LeVeque type inequalities and discrepancy estimates for minimal energy configurations on spheres

Let ${\mathbb{S}}^d$ denote the unit sphere in the Euclidean space ${\mathbb{R}}^{d+1}(d\ge 1)$. We develop LeVeque type inequalities for the discrepancy between the rotationally invariant probability measure and the normalized counting measures on ${\mathbb{S}}^d$. We obtain both upper bound and lower bound estimates. We then use these inequalities to estimate the discrepancy of the normalized counting measures associated with minimal energy configurations on ${\mathbb{S}}^d$.     

Mayer control problem with probabilistic uncertainty on initial positions

In this paper we introduce and study an optimal control problem in the Mayer's form in the space of probability measures on ${\mathbb{R}}^n$ endowed with the Wasserstein distance. Our aim is to study optimality conditions when the knowledge of the initial state and velocity is subject to some uncertainty, which are modeled by a probability measure on ${\mathbb{R}}^d$ and by a vector-valued measure on ${\mathbb{R}}^d$, respectively. We provide a characterization of the value function of such a problem as unique solution of an Hamilton–Jacobi–Bellman equation in the space of measures in a suitable viscosity sense. Some applications to a pursuit-evasion game with uncertainty in the state space is also discussed, proving the existence of a value for the game.     

Regularity and decay of solutions of nonlinear harmonic oscillators

We prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators. Namely, we show holomorphic extension to a sector in the complex domain, with a corresponding Gaussian decay, according to the basic properties of the Hermite functions in ${\mathbb{R}}^d$. Our results apply, in particular, to nonlinear eigenvalue problems for the harmonic oscillator associated to a real-analytic scattering, or asymptotically conic, metric in ${\mathbb{R}}^d$, as well as to certain perturbations of the classical harmonic oscillator.     

An Lp-theory for almost sure local well-posedness of the nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equations (NLS) on ${\mathbb{R}}^d$ with random and rough initial data. By working in the framework of $L^p({\mathbb{R}}^d)$ spaces, $p>2$, we prove almost sure local well-posedness for rougher initial data than those considered in the existing literature. The main ingredient of the proof is the dispersive estimate.     

Quasi-linear PDEs and forward–backward stochastic differential equations: Weak solutions

In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space $H_{\rho }^1({\mathbb{R}}^d)$. For this, we study first the solutions of forward–backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space $L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^d)?L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^k)?L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^{k\times d})$. This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon.     

Spread of a catalytic branching random walk on a multidimensional lattice

For a supercritical catalytic branching random walk on ${\mathbb{Z}}^d$, $d\in \mathbb{N}$, with an arbitrary finite catalysts set we study the spread of particles population as time grows to infinity. It is shown that in the result of the proper normalization of the particles positions in the limit there are a.s. no particles outside the closed convex surface in ${\mathbb{R}}^d$ which we call the propagation front and, under condition of infinite number of visits of the catalysts set, a.s. there exist particles on the propagation front. We also demonstrate that the propagation front is asymptotically densely populated and derive its alternative representation.     

Regularity of random attractors for fractional stochastic reaction–diffusion equations on Rn

We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in $H^s({\mathbb{R}}^n)$ with $s\in (0,1)$. We prove the existence and uniqueness of the tempered random attractor that is compact in $H^s({\mathbb{R}}^n)$ and attracts all tempered random subsets of $L^2({\mathbb{R}}^n)$ with respect to the norm of $H^s({\mathbb{R}}^n)$. The main difficulty is to show the pullback asymptotic compactness of solutions in $H^s({\mathbb{R}}^n)$ due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains.