Filter off: No filter for categories.
In my talk, we start with introducing gradient systems and EDP-convergence. A gradient system (X, E, R) consists of a state space Q ⊂ X (X is a separable, reflexive Banach space), an energy functional E : Q → R ∪ {∞} and a dissipation potential R : Q × X → [0, ∞[, which is convex, lower semicontinuous and satisfies R(·, 0) = 0. The associated gradient-flow equation is then given by 0 ∈ ∂R(u(t), u′(t)) + DE(u(t)) or equivalently u′(t) ∈ ∂R∗(u(t), −DE(u(t))). The energy-dissipation principle (EDP) then states, that the gradient-flow evolution equation is equivalent to the energy-dissipation balance of the form E(u(t)) + D(u) = E(u(0)), D(u) = ∫ T 0 R(u, ̇u) + R∗(u, −DE(u)) dt. For families of gradient systems (Q, E, R), a structural convergence, the so-called ”EDP-convergence” has been developed in recent years. It provides many interesting features from the mathematical as well as the modeling perspective. As an application of the general theory, we consider reaction and reaction-diffusion systems satisfying mass-action kinetics in the singular limit of slow and fast reactions. Instead of investigating solely the evolution equations, our analysis uses methods from the calculus of variations and relies on the energy-dissipation principle of the associated gradient systems. We show that an effective gradient structure can be rigorously derived via EDP-convergence.
One of the archetypical aggregation-diffusion models is the so-called classical parabolic- elliptic Patlak-Keller-Segel (PKS for short) model. This model was classically introduced as the simplest description for chemotactic bacteria movement in which linear diffusion tendency to spread fights the attraction due to the logarithmic kernel interaction in two dimensions. For this model there is a well-defined critical mass. In fact, here a clear dichotomy arises: if the total mass of the system is less than the critical mass, then the long time asymptotics are described by a self-similar solution, while for a mass larger than the critical one, there is finite time blow-up. In this talk we will first give an overview about some results obtained in the papers [1]-[2] concerning the characterization of the stationary states for a nonlinear variant of the PKS model, of the form (1) ∂tρ = ∆ρm + ∇ · (ρ∇(W ∗ ρ)), being W ∈ C1(Rd \ {0}) a Riesz kernel aggregation, namely W(x) = cd,s|x|2s−d for s ∈ (0, d/2), in the assumptions of dominated diffusion, i.e. when i.e. for m > 2 − (2s)/d. In particular, all stationary states of the model are shown to be radially symmetric decreasing and uniquely identified with global minimizers of the associated free energy functionals. In the second part of the talk we will discuss the recent results established in the joint paper [3], in which an addition of a quadratic diffusion term in equation (1) produces a more precise competition with the aggregation term for small s, as they have the same scaling if s = 0. We characterize the asymptotic behavior of the stationary states behavior as s goes to zero. Finally, we establish the existence of gradient flow solutions to the evolution problem by applying the JKO scheme. References [1] J. A. Carrillo, F. Hoffmann, E. Mainini, B. Volzone. Ground States in the Diffusion-Dominated Regime, Calc. Var. Partial Differ. Equ. 57, No. 5, Paper No. 127, 28 p. (2018). [2] H. Chan, M. Gonz´alez, Y. Huang, E. Mainini, B. Volzone. Uniqueness of entire ground states for the fractional plasma problem., Calc. Var. Partial Differ. Equ. 59, No. 6, Paper No. 195, 41 p. (2020). [3] Y. Huang, E. Mainini, J. L. V´azquez, B. Volzone. Nonlinear aggregation-diffusion equations with Riesz potentials, arXiv:2205.13520 [math.AP] (2022)
Based on ordinary differential equations, we consider various classes of stochastic integral equations that may have path- or distribution-dependent coefficients or that are of Volterra type. By extending different methods, we derive unique strong solutions to such equations, determine the Hölder regularity of their paths, characterise the supports of their distributions and establish their moment and pathwise stability.
