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Due to their flexibility Gaussian processes are a well-known Bayesian framework for nonparametric function estimation. Integrating inequality constraints, such as monotonicity, convexity, and boundedness, into Gaussian process models significantly improves prediction accuracy and yields more realistic credible intervals in various real-world data applications. The Gaussian process approximation, originally proposed in [22] is considered. It satisfies interpolation conditions and handles a wide range of inequality constraints everywhere. Our contribution in this paper is threefold. First, we extend this approach to handle noisy observations and multiple, more general convex and non-convex constraints. Second, we propose new basis functions in order to extend the smoothness of sample paths to differentiability of class C^p , for any p ≥ 1. Third, we examine its behavior in specific scenarios such as monotonicity with flat regions and boundedness near lower and/or upper bounds. In that case, we show that, unlike the Maximum a posteriori (MAP) estimate, the mean a posteriori (mAP) estimate fails to capture flat regions. To address this issue, we propose incorporating multiple constraints, such as monotonicity with bounded slope constraints. According to the theoretical convergence and based on a variety of numerical experiments, the MAP estimate behaves well and outperforms the mAP estimate in terms of prediction accuracy. The performance of the proposed approach is confirmed through real-world data studies.

In this paper, we extend the correspondence between Bayesian estima- tion and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) by adding convex constraints to the problem. Through a sequence of approxi- mating Hilbertian subspaces and a discretized model, we prove that the Max- imum a posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalization of the paper [15], where it is proved that the optimal smooth- ing solution is the mean of the posterior distribution. Synthetic and real data studies confirm the correspondence established in this paper.

International audience; Gaussian processes have become essential for non-parametric function estimation and widely used in many fields like machine learning. In this paper, large scale Gaussian process regression (GPR) is investigated. This problem is related to the simulation of high dimensional Gaussian vectors truncated on the intersection of a set of hyperplanes. The main idea is to combine both Matheron's update rule (MUR) and Karhunen-Lovève expansion (KLE). First, by the MUR we show how simulating from the posterior distribution is possible without computing the posterior covariance matrix and its decomposition. Second, by splitting the input domain in smallest nonoverlapping subdomains, the KLE coefficients are conditioned in order to guarantee the correlation structure in the entire domain. The parallelization of this technique is developed and the advantages are highlighted. By this, the computational complexity is drastically reduced. The mean-square block error is computed. It provides accurate results when using a family of covariance functions with compact support. Some numerical examples to study the performance of the proposed approach are included.

We show that the Jouanolou foliation of degree 2 on the complex projective plane is structurally stable. Moreover, its Fatou set is a fibration on the Klein quartic with the structure of a smooth fiber bundle in disks. In particular, there is no dense leaf., Comment: in French language

We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $\lambda\beta e^{\beta u }$, forced by an additive space-time white noise. We prove local and global well-posedness of these equations, depending on the sign of $\lambda$ and the size of $\beta^2 > 0$, and invariance of the associated Gibbs measures. See the abstract of the paper for a more precise abstract. (Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here.), Comment: 70 pages. To appear in Comm. Math. Phys. Minor typos corrected

Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced by Ditlevsen and Löcherbach (Stoch. Process. Appl., 2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. In this paper, first, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity, and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.

Making use of the localised Putnam theory developed in [GJ1], applied in [JM],..., we show the limiting absorption principle for Dirac operators with a perturbed Wigner-Von Neumann potential on appropriate energy intervals. Furthermore we can allow longrange and short-range components.

International audience; We study the semilinear elliptic equation −∆u + g(u)σ = µ with Dirichlet boundary conditions in a smooth bounded domain where σ is a nonnegative Radon measure, µ a Radon measure and g is an absorbing nonlinearity. We show that the problem is well posed if we assume that σ belongs to some Morrey class. Under this condition we give a general existence result for any bounded measure provided g satisfies a subcritical integral assumption. We study also the supercritical case when g(r) = |r| q−1 r, with q > 1 and µ satisfies an absolute continuity condition expressed in terms of some capacities involving σ. 2010 Mathematics Subject Classification. 35 J 61; 31 B 15; 28 C 05 .

