Vlasov Equation Research Articles

On optimal control of coupled mean-field forward-backward stochastic equations

Abstract We consider a control problem for a mean-field coupled forward-backward stochastic differential equations, called also McKean–Vlasov equation (MF-FBSDE). For this type of equations, the coefficients depend not only on the state of the system, but also on its marginal distributions. They arise naturally in mean-field control problems and mean-field games. We consider the relaxed control problem where admissible controls are measure-valued processes. We prove the existence of a relaxed optimal control by using a suitable form of Skorokhod representation theorem and Jakubowski’s topology, on the space of càdlàg functions. We use martingale measure to define the relaxed state process. Our results extend to MF-FBSDEs those already known for forward and backward stochastic equations of Itô type.

On a Vlasov equation model for two-phase fluid flow

We consider the problem of water–air two-phase flow, which occurs during water aeration by air injection. We present and analyze a new model incorporating a nonlocal boundary condition, integrating the effect of the air phase through a source term applied to the water phase. This term is described using a kinetic approach based on the Vlasov equation. A numerical study is conducted, and numerical experiments are presented to demonstrate the effectiveness of the presented model

A hybrid Eulerian-Lagrangian Vlasov method for nonlinear wave-particle interaction in weakly inhomogeneous magnetic field

We present a hybrid Eulerian-Lagrangian (HEL) Vlasov method for nonlinear resonant wave-particle interactions in weakly inhomogeneous magnetic field. The governing Vlasov equation is derived from a recently proposed resonance tracking Hamiltonian theory. It gives the evolution of the distribution function with a scale-separated Hamiltonian that contains the fast-varying coherent wave-particle interaction and slowly-varying motion about the resonance frame of reference. The hybrid scheme solves the fast-varying phase space evolution on Eulerian grid with an adaptive time step and then advances the slowly-varying dynamics by Lagrangian method along the resonance trajectory. We apply the HEL method to study the frequency chirping of whistler-mode chorus wave in the magnetosphere and the self-consistent simulations reproduce the chirping chorus wave and give high-resolution phase space dynamics of energetic particles at low computational cost. The scale-separated HEL approach could provide additional insights of the wave instabilities and wave-particle nonlinear coherence compared to the conventional Vlasov and particle-in-cell methods.

Stability, Uniqueness and Existence of Solutions to McKean–Vlasov Stochastic Differential Equations in Arbitrary Moments

We deduce stability and pathwise uniqueness for a McKean–Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz continuous drift and includes moment estimates for random Itô processes that are of independent interest. For deterministic coefficients, we provide unique strong solutions even if the drift fails to be of affine growth. The theory that we develop rests on Itô’s formula and leads to pth moment and pathwise exponential stability for p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 2$$\\end{document} with explicit Lyapunov exponents.

A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean–Vlasov models

Motivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean–Vlasov equation dXt=σ(t,Xt)XtvtE[vt|Xt]dWt,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ d X_{t}= \\sigma (t,X_{t}) X_{t} \\frac{\\sqrt{v}_{t}}{\\sqrt{\\mathbb{E}[v_{t}|X_{t}]}}dW_{t}, $$\\end{document} where W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W$\\end{document} is a Brownian motion and v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$v$\\end{document} is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model. While such models are quite popular among practitioners, its well-posedness has unfortunately not yet been fully understood and in general is possibly not guaranteed at all. We develop a novel regularisation approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularised model is well posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularised model is able to perfectly replicate option prices coming from typical local volatility models. Our results are also applicable to more general McKean–Vlasov equations.

A generalized Dean–Kawasaki equation for an interacting Brownian gas in a partially absorbing medium

The Dean–Kawasaki (DK) equation is a stochastic partial differential equation (SPDE) for the global density ρ = N − 1 ∑ j = 1 N δ ( x − X j ( t ) ) of a gas of N over-damped Brownian particles, where X j ( t ) is the position of the j th particle. In the thermodynamic limit N → ∞ with weak pairwise interactions, the expectation ⟨ ρ ⟩ with respect to the white noise processes converges in distribution to the solution of a McKean–Vlasov (MV) equation. In this article, we use an encounter-based approach to derive a generalized DK equation for an interacting Brownian gas on the half-line with a partially absorbing boundary at x = 0 . Each particle is independently absorbed when its local time L j ( t ) at x = 0 exceeds a random threshold ℓ ^ j . The global density is now summed over the set of particles that have not yet been absorbed, and expectations are taken with respect to the Gaussian noise and the random thresholds ℓ ^ j . Assuming the DK equation has a well-defined mean-field limit, we derive the corresponding MV equation on the half-line. We illustrate the theory by (i) analysing stationary solutions for a Curie–Weiss (quadratic) interaction potential and a totally reflecting boundary; and (ii) calculating the effective rate of particle loss in the weak absorption limit. Extensions to finite intervals and partially absorbing traps are also considered.

Anti-symmetric and positivity preserving formulation of a spectral method for Vlasov-Poisson equations

We analyze the anti-symmetric properties of a spectral discretization for the one-dimensional Vlasov-Poisson equations. The discretization is based on a spectral expansion in velocity with the symmetrically weighted Hermite basis functions, central finite differencing in space, and an implicit Runge Kutta integrator in time. The proposed discretization preserves the anti-symmetric structure of the advection operator in the Vlasov equation, resulting in a stable numerical method. We apply such discretization to two formulations: the canonical Vlasov-Poisson equations and their continuously transformed square-root representation. The latter preserves the positivity of the particle distribution function. We derive analytically the conservation properