quality of life than FC and NRC. Digital rectal examination was often not performed but at least one diagnostic test was prescribed to most patients. Colonoscopy and blood tests were the "first line" diagnostic tools. Macrogol was the most prescribed laxative, and prucalopride and pelvic floor rehabilitation represented a "second line" approach. Diagnostic tests and prescribed therapies increased by increasing CC severity

Clinical features and outcomes of elderly hospitalised patients with chronic obstructive pulmonary disease, heart failure or both

Background and objective: Chronic obstructive pulmonary disease (COPD) and heart failure (HF) mutually increase the risk of being present in the same patient, especially if older. Whether or not this coexistence may be associated with a worse prognosis is debated. Therefore, employing data derived from the REPOSI register, we evaluated the clinical features and outcomes in a population of elderly patients admitted to internal medicine wards and having COPD, HF or COPD + HF. Methods: We measured socio-demographic and anthropometric characteristics, severity and prevalence of comorbidities, clinical and laboratory features during hospitalization, mood disorders, functional independence, drug prescriptions and discharge destination. The primary study outcome was the risk of death. Results: We considered 2,343 elderly hospitalized patients (median age 81 years), of whom 1,154 (49%) had COPD, 813 (35%) HF, and 376 (16%) COPD + HF. Patients with COPD + HF had different characteristics than those with COPD or HF, such as a higher prevalence of previous hospitalizations, comorbidities (especially chronic kidney disease), higher respiratory rate at admission and number of prescribed drugs. Patients with COPD + HF (hazard ratio HR 1.74, 95% confidence intervals CI 1.16-2.61) and patients with dementia (HR 1.75, 95% CI 1.06-2.90) had a higher risk of death at one year. The Kaplan-Meier curves showed a higher mortality risk in the group of patients with COPD + HF for all causes (p = 0.010), respiratory causes (p = 0.006), cardiovascular causes (p = 0.046) and respiratory plus cardiovascular causes (p = 0.009). Conclusion: In this real-life cohort of hospitalized elderly patients, the coexistence of COPD and HF significantly worsened prognosis at one year. This finding may help to better define the care needs of this population