Strong uniqueness for stochastic evolution equations with unbounded measurable drift term

We consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term $B$ and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in the class of global solutions. This paper extends our previous paper (Da Prato, Flandoli, Priola and M. Rockner, Annals of Prob., published online in 2012) which generalized Veretennikov's fundamental result to infinite dimensions assuming boundedness of the drift term. As in our previous paper pathwise uniqueness holds for a large class, but not for every initial condition. We also include an application of our result to prove existence of strong solutions when the drift $B$ is only measurable, locally bounded and grows more than linearly.Comment: The paper will be published in Journal of Theoretical Probability. arXiv admin note: text overlap with arXiv:1109.036

The Symplectic Penrose Kite

The purpose of this article is to view the Penrose kite from the perspective of symplectic geometry.Comment: 24 pages, 7 figures, minor changes in last version, to appear in Comm. Math. Phys

Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise

We prove that every Markov solution to the three dimensional Navier-Stokes equation with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast. Moreover, we give a well-posedness criterion for the equations in terms of invariant measures. We also analyse the energy balance and identify the term which ensures equality in the balance.Comment: 32 page

Dimension-independent Harnack inequalities for subordinated semigroups

Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the Bakry-Emery curvature condition, the subordinate semigroup with power $\alpha$ satisfies a dimension-free Harnack inequality provided $\alpha \in(1/2, 1)$, and it satisfies the log-Harnack inequality for all $\alpha \in (0,1).$ Some infinite-dimensional examples are also presented

Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian

We consider Hamilton Jacobi Bellman equations in an inifinite dimensional Hilbert space, with quadratic (respectively superquadratic) hamiltonian and with continuous (respectively lipschitz continuous) final conditions. This allows to study stochastic optimal control problems for suitable controlled Ornstein Uhlenbeck process with unbounded control processes

Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose non-linear drift parts are sums of the subdifferential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and non-degenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also prove weak existence, hence obtain unique strong solutions by the Yamada-Watanabe theorem. The proofs are based in part on a recent maximal regularity result in infinite dimensions, the theory of quasi-regular Dirichlet forms and an infinite dimensional version of a Zvonkin-type transformation. As a main application we show pathwise uniqueness for stochastic reaction diffusion equations perturbed by a Borel measurable bounded drift. Hence such SDE have a unique strong solution

Convex Polytopes and Quasilattices from the Symplectic Viewpoint

We construct, for each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e. each of these spaces is a collection of quasifolds glued together in an suitable way. A quasifold is a space locally modelled on $\R^k$ modulo the action of a discrete, possibly infinite, group. The way strata are glued to each other also involves the action of an (infinite) discrete group. Each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the original polytope back. These spaces may be viewed as a natural generalization of symplectic toric varieties to the nonrational setting.Comment: LaTeX, 29 pages. Revised version: TITLE changed, reorganization of notations and exposition, added remarks and reference

Deep 10 and 18 micron Imaging of the HR 4796A Circumstellar Disk: Transient Dust Particles & Tentative Evidence for a Brightness Asymmetry

We present new 10.8 and 18.2 micron images of HR 4796A, a young A0V star that was recently discovered to have a spectacular, nearly edge-on, circumstellar disk prominent at ~20 microns (Jayawardhana et al. 1998; Koerner et al. 1998). These new images, obtained with OSCIR at Keck II, show that the disk's size at 10 microns is comparable to its size at 18 microns. Therefore, the 18 micron-emitting dust may also emit some, or all, of the 10 micron radiation. Using these multi-wavelength images, we determine a "characteristic" diameter of 2-3 microns for the mid-infrared-emitting dust particles if they are spherical and composed of astronomical silicates. Particles this small are expected to be blown out of the system by radiation pressure in a few hundred years, and therefore these particles are unlikely to be primordial. Dynamical modeling of the disk (Wyatt et al. 2000) indicates that the disk surface density is relatively sharply peaked near 70 AU, which agrees with the mean annular radius deduced by Schneider et al. (1999) from their NICMOS images. We present evidence (~1.8 sigma significance) for a brightness asymmetry that may result from the presence of the hole and the gravitational perturbation of the disk particle orbits by the low-mass stellar companion or a planet. This "pericenter glow," which must still be confirmed, results from a very small (a few AU) shift of the disk's center of symmetry relative to the central star HR 4796A; one side of the inner boundary of the annulus is shifted towards HR 4796A, thereby becoming warmer and more infrared-emitting. The possible detection of pericenter glow implies that the detection of even complex dynamical effects of planets on disks is within reach.Comment: 18 pages. 9 GIF images. Total size ~800 kB. High resolution images available upon request. Accepted for publication in the Astrophysical Journal (scheduled for January 10, 2000

Statistical properties of stochastic 2D Navier-Stokes equations from linear models

A new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence has been proposed and tested through numerical simulations. This is achieved by constructing, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. In this paper, we investigate this conjecture for the 2D Navier-Stokes equations driven by an additive noise. In order to check this conjecture, we analyze the coupled system Navier-Stokes/linear advection system in the unknowns $(u,w)$. We introduce a parameter $\lambda$ which gives a system $(u^\lambda,w^\lambda)$; this system is studied for any $\lambda$ proving its well posedness and the uniqueness of its invariant measure $\mu^\lambda$. The key point is that for any $\lambda \neq 0$ the fields $u^\lambda$ and $w^\lambda$ have the same scaling exponents, by assuming universality of the scaling exponents to the force. In order to prove the same for the original fields $u$ and $w$, we investigate the limit as $\lambda \to 0$, proving that $\mu^\lambda$ weakly converges to $\mu^0$, where $\mu^0$ is the only invariant measure for the joint system for $(u,w)$ when $\lambda=0$.Comment: 23 pages; improved versio

Technical standardization of MIS management of children with pilonidal sinus disease using pediatric endoscopic pilonidal sinus treatment (PEPSiT) and laser epilation

This study aimed to standardize the technique of pediatric endoscopic pilonidal sinus treatment (PEPSiT) associated with laser epilation. METHODS: All pediatric patients presenting with acute or chronic pilonidal sinus disease (PSD) who underwent PEPSiT in our institution over a 36-month period (July 2015-July 2018), were included in the study. Pre- and postoperative management, recurrence rate, postoperative pain, hospital stay, analgesic requirements, and patient satisfaction levels were evaluated. RESULTS: A total of 59 patients (23 girls and 36 boys) underwent PEPSiT during the study period. Ten/59 patients (16.9%) had recurrent PSD after open repair, and 4/59 (6.7%) presented a concomitant pilonidal cyst. All children underwent laser epilation pre- and postoperatively over the last 15 months. The average length of surgery was 27.5 min (range 20-45). The average pain score during the first 48 postoperative hours was 2.7 (range 2-5), and the average analgesic requirement was 20 h (range 16-24). The average hospitalization was 22.4 h (range 18-36). At 1 month postoperatively, external openings were healed in all patients. During follow-up, 1 recurrence (1.6%) was recorded and successfully re-treated with PEPSiT. CONCLUSIONS: We believe that PEPSiT represents the technique of choice for treatment of PSD in the pediatric population. It is crucial to standardize the technique consisting of pre- and postoperative laser epilation, PEPSiT, and accurate postoperative wound management with eosin and sulfadiazine spray