Título: Approximations of the covariance operators of solutions of fractional elliptic SPDEs driven by Gaussian white noise
Palestrante: Alexandre de Bustamante Simas (UFPB & Kaust)
Data: 13/09/2021
Horário: 15:00h
Local: Transmissão online.
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Resumo: In this talk we will briefly present the model we are interested in, which is a fractional elliptic stochastic partial differential equation driven by Gaussian white noise. There is in the literature a standard way to approximate the covariance operator of the solution of such equations, the so-called rational approximation (Bolin and Kirchner, 2020), however this approach uses the solution to build such an approximation. By considering directly the covariance operator, we are able to provide a more computationally efficient approximation. We compute the rate of this approximation in terms of the Hilbert-Schmidt norm. Furthermore, we also obtain, rigorously, the rate of approximation of the so-called lumped mass method. This method is widely used by practitioners and is essential to make it computationally feasible to fit some models in spatial statistics. We obtain the rate of approximation of the lumped mass method in terms of the operator's norm as well as, under some additional restrictions, the Hilbert-Schmidt norm. Finally, we present the usage of these approximations in maximum likelihood estimation. Joint work with David Bolin and Zhen Xiong.
All the talks are held in English.
The videos of the online seminars are available:
2020
2021-1
For the second semester, a few days after each meeting the video should be available at HERE.
Título: Dyson models with random boundary conditions
Palestrante: Aernout van Enter (Groningen University)
Data: 06/09/2021
Horário: 15:00h
Local: Transmissão online.
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Resumo: I discuss the low-temperature behaviour of Dyson models (polynomially decaying long-range Ising models in one dimension) in the presence of random boundary conditions. As for typical random (i.i.d.) boundary conditions Chaotic Size Dependence occurs, that is, the pointwise thermodynamic limit of the finite-volume Gibbs states for increasing volumes does not exist, but the sequence of states moves between various possible limit points, as a consequence it makes sense to study distributional limits, the so-called "metastates" which are measures on the possible limiting Gibbs measures.
The Dyson model is known to have a phase transition for decay parameters α between 1 and 2. We show that the metastate changes character at α =3/2. It is dispersed in both cases, but it changes between being supported on two pure Gibbs measures when α is less than 3/2 to being supported on mixtures thereof when α is larger than 3/2.
Joint work with Eric Endo and Arnaud Le Ny
All the talks are held in English.
The videos of the online seminars are available:
2020
2021-1
For the second semester, a few days after each meeting the video should be available at HERE.
Título: Quasi-static hydrodynamic limits
Palestrante: Stefano Olla, Paris Dauphine
Data: 25/08/2021
Horário: 13:00h
Local: Transmissão online.
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ID da reunião: 958 0581 3232
Resumo: The quasi-static scaling limit corresponds to changes of the boundary conditions (boundary tension, temperature heat bath, density of particle reservoirs) on a time scale that is slower (i.e. larger) than the equilibrium relaxation scale of the dynamics in the bulk. In this very large time scale, the system is always very close to the global stationary state corresponding to the (time varying) tension, temperatures or densities applied at the boundaries. These quasi-static evolutions are usually presented as idealization of real thermodynamic transformations. On the other hand they are necessary concepts in order to construct thermodynamic potentials, for example to define thermodynamic entropy from Carnot cycles. The existence of the quasi-static transformations can be seen as another thermodynamic principle that needs to be derived from the microscopic dynamics under a proper space-time scaling that we call quasi-static hydrodynamic limit. We are particularly interested in studying quasi-static transformations among non-equilibrium stationary states (NESS). I will expose some results concerning quasi-static hydrodynamic limits for dynamics with diffusive behavior (like symmetric simple exclusion and anharmonic chains in contact with heat bath in the bulk and tension at the boundary) and hyperbolic/ballistic behavior (like the asymmetric simple exclusion). We also studies the large deviations from these quasi-static limits and the relation with the large deviations in the corresponding NESS. Works in collaboration with Anna de Masi, Lu Xu, Stefano Marchesani.
Título: The Widom-Rowlinson model: metastability, mesoscopic and microscopic fluctuations for the critical droplet
Palestrante: Elena Pulvirenti (Delft University of Technology)
Data: 30/08/2021
Horário: 15:00h
Local: Transmissão online.