International audience

We study the Gross-Pitaevskii equation in dimension two with periodic conditions in one direction, or equivalently on the product space $\mathbb{R} \times \mathbb{T}_L$ where $L > 0$ and $\mathbb{T}_L = \mathbb{R} / L \mathbb{Z}.$ We focus on the variational problem consisting in minimizing the Ginzburg-Landau energy under a fixed momentum constraint. We prove that there exists a threshold value for $L$ below which minimizers are the one-dimensional dark solitons, and above which no minimizer can be one-dimensional.

We study the weak universality of the two-dimensional fractional nonlinear wave equation. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional $\Phi_2^4$, we first establish a \emph{sufficient and almost necessary} criteria for the convergence of invariant measures to the fractional $\Phi_2^4$. Then we prove the convergence result for the sequence of associated wave dynamics to the (renormalized) cubic wave equation. Our constraint on the fractional index is independent of the degree of the nonlinearity. This extends the result of Gubinelli-Koch-Oh [Renormalisation of the two-dimensional stochastic nonlinear wave equations, Trans. Amer. Math. Soc. 370 (2018)] to a situation where we do not have a local Cauchy theory with highly supercritical nonlinearities., Comment: Comments are welcome

In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) adding a convex constraint on the solution. Through a sequence of approximating Hilbertian spaces and a discretized model, we prove that the Maximum A Posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalization of the paper [11] of Kimeldorf-Wahba where it is proved that the optimal smoothing solution is the mean of the posterior distribution. This is also a generalization of the paper [4] where the case of constrained optimal interpolation is treated. Here we relax the interpolation by introducing noise effect in the data. A numerical example is given to illustrate the theoretical result of this paper.

International audience; We study a class of dynamical semigroups $(\mathbb{L}^n)_{n\in\mathbb{N}}$ that emerge, by a Feynman--Kac type formalism, from a random quantum dynamical system $(\mathcal{L}_{\omega_n}\circ\cdots\circ\mathcal{L}_{\omega_1}(\rho_{\omega_0}))_{n\in\mathbb{N}}$ driven by a Markov chain $(\omega_n)_{n\in\mathbb{N}}$. We show that the almost sure large time behavior of the system can be extracted from the large $n$ asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator $\mathbb{L}$. As a physical application, we consider the case where the $\mathcal{L}_\omega$'s are the reduced dynamical maps describing the repeated interactions of a system $\mathcal{S}$ with thermal probes $\mathcal{C}_\omega$. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.

International audience; Statistical researchers have shown increasing interest in generating conditional multivariate normal distributions. In this paper, we discuss several existing methods for the simulation of multivariate normal distribution truncated on the intersection of a set of hyperplanes. We also propose an approach based on the consideration of an orthonormal basis on the set of constraints. Contrarily to the standard approaches, we do not need to compute the covariance matrix of the posterior distribution and its decomposition. The interest of the proposed approach is shown through numerical examples.

Generating multivariate normal distributions is widely used in many fields (such as engineering). In this paper, simulating large multivariate normal distributions truncated on the intersection of a set of hyperplanes is investigated. The proposed methodology focuses on Gaussian vectors extracted from a Gaussian process (GP) in one dimension. It is based on combining both Karhunen-Loève expansions (KLE) and Matheron's update rules (MUR). The KLE requires the computation of the decomposition of the covariance matrix of the random variables. This step becomes expensive when the random vector is too large. To deal with this issue, the input domain is split in smallest subdomains where the eigendecomposition can be computed. By this strategy, the computational complexity is drastically reduced. The mean-square truncation and block errors have been calculated. Some numerical experiments are presented in order to study the efficiency of the proposed approach. Keywords Large scale random fields • hyperplane-truncated • Karhunen-Loève expansion • Matheron's update rule.

A complex Hadamard matrix is a square matrix $H\in M_N(\mathbb C)$ whose entries are on the unit circle, $|H_{ij}|=1$, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, $F_N=(w^{ij})$ with $w=e^{2\pi i/N}$. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects.

Given a morphism $\varphi : G \to A \wr B$ from a finitely presented group $G$ to a wreath product $A \wr B$, we show that, if the image of $\varphi$ is a sufficiently large subgroup, then $\mathrm{ker}(\varphi)$ contains a non-abelian free subgroup and $\varphi$ factors through an acylindrically hyperbolic quotient of $G$. As direct applications, we classify the finitely presented subgroups in $A \wr B$ up to isomorphism and we deduce that a group having a wreath product $(\text{non-trivial}) \wr (\text{infinite})$ as a quotient must be SQ-universal (extending theorems of Baumslag and Cornulier-Kar). Finally, we exploit our theorem in order to describe the structure of the automorphism groups of several families of wreath products, highlighting an interesting connection with the Kaplansky conjecture on units in group rings., Comment: 20 pages, 2 figures. Comments are welcome!

