Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite dimensional convex domains

Let $X$ be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure $\gamma$ and let $\lambda_1$ be the maximum eigenvalue of the covariance operator associated with $\gamma$. The associated Cameron--Martin space is denoted by $H$. For a sufficiently regular convex function $U:X\to\mathbb{R}$ and a convex set $\Omega\subseteq X$, we set $\nu:=e^{-U}\gamma$ and we consider the semigroup $(T_\Omega(t))_{t\geq 0}$ generated by the self-adjoint operator defined via the quadratic form $$(\varphi,\psi)\mapsto \int_\Omega\langle D_H\varphi,D_H\psi\rangle_Hd\nu,$$ where $\varphi,\psi$ belong to $D^{1,2}(\Omega,\nu)$, the Sobolev space defined as the domain of the closure in $L^2(\Omega,\nu)$ of $D_H$, the gradient operator along the directions of $H$. A suitable approximation procedure allows us to prove some pointwise gradient estimates for $(T_\Omega(t))_{t\ge 0}$. In particular, we show that $$|D_H T_\Omega(t)f|_H^p\le e^{- p \lambda_1^{-1} t}(T_\Omega(t)|D_H f|^p_H), \qquad\, t>0,\ \nu\textrm{ -a.e. in }\Omega,$$ for any $p\in [1,+\infty)$ and $f\in D^{1,p}(\Omega ,\nu)$. We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincar\'e inequality in $\Omega$ for the measure $\nu$ and some improving summability properties for $(T_\Omega(t))_{t\geq 0}$. In addition we prove that if $f$ belongs to $L^p(\Omega,\nu)$ for some $p\in(1,\infty)$, then $$|D_H T_\Omega(t)f|^p_H \leq K_p t^{-\frac{p}{2}} T_\Omega(t)|f|^p,\qquad \, t>0,\ \nu\text{-a.e. in }\Omega,$$ where $K_p$ is a positive constant depending only on $p$. Finally we investigate on the asymptotic behaviour of the semigroup $(T_\Omega(t))_{t\geq 0}$ as $t$ goes to infinity

LpL^p-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space $L^p({\mathbb R}^d;{\mathbb R}^m)$, $(d,m \ge 1)$ with $p\in [1,+\infty)$. Sufficient conditions for the associated evolution operator ${\bf G}(t,s)$ in $C_b({\mathbb R}^d;{\mathbb R}^m)$ to extend to a strongly continuous operator in $L^p({\mathbb R}^d;{\mathbb R}^m)$ are given. Some $L^p$-$L^q$ estimates are also established together with $L^p$ gradient estimates

Two characterization of BV functions on Carnot groups via the heat semigroup

In this paper we provide two different characterizations of sets with finite perimeter and functions of bounded variation in Carnot groups, analogous to those which hold in Euclidean spaces, in terms of the short-time behaviour of the heat semigroup. The second one holds under the hypothesis that the reduced boundary of a set of finite perimeter is rectifiable, a result that presently is known in Step 2 Carnot groups

Structural Prediction of Protein–Protein Interactions by Docking: Application to Biomedical Problems

A huge amount of genetic information is available thanks to the recent advances in sequencing technologies and the larger computational capabilities, but the interpretation of such genetic data at phenotypic level remains elusive. One of the reasons is that proteins are not acting alone, but are specifically interacting with other proteins and biomolecules, forming intricate interaction networks that are essential for the majority of cell processes and pathological conditions. Thus, characterizing such interaction networks is an important step in understanding how information flows from gene to phenotype. Indeed, structural characterization of protein–protein interactions at atomic resolution has many applications in biomedicine, from diagnosis and vaccine design, to drug discovery. However, despite the advances of experimental structural determination, the number of interactions for which there is available structural data is still very small. In this context, a complementary approach is computational modeling of protein interactions by docking, which is usually composed of two major phases: (i) sampling of the possible binding modes between the interacting molecules and (ii) scoring for the identification of the correct orientations. In addition, prediction of interface and hot-spot residues is very useful in order to guide and interpret mutagenesis experiments, as well as to understand functional and mechanistic aspects of the interaction. Computational docking is already being applied to specific biomedical problems within the context of personalized medicine, for instance, helping to interpret pathological mutations involved in protein–protein interactions, or providing modeled structural data for drug discovery targeting protein–protein interactions.Spanish Ministry of Economy grant number BIO2016-79960-R; D.B.B. is supported by a predoctoral fellowship from CONACyT; M.R. is supported by an FPI fellowship from the Severo Ochoa program. We are grateful to the Joint BSC-CRG-IRB Programme in Computational Biology.Peer ReviewedPostprint (author's final draft

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups

Functional inequalities for some generalised Mehler semigroups

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

Su alcune generalizzazioni del teorema di Meyers-Serrin

We present a generalisation of Meyers-Serrin theorem, in which we replace the standard weak derivatives in open subsets of ℝm with finite families of linear differential operators defined on smooth sections of vector bundles on a (not necessarily compact) manifold X.Presentiamo una generalizzazione del teorema di Meyers-Serrin, in cui sostituiamo le derivate deboli in sottoinsiemi aperti di ℝm con famiglie finite di operatori differenziali lineari, definiti su sezioni regolari di fibrati vettoriali su una varietà (non necessariamente compatta) X