The Moment-SOS hierarchy and the Christoffel-Darboux kernel

We consider the global minimization of a polynomial on a compact set B. We show that each step of the Moment-SOS hierarchy has a nice and simple interpretation that complements the usual one. Namely, it computes coefficients of a polynomial in an orthonormal basis of L 2 (B,μ) where μ is an arbitrary reference measure whose support is exactly B. The resulting polynomial is a certain density (with respect to μ) of some signed measure on B. When some relaxation is exact (which generically takes place) the coefficients of the optimal polynomial density are values of orthonormal polynomials at the global minimizer and the optimal (signed) density is simply related to the Christoffel-Darboux (CD) kernel and the Christoffel function associated with μ. In contrast to the hierarchy of upper bounds which computes positive densities, the global optimum can be achieved exactly as integration against a polynomial (signed) density because the CD-kernel is a reproducing kernel, and so can mimic a Dirac measure (as long as finitely many moments are concerned).

Value distribution of derivatives in polynomial dynamics

For every $m\in \mathbb {N}$ , we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$ th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $ . We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on $\mathbb {P}^1(\overline {k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb {C}^2$ has a given eigenvalue.

Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus

<p style='text-indent:20px;'>In this paper, we study an elliptic system arising from the U(1)<inline-formula><tex-math id="M2">\begin{document}$ \times $\end{document}</tex-math></inline-formula>U(1) Abelian Chern-Simons Model[<xref ref-type="bibr" rid="b25">25</xref>,<xref ref-type="bibr" rid="b37">37</xref>] of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE123"> \begin{document}$ \begin{equation} \left\{\begin{split} \Delta u = &amp;\lambda \left(a(b-a)e^{u}-b(b-a)e^{v}+a^2e^{2u} -abe^{2v}+b(b-a)e^{u+v}\right)\\ &amp; +4\pi \sum\limits_{j = 1}^{k_1}m_j\delta_{p_j}, \\ \Delta v = &amp;\lambda \left(-b(b-a)e^{u}+a(b-a)e^{v}-abe^{2u} +a^2e^{2v}+b(b-a)e^{u+v}\right)\\ &amp; +4\pi \sum\limits_{j = 1}^{k_2}n_j\delta_{q_j}, \end{split}\right. \quad\quad\quad\quad (1)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>which are defined on a parallelogram <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> with doubly periodic boundary conditions. Here, <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> are interaction constants, <inline-formula><tex-math id="M7">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is related to coupling constant, <inline-formula><tex-math id="M8">\begin{document}$ m_j&gt;0(j = 1,\cdots,k_1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ n_j&gt;0(j = 1,\cdots,k_2) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \delta_{p} $\end{document}</tex-math></inline-formula> is the Dirac measure, <inline-formula><tex-math id="M11">\begin{document}$ p $\end{document}</tex-math></inline-formula> is called vortex point. Concerning the existence results of this system over <inline-formula><tex-math id="M12">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>, only the cases <inline-formula><tex-math id="M13">\begin{document}$ (a,b) = (0,1) $\end{document}</tex-math></inline-formula>[<xref ref-type="bibr" rid="b28">28</xref>] and <inline-formula><tex-math id="M14">\begin{document}$ a&gt;b&gt;0 $\end{document}</tex-math></inline-formula>[<xref ref-type="bibr" rid="b14">14</xref>] were studied in the literature. The solvability of this system (1) is still an open problem as regards other parameters <inline-formula><tex-math id="M15">\begin{document}$ (a,b) $\end{document}</tex-math></inline-formula>. We show that the system (1) admits topological solutions provided <inline-formula><tex-math id="M16">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is large and <inline-formula><tex-math id="M17">\begin{document}$ b&gt;a&gt;0 $\end{document}</tex-math></inline-formula> Our arguments are based on a iteration scheme and variational formulation.</p>

The Hopf Lemma for the Schrödinger Operator

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator {-\Delta+V} with a nonnegative potential V which merely belongs to {L_{\mathrm{loc}}^{1}(\Omega)}. More precisely, if {u\in W_{0}^{1,2}(\Omega)\cap L^{2}(\Omega;V\mathop{}\!\mathrm{d}{x})} satisfies {-\Delta u+Vu=f} on Ω for some nonnegative datum {f\in L^{\infty}(\Omega)}, {f\not\equiv 0}, then we show that at every point {a\in\partial\Omega} where the classical normal derivative {\frac{\partial u(a)}{\partial n}} exists and satisfies the Poisson representation formula, one has {\frac{\partial u(a)}{\partial n}>0} if and only if the boundary value problem\begin{dcases}\begin{aligned} \displaystyle-\Delta v+Vv&\displaystyle=0&&% \displaystyle\phantom{}\text{in ${\Omega}$,}\\ \displaystyle v&\displaystyle=\nu&&\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\end{dcases}involving the Dirac measure {\nu=\delta_{a}} has a solution. More generally, we characterize the nonnegative finite Borel measures ν on {\partial\Omega} for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.

Quasi-stationarity for one-dimensional renormalized Brownian motion

We are interested in the quasi-stationarity for the time-inhomogeneous Markov process $$X_t = \frac{B_t}{(t+1)^\kappa},$$ where (Bt)t≥0 is a one-dimensional Brownian motion and κ ∈ (0, ∞). We first show that the law of Xt conditioned not to go out from (−1, 1) until time t converges weakly towards the Dirac measure δ0 when κ>½, when t goes to infinity. Then, we show that this conditional probability measure converges weakly towards the quasi-stationary distribution for an Ornstein-Uhlenbeck process when κ=½. Finally, when κ<½, it is shown that the conditional probability measure converges towards the quasi-stationary distribution for a Brownian motion. We also prove the existence of a Q-process and a quasi-ergodic distribution for κ=½ and κ<½.

A new class of costs for optimal transport planning

We study a class of optimal transport planning problems where the reference cost involves a non-linear function G(x, p) representing the transport cost between the Dirac measure δx and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case ($G(x,p)=\int c(x,y)dp$) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory.

Characterization of the generalized Chebyshev-type polynomials of first kind

<p>Orthogonal polynomials have very useful properties in the mathematical problems, so recent years have seen a great deal in the field of approximation theory using orthogonal polynomials. In this paper, we characterize a sequence of the generalized Chebyshev-type polynomials of the first kind \(\left\{\mathscr{T}_{n}^{(M,N)}(x)\right\}_{n\in\mathbb{N}\cup\{0\}},\) which are orthogonal with respect to the measure \(\frac{\sqrt{1-x^{2}}}{\pi}dx+M\delta_{-1}+N\delta_{1},\) where \(\delta_{x}\) is a singular Dirac measure and \(M,N\geq 0.\) Then we provide a closed form of the constructed polynomials in term of the Bernstein polynomials \(B_{k}^{n}(x).\)</p><p>We conclude the paper with some results on the integration of the weighted generalized Chebyshev-type with the Bernstein polynomials.</p>