Generalized metric spaces are natural and interesting geometric objects, which were firstly studied in depth by H. Busemann. In this paper, we are interested in the formulation and the study of hyperbolic generalized metric spaces. We construct some examples of this kind of spaces, which have infinite reversibility. Therefore, they cannot be quasi-isometrically embedded into any metric space. We define and study the boundary at infinity of hyperbolic generalized metric spaces, by sticking together their natural forward and backward boundaries. Finally, via a concrete example, we show that in some cases the boundary admits a couple of natural and non-equivalent quasi-conformal structures, which seems worthy of further study.

The purpose of this article is to show that the spectral projector estimates for Laplace operators can be deduced from Logunov-Malinnikova's Propagation of smallness estimates for harmonic functions [11, 10, 9]. The main point is to pass from the local estimates obtained in [4] (on a compact manifold) to global estimates. We also state classical consequences in terms of observability and control for heat equations, which are direct consequences of these spectral projector estimates..

We consider an electronic bound state of the usual, non-relativistic, molecular Hamiltonian with Coulomb interactions and fixed nuclei. Away from appropriate collisions, we prove the real analyticity of all the reduced densities and density matrices, that are associated to this bound state. We provide a similar result for the associated reduced current density.Published: J. Math. Phys. 63, 013509 (2022); https://doi.org/10.1063/5.0056488, Comment: Le co-auteur de l'article Camille No{\^u}s a {\'e}t{\'e} refus{\'e} par l'{\'e}diteur du journal ''Journal of mathematical physics'' qui a publi{\'e} l'article. La raison invoqu{\'e}e par le journal {\'e}tait que cet auteur est ''fictif''

The permutation group $S_N$ has a quantum analogue $S_N^+$, which is infinite at $N\geq4$. We review the known facts regarding $S_N^+$, and its versions $S_F^+$, with $F$ being a finite quantum space. We discuss then the structure of the closed subgroups $G\subset S_N^+$ and $G\subset S_F^+$, with particular attention to the quantum reflection groups.

The space $\mathbb C^N$ has no free analogue, but we can talk instead about the free sphere $S^{N-1}_{\mathbb C,+}$, as the manifold defined by the equations $\sum_ix_ix_i^*=\sum_ix_i^*x_i=1$. We discuss here the structure and hierarchy of the submanifolds $X\subset S^{N-1}_{\mathbb C,+}$, with particular attention to the manifolds having an integration functional $tr:C(X)\to\mathbb C$.

In this paper, we study the amount of information contained in the Steklov spectrum of some compact manifolds with connected boundary equipped with a warped product metric. Examples of such manifolds can be thought of as deformed balls in $${\mathbb {R}}^d$$ . We first prove that the Steklov spectrum determines uniquely the warping function of the metric. We show in fact that the approximate knowledge (in a given precise sense) of the Steklov spectrum is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, we provide stability estimates of log-type on the warping function from the Steklov spectrum. The key element of these stability results relies on a formula that, roughly speaking, connects the inverse data (the Steklov spectrum) to the Laplace transform of the difference of the two warping factors.

We study the asymptotic behavior of Masur–Veech volumes as the genus goes to infinity. We show the existence of a complete asymptotic expansion of these volumes that depends only on the genus and the number of singularities. The computation of the 1st term of this asymptotics expansion was a long-standing problem. This problem was recently solved in [2] using purely combinatorial arguments and then in [3] using algebro-geometric insights. Our proof relies on a combination of both methods.