Confira AQUI o link para transmissão.
Resumo: We introduce the equilibrium Widom-Rowlinson model on a two-dimensional finite torus in which the energy of a particle configuration is attractive and determined by the union of small discs centered at the positions of the particles. We then discuss the metastable behaviour of a dynamic version of the WR model. This means that the particle configuration is viewed as a continuous time Markov process where particles are randomly created and annihilated as if the outside of the torus were an infinite reservoir with a given chemical potential. In particular, we start with the empty torus and are interested in the first time when the torus is fully covered by discs in the regime at low temperature and when the chemical potential is supercritical. In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet, called critical droplet, which triggers the crossover. We compute the distribution of the crossover time and identify the size and the shape of the critical droplet. The analysis relies on a mesoscopic and microscopic description of the surface of the critical droplet. It turns out that the critical droplet is close to a disc of a certain deterministic radius, with a boundary that is random and consists of a large number of small discs that stick out by a small distance. We will show how the analysis of surface fluctuations in the WR model allows us to derive the leading order term of the condensation time and also the correction order term. This is a joint work with Frank den Hollander (Leiden), Sabine Jansen (Munich) and Roman Kotecky (Prague & Warwick).
All the talks are held in English.
The videos of the online seminars are available:
2020
2021-1
For the second semester, a few days after each meeting the video should be available at HERE.
Títulos: The Parabolic Anderson Model on a Galton-Watson Tree e Local Scaling Limits of Lévy Driven Fractional Random Fields
Palestrantes: Frank den Hollander (Leiden University) e Donatas Surgailis (Vilnius University)
Data: 23/08/2021
Horário: 14:00h
Local: Transmissão online.
Confira AQUI o link para transmissão.
Resumos:
Frank den Hollander (Leiden University) - The Parabolic Anderson Model on a Galton-Watson Tree
We consider the parabolic Anderson model on a supercritical Galton-Watson tree with an i.i.d.\ random potential whose marginal distribution is close to the double-exponential. Under the assumption that the degree distribution has a sufficiently thin tail, we derive an asymptotic expansion for large times of the total mass of the solution given that initially a unit mass sits at the root. We derive the expansion both under the quenched law (i.e., conditional on the realisation of the random tree and the random potential) and under the half-annealed law (i.e., conditional on the realisation of the random tree but averaged over the random potential). The two expansions turn out to be different, but both contain a coefficient that is given by a variational formula indicating that the solution concentrates on a subtree with minimal degree according to a computable profile. A key tool in the analysis is the large deviation principle for the empirical distribution of a Markov renewal process.
Joint work with Wolfgang König (Berlin), Renato dos Santos (Belo Horizonte), Daoyi Wang (Leiden).
Donatas Surgailis (Vilnius University) - Local Scaling Limits of Lévy Driven Fractional Random Fields
We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields $X$ on ${\mathbb{R}}^2$ written as stochastic integral with respect to an infinitely divisible random measure. The scaling procedure involves increments of $X$ over points the distance between which in the horizontal and vertical directions shrinks as $O(\lambda) $ and $O(\lambda^\gamma)$ respectively as $\lambda \downarrow 0$, for some $\gamma>0$. We consider two types of increments of $X$: usual increment and rectangular increment, leading to the respective concepts of $\gamma$-tangent and $\gamma$-rectangent random fields. We prove that for above $X$ both types of local scaling limits exist for any $\gamma>0$ and undergo a transition, being independent of $\gamma>\gamma_0$ and $\gamma<\gamma_0$, for some $\gamma_0>0$; moreover, the `unbalanced' scaling limits ($\gamma\ne\gamma_0$) are $(H_1,H_2)$-multi self-similar with one of $H_i$, $i=1,2$, equal to $0$ or $1$. The paper extends Pilipauskaite and Surgailis (2017) and Surgailis (2020) on large-scale anisotropic scaling of random fields on ${\mathbb{Z}}^2$ and Benassi et al. (2004) on $1$-tangent limits of isotropic fractional Lévy random fields.
This is joint work with Vytaute Pilipauskaite (University of Luxembourg)