To understand mechanical origin of probability in statistical and continuum mechanics, it is useful to study hydrodynamic limit for interacting particles following deterministic Hamiltonian dynamics. Traditional approach on such a program faces many difficulties. One of them is about rigorous justification of canonical type ensembles. This is because that relevant deterministic ergodic theory is still largely out of reach. Another huge barrier is on making rigorous sense out of hyperbolic conservation laws. Such PDEs are used to express F=ma and thermodynamic relations in the continuum. We examine a new line of thoughts by formulating the hydrodynamic limit program as a multi-scale abstract Hamilton-Jacobi theory in space of probability measures. This talk will focus on derivation of an isentropic model. Through mass transport calculus, we develop tools to reduce the hydrodynamic problem to known results on finite dimensional weak KAM (Kolmogorov-Arnold-Moser) theory, showing sufficiency of using a weak version of ergodic results on micro-canonical type ensembles, instead of the canonical ones. We will also reply on recent progress of viscosity solution theory for abstract Hamilton-Jacobi equation in space of probability measures (an example of Alexandrov space). Such approach gives a weak and indirect characterization on evolution of the limiting continuum model using generating-function formalism at the level of canonical transformation in calculus of variations. It avoids the use of hyperbolic systems of PDEs, which operates at the level of abstract Euler-Lagrange equations from the action functionals. All together, these techniques enable us to realize a weaker but rigorous version of the hydrodynamic limit program for some nontrivial cases. This is a joint work with Toshio Mikami from Tsuda University, Tokyo, Japan.
The concept of time-correlated noise is an important tool of both discrete- and continuous-time stochastic modelling but approaches to its implementation are manifold. More specifically, while the discrete-time notion of so-called red noise is often synonymous with AR(1)-noise, there has not yet been a comprehensive motivation for the use of any specific continuous-time analogue. We discuss this generalisation to the continuous-time case and its inherent ambiguities. The implications of carrying certain attributes like the autocovariance structure from the discrete-time to the continuous-time setting are explored. We then exploit the characterisation of red noise via its power spectral density to narrow down the range of feasible model choices. We find that the attribute of a power spectral density decaying as $S(\omega)\sim\omega^{-2}$ commonly ascribed to the notion of red noise has far reaching consequences when posited in the continuous-time stochastic differential setting. In particular, any such It\^{o}-differential $\mathrm{d} Y_t=\alpha_t \mathrm{d} t+\beta_t \mathrm{d} W_t$ with continuous, square-integrable integrands must have a vanishing martingale part, i.e. $\mathrm{d} Y_t=\alpha_t\mathrm{d} t$ for almost all $t\geq 0$. We further argue that $\alpha$ should be an Ornstein-Uhlenbeck process.
TBA
Quantization in general refers to the transition from a classical to a corresponding quantum theory. The inverse issue, called the classical limit of quantum theories, is considered a much more difficult problem. A rigorous and natural framework that addresses this problem exists under the name strict (or C*-algebraic) deformation quantization. In this talk, I will first introduce this concept by means of the relevant definitions.Subsequently, I will show how this can be applied as a tool to study of the classical limit of quantum theories. More precisely, the so-called quantization maps allow one to take the limit ofsuitable sequence of algebraic states indexed by a semi-classical parameter in which the sequence typically converges to a probability measure on the pertinent phase space, as this paramters approaches zero. In addition, since this C*-algebraic approach allows for both quantum and classical theories, it provides a convenient way to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off this parameter. These ideas are illustrated with several physical models, e.g. Schrodinger operators labeled by Planck's constant and mean-field quantum spin systems indexed by the number of lattice sites. Finally, a short summary on symmetry breaking in real materials is provided.
In the last 10-13 years many mathematical models of honey bee population and disease dynamics have appeared in the literature. In most instances, these studies consider a single colony, often a generic rather than a specific disease, and are formulated as a system of autonomuous ordinary differential equations that permit the application of classical infectious disease modeling tools. We will present an apiary level model where the disease is spread between hives by drifting foragers. The model considers the seasonality of honey bee biology, and two routes of transmission that both have been proposed for Nosemosis, a contagious disease that weakens colonies. The resulting eco-epidemiological metapopulation model consists of 80 non-autonomuous ODEs (5 per colony; 16 colonies in an apiary) that we investigate in painstaking computer simulations. Originally to our surprise, we find highly complex and unexpected behavior such as period doubling and chaos in some parameter regimes. This seems to be caused by an intricate interplay of seasonality and population strength requirement for brood/hive maintenance. Our simulations also suggest that higher forager drifting rates may help survive individual colonies. This is joint work with Nasim Muhammad (Mohawk College, Hamilton ON).