We consider the Cauchy problem for the nonlinear wave equation $$u_{tt} - \Delta _x u +q(t, x) u + u^3 = 0$$ with smooth potential $$q(t, x) \ge 0$$ having compact support with respect to x. The linear equation without the nonlinear term $$u^3$$ and potential periodic in t may have solutions with exponentially increasing $$H^1(\mathbb {R}^{3}_{x})$$ norm as $$t\rightarrow \infty $$. In Petkov and Tzvetkov (IMRN, https://doi.org/10.1093/imrn/rnz014), it was established that by adding the nonlinear term $$u^3$$, the $$H^1(\mathbb {R}^{3}_{x})$$ norm of the solution is polynomially bounded for every choice of q. In this paper, we show that the $$H^k({{\mathbb {R}}}^3_x)$$ norm of this global solution is also polynomially bounded. To prove this, we apply a different argument based on the analysis of a sequence $$\{Y_k(n\tau _k)\}_{n = 0}^{\infty }$$ with suitably defined energy norm $$Y_k(t)$$ and $$0< \tau _k

We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve., 26 pages, 1 figure, comments are welcome

In this paper, we extend the correspondence between Bayesian estima- tion and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) by adding convex constraints to the problem. Through a sequence of approxi- mating Hilbertian subspaces and a discretized model, we prove that the Max- imum a posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalization of the paper [15], where it is proved that the optimal smooth- ing solution is the mean of the posterior distribution. Synthetic and real data studies confirm the correspondence established in this paper.

The partial automorphisms of a graph $X$ having $N$ vertices are the bijections $\sigma:I\to J$ with $I,J\subset\{1,\ldots,N\}$ which leave invariant the edges. These bijections form a semigroup $\widetilde{G}(X)$, which contains the automorphism group $G(X)$. We discuss here the quantum analogue of this construction, with a definition and basic theory for the quantum semigroup of quantum partial automorphisms $\widetilde{G}^+(X)$, which contains both $G(X)$, and the quantum automorphism group $G^+(X)$. We comment as well on the case $N=\infty$, which is of particular interest, due to the fact that $\widetilde{G}^+(X)$ is well-defined, while its subgroup $G^+(X)$, not necessarily, at least with the currently known methods., Comment: Withdrawn by the author - this paper is now part of arXiv:2012.10975

This article is dedicated to the asymptotic geometry of wreath products $F\wr H := \left( \bigoplus_H F \right) \rtimes H$ where $F$ is a finite group and $H$ a one-ended finitely presented group. Our main result is a complete classification of these groups up to quasi-isometry. More precisely, given two finite groups $F_1,F_2$ and two finitely presented one-ended groups $H_1,H_2$, we show that $F_1 \wr H_1$ and $F_2 \wr H_2$ are quasi-isometric if and only if either (i) $H_1,H_2$ are non-amenable quasi-isometric groups and $|F_1|,|F_2|$ have the same prime divisors, or (ii) $H_1,H_2$ are amenable, $|F_1|=k^{n_1}$ and $|F_2|=k^{n_2}$ for some $k,n_1,n_2 \geq 1$, and there exists a quasi-$(n_2/n_1)$-to-one quasi-isometry $H_1 \to H_2$. The article also contains algebraic information on groups quasi-isometric to such wreath products. This can be seen as far reaching extension of a celebrated work of Eskin-Fisher-Whyte who treated the case of $H=\mathbb{Z}$. Our approach is however fundamentally different, as it crucially exploits the assumption that $H$ is one-ended. Our central tool is a new geometric interpretation of lamplighter groups involving natural families of quasi-median spaces., 58 pages, 7 figures. Comments are welcome

It is known that for some time-periodic potentials $q(t, x) \geq 0$ having compact support with respect to $x$ some solutions of the Cauchy problem for the wave equation $\partial _t^2 u - \Delta _x u + q(t,x)u = 0$ have exponentially increasing energy as $t \to \infty $. We show that if one adds a nonlinear defocusing interaction $|u|^ru, 2\leq r < 4,$ then the solution of the nonlinear wave equation exists for all $t \in{\mathbb R}$, and its energy is polynomially bounded as $t \to \infty $ for every choice of $q$. Moreover, we prove that the zero solution of the nonlinear wave equation is instable if the corresponding linear equation has the property mentioned above.

International audience; In this paper, we consider stochastic master equations describing the evolution of quantum spin-1/2 systems interacting with electromagnetic fields undergoing continuous-time measurements. We suppose that the initial states and the exact values of the physical parameters are unknown. We prove that the feedback stabilization strategy considered in [16] is robust to these imperfections. This is shown by studying the asymptotic behavior of the coupled stochastic master equations describing the evolutions of the actual state and the estimated one under appropriate assumptions on the feedback controller. We provide sufficient conditions on the feedback controller and a valid domain of estimated parameters which ensure exponential stabilization of the coupled system. Furthermore, our results allow us to answer positively to [15, Conjecture 4.4] in the case of spin-1/2 systems with unknown initial states, even in presence of imprecisely known physical parameters.

In this article, we establish the logarithmic foundation for compactifying the moduli stacks of the gauged linear sigma model using stable log maps of Abramovich-Chen-Gross-Siebert. We then illustrate our method via the key example of Witten's $r$-spin class to construct a proper moduli stack with a reduced perfect obstruction theory whose virtual cycle recovers the $r$-spin virtual cycle of Chang-Li-Li. Indeed, our construction of the reduced virtual cycle is built upon the work of Chang-Li-Li by appropriately extending and modifying the Kiem-Li cosection along certain logarithmic boundary. In the subsequent article, we push the technique to a general situation. One motivation of our construction is to fit the gauged linear sigma model in the broader setting of Gromov-Witten theory so that powerful tools such as virtual localization can be applied. A project along this line is currently in progress leading to applications including computing loci of holomorphic differentials, and calculating higher genus Gromov-Witten invariants of quintic threefolds., Comment: v2: agrees with published version

We describe a conjectural formula via intersection numbers for the Masur-Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to compute the top Segre class of the quadratic Hodge bundle, which can be further simplified to certain linear Hodge integrals. An appendix proves that the intersection of this class with $\psi$-classes can be computed by Eynard-Orantin topological recursion. As applications, we analyze numerical properties of Masur-Veech volumes, area Siegel-Veech constants and sums of Lyapunov exponents of the principal strata for fixed genus and varying number of zeros, which settles the corresponding conjectures due to Grivaux-Hubert, Fougeron, and elaborated in [the7]. We also describe conjectural formulas for area Siegel-Veech constants and sums of Lyapunov exponents for arbitrary affine invariant submanifolds, and verify them for the principal strata.

We consider the energy supercritical defocusing nonlinear Schrödinger equation $$\begin{aligned} i\partial _tu+\Delta u-u|u|^{p-1}=0 \end{aligned}$$ i ∂ t u + Δ u - u | u | p - 1 = 0 in dimension $$d\ge 5$$ d ≥ 5 . In a suitable range of energy supercritical parameters (d, p), we prove the existence of $${\mathcal {C}}^\infty $$ C ∞ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of $${\mathcal {C}}^\infty $$ C ∞ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.

This paper aims at providing statistical guarantees for a kernel-based estimation of time-varying parameters driving the dynamic of infinite memory processes introduced by Doukhan and Wintenberger \cite{DW}. We then extend the results of Dahlhaus {\it et al.} \cite{DRW} on local stationary Markov processes to other important models such as the GARCH model. The estimators are computed as localized M-estimators of any contrast satisfying appropriate regularity conditions. % as in Bardet and Wintenberger \cite{BW}. We prove the uniform consistency and the pointwise asymptotic normality of such kernel-based estimators. We apply our results to usual contrasts such as least-square, least absolute value, or quasi-maximum likelihood contrasts. Various time-varying models such as AR$(\infty$), ARCH$(\infty)$ and LARCH$(\infty)$ are considered. We discuss their approximation of locally stationary ARMA and GARCH models under contraction conditions. Numerical experiments demonstrate the efficiency of the estimators on both simulated and real data sets.

International audience; We prove polynomial upper bounds on the growth of solutions to 2d cubic NLS where the Laplacian is confined by the harmonic potential. Due to better bilinear effects our bounds improve on those available for the $2d$ cubic NLS in the periodic setting: our growth rate for a Sobolev norm of order s=2k, $k\in \mathbb{N}$, is $t^{2(s-1)/3+\varepsilon}$. In the appendix we provide an direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.

We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg-Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg-Landau energy is well-defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross-Pitaevskii equation, the natural Hamiltonian evolution equation associated to the Ginzburg-Landau energy.

We pursue our work on the asymptotic regimes of the Landau–Lifshitz equation. We put the focus on the cubic Schrodinger equation, which is known to describe the dynamics of uniaxial ferromagnets in a regime of strong easy-axis anisotropy. In any dimension, we rigorously prove this claim for solutions with sufficient regularity. In this regime, we additionally classify the one-dimensional solitons of the Landau–Lifshitz equation and quantify their convergence towards the solitons of the one-dimensional cubic Schrodinger equation. Our proof is based on a nonlinear rewriting of the Landau–Lifshitz equation for uniaxial ferromagnets as a nonlinear Schrodinger equation. This latter equation turns out to be consistent with the cubic Schrodinger equation in a regime of strong easy-axis anisotropy. We rely on this consistency to perform suitable energy estimates and then control the difference between the solutions of the two equations.

In the first part of this thesis we are studying the link between individual-based stochastic models (birth-and-death processes, Hawkes processes) and their respective continuous approximations (stochastic diffusions, partial differential equations), obtained at a larger scale. In particular, we are tackling the question of the numerical simulation of stochastic and deterministic processes with the help of splitting and implicit numerical schemes, which preserve the asymptotic behavior of the process. On the applied level, we consider mathematical models of interacting networks of biological neurons, as well as bacteria populations. In the second part of the manuscript we are dealing with statistics for stochastic differential equations, such as parametric inference for diffusions with degenerate noise, and hypothesis testing for the covariance matrix rank from discrete observations. For the parametric estimation we use quasi-maximum likelihood estimators (also known as contrast estimators), where the contrast is built on the density approximated with the local linearization scheme. For the second problem, we study a non-asymptotic regime (i.e., the case when the observations are available with a fixed time step). We consider the case when the distribution of the test statistics can be written explicitly (for example, when the drift is known and the dimension is 1 or 2). Then, we use concentration inequalities to derive the tail properties and use them to obtain a non-asymptotic control of the Type I and Type II errors.; Dans la première partie de cette thèse, nous étudions le lien entre les modèles stochastiques indivu-centrés (processus de naissance et de mort, processus de Hawkes) et leurs approximations continues respectives (diffusions stochastiques, l'équations aux dérivées partielles), obtenues à plus grande échelle. En particulier, nous abordons la question de la simulation numérique des processus stochastiques et déterministes à l'aide de schémas par fractionnement (splitting) et numériques implicites, qui préservent le comportement asymptotique du processus. Au niveau appliqué, nous considérons des modèles mathématiques d'interaction de réseaux de neurones biologiques, ainsi que des populations de bactéries.Dans la deuxième partie du manuscrit, nous traitons de statistique pour les équations différentielles stochastiques, telles que l'inférence paramétrique pour les diffusions avec bruit dégénéré, et le test d'hypothèses pour le rang de la matrice de covariance à partir d'observations discrètes. Pour l'estimation paramétrique, nous utilisons des estimateurs de quasi-maximum de vraisemblance (également appelés estimateurs de contraste), où le contraste est construit sur la densité approchée avec le schéma de linéarisation locale. Pour le deuxième problème, nous étudions un régime non asymptotique (c'est-à-dire le cas où les observations sont disponibles avec un pas de temps fixe). On considère le cas où la distribution des statistiques de test peut être écrite explicitement (par exemple, lorsque la dérive est connue et que la dimension est 1 ou 2). Ensuite, nous utilisons des inégalités de concentration pour évaluer les erreurs du 1ère et 2ème espèces du test.

In the first part of this thesis we are studying the link between individual-based stochastic models (birth-and-death processes, Hawkes processes) and their respective continuous approximations (stochastic diffusions, partial differential equations), obtained at a larger scale. In particular, we are tackling the question of the numerical simulation of stochastic and deterministic processes with the help of splitting and implicit numerical schemes, which preserve the asymptotic behavior of the process. On the applied level, we consider mathematical models of interacting networks of biological neurons, as well as bacteria populations. In the second part of the manuscript we are dealing with statistics for stochastic differential equations, such as parametric inference for diffusions with degenerate noise, and hypothesis testing for the covariance matrix rank from discrete observations. For the parametric estimation we use quasi-maximum likelihood estimators (also known as contrast estimators), where the contrast is built on the density approximated with the local linearization scheme. For the second problem, we study a non-asymptotic regime (i.e., the case when the observations are available with a fixed time step). We consider the case when the distribution of the test statistics can be written explicitly (for example, when the drift is known and the dimension is 1 or 2). Then, we use concentration inequalities to derive the tail properties and use them to obtain a non-asymptotic control of the Type I and Type II errors., Dans la première partie de cette thèse, nous étudions le lien entre les modèles stochastiques indivu-centrés (processus de naissance et de mort, processus de Hawkes) et leurs approximations continues respectives (diffusions stochastiques, l'équations aux dérivées partielles), obtenues à plus grande échelle. En particulier, nous abordons la question de la simulation numérique des processus stochastiques et déterministes à l'aide de schémas par fractionnement (splitting) et numériques implicites, qui préservent le comportement asymptotique du processus. Au niveau appliqué, nous considérons des modèles mathématiques d'interaction de réseaux de neurones biologiques, ainsi que des populations de bactéries.Dans la deuxième partie du manuscrit, nous traitons de statistique pour les équations différentielles stochastiques, telles que l'inférence paramétrique pour les diffusions avec bruit dégénéré, et le test d'hypothèses pour le rang de la matrice de covariance à partir d'observations discrètes. Pour l'estimation paramétrique, nous utilisons des estimateurs de quasi-maximum de vraisemblance (également appelés estimateurs de contraste), où le contraste est construit sur la densité approchée avec le schéma de linéarisation locale. Pour le deuxième problème, nous étudions un régime non asymptotique (c'est-à-dire le cas où les observations sont disponibles avec un pas de temps fixe). On considère le cas où la distribution des statistiques de test peut être écrite explicitement (par exemple, lorsque la dérive est connue et que la dimension est 1 ou 2). Ensuite, nous utilisons des inégalités de concentration pour évaluer les erreurs du 1ère et 2ème espèces du test.

International audience; In this paper, we consider stochastic master equations describing the evolution of a multi-qubit system interacting with electromagnetic fields undergoing continuous-time measurements. By considering multiple z-type (Pauli z matrix on different qubits) and x-type (Pauli x matrix on all qubits) measurements and one control Hamiltonian, we provide general conditions on the feedback controller and the control Hamiltonian ensuring almost sure exponential convergence to a predetermined Greenberger-Horne-Zeilinger (GHZ) state, which is assumed to be a common eigenstate of the measurement operators. We provide explicit expressions of feedback controllers satisfying such conditions. We also consider the case of only z-type measurements and multiple control Hamiltonians, and we discuss asymptotic convergence towards a predetermined GHZ state. Finally, we demonstrate the effectiveness of our methodology for a three-qubit system through numerical simulations.

We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an instability phenomenon, in the following sense: if some parameter associated to the model is rational (resp.\ non-rational), then the Martin boundary is countable, homeomorphic to $\mathbb Z\cup\{\pm\infty\}$ (resp.\ uncountable, homeomorphic to $\mathbb R\cup\{\pm\infty\}$). Such instability phenomena are very rare in the literature. Along the way of proving this result, we obtain several precise estimates for the Green functions of reflected random walks with escape probabilities along the boundary axes and an arbitrarily large number of inhomogeneity domains. Our methods mix probabilistic techniques and an analytic approach for random walks with large jumps in dimension two., Nous introduisons une famille de marches al\'eatoires en dimension deux, r\'efl\'echies au bord du quart de plan positif, et \'etudions leur fronti\`ere de Martin. Tandis que leur fronti\`ere minimale est syst\'ematiquement une union de deux points, nous montrons que la fronti\`ere de Martin compl\`ete est intrins\`equement instable, au sens suivant : lorsqu'un certain param\`etre associ\'e au mod\`ele s'av\`ere rationnel (respectivement non rationnel), la fronti\`ere de Martin est alors d\'enombrable et hom\'eomorphe \`a $\mathbb Z\cup\{\pm\infty\}$ (respectivement non d\'enombrable et hom\'eomorphe \`a $\mathbb R\cup\{\pm\infty\}$). De tels ph\'enom\`enes d'instabilit\'e sont rares dans la litt\'erature. Les d\'emonstrations contiennent plusieurs estim\'ees pr\'ecises pour des fonctions de Green de marches al\'eatoires r\'efl\'echies avec probabilit\'e de fuite le long des axes, poss\'edant en outre un nombre infini de domaines d'inhomog\'en\'eit\'e. Nos m\'ethodes m\'elangent des techniques probabilistes avec une approche analytique pour des marches al\'eatoires avec grands pas en dimension deux.

We continue our study of the cubic fractional NLS with very weak dispersion $\alpha>1$ and data distributed according to the Gibbs measure. We construct the natural strong solutions for $\alpha>\alpha_0=\frac{31-\sqrt{233}}{14}\approx 1.124$ which is strictly smaller than $\frac{8}{7}$, the threshold beyond which the first nontrivial Picard iteration has no longer the Sobolev regularity needed for the deterministic well-posedness theory. This also improves our previous result in Sun-Tzvetkov \cite{Sun-Tz2}. We rely on recent ideas of Bringmann \cite{Bringmann} and Deng-Nahmod-Yue \cite{Deng2}. In particular we adapt to our situation the new resolution ansatz in \cite{Deng2} which captures the most singular frequency interaction parts in the $X^{s,b}$ type space. To overcome the difficulties caused by the weakly dispersive effect, our specific strategy is to benefit from the "almost" transport effect of these singular parts and to exploit their $L^{\infty}$ as well as the Fourier-Lebesgue property in order to inherit the random feature from the linear evolution of high frequency portions., Comment: 84 pages

We provide a variational construction of special solutions to the generalized surface quasi-geostrophic equations. These solutions take the form of N vortex patches with N-fold symmetry , which are steady in a uniformly rotating frame. Moreover, we investigate their asymptotic properties when the size of the corresponding patches vanishes. In this limit, we prove these solutions to be a desingularization of N Dirac masses with the same intensity, located on the N vertices of a regular polygon rotating at a constant angular velocity.

In this article, we study the observability (or equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the ‘degenerated directions’ of the subelliptic structure.First, for any $\gamma \geq 1$ , we establish a resolvent estimate for the Baouendi–Grushin-type operator $\Delta _{\gamma }=\partial _x^2+\left \lvert x\right \rvert ^{2\gamma }\partial _y^2$ , which has step $\gamma +1$ . We then derive consequences for the observability of the Schrödinger-type equation $i\partial _tu-\left (-\Delta _{\gamma }\right )^{s}u=0$ , where $s\in \mathbb N$ . We identify three different cases: depending on the value of the ratio $(\gamma +1)/s$ , observability may hold in arbitrarily small time or only for sufficiently large times or may even fail for any time.As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial _tu+\left (-\Delta _{\gamma }\right )^su=0$ and establish a decay rate for the damped wave equation associated with $\Delta _{\gamma }$ .

We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel-Veech constants for all strata. We also show that the same results hold for the spin and hyper-elliptic components of the strata., Comment: 67 pages, 3 figures

International audience; In this paper, we consider $N$-level quantum angular momentum systems interacting with electromagnetic fields undergoing quantum nondemolition measurements in continuous-time. We suppose unawareness of the initial state and physical parameters, entailing the introduction of an additional state representing the estimated quantum state. The evolution of the quantum state and its estimation is described by a coupled stochastic master equation. Here, we study the asymptotic behavior of such a system in the presence of a feedback controller. We provide sufficient conditions on the feedback controller and on the estimated parameters that guarantee exponential stabilization of the coupled stochastic system toward an eigenstate of the measurement operator. Furthermore, we estimate the corresponding rate of convergence. We also provide parametrized feedback laws satisfying such conditions. Our results show the robustness of the feedback stabilization strategy considered in [W. Liang, N. H. Amini, and P. Mason, SIAM J. Control Optim., 57 (2019), pp. 3939--3960] in the case of imprecise initialization of the estimated state and with respect to the unknown physical parameters.

In this paper we study a particular class of Piecewise deterministic Markov processes (PDMP's) which are semi-stochastic catastrophe versions of deterministic population growth models. In between successive jumps the process follows a flow describing deterministic population growth. Moreover, at random jump times, governed by state-dependent rates, the size of the population shrinks by a random amount of its current size, an event possibly leading to instantaneous local (or total) extinction. A special separable shrinkage transition kernel is investigated in more detail, including the case of total disasters. We discuss conditions under which such processes are recurrent (positive or null) or transient. To do so, we introduce a modified scale function which is used to compute, when relevant, the law of the height of excursions and to decide if the process is recurrent or not. The question of the finiteness of the time to extinction is investigated together with the evaluation of the mean time to extinction when the last one is finite. Some information on the embedded jump chain of the PDMP is also required when dealing with the classification of states 0 and infinity that we exhibit., 2 figures

International audience; We study the quasi-mode of the Stokes system posed on a smooth bounded domain X with Dirichlet boundary condition. We prove that the semi-classical defect measure associated with a sequence of solutions concentrates on the bicharacteristics of Laplacian as a matrix-valued Radon measure. Moreover, we show that the support of the measure is invariant under the Melrose-Sj\"ostrand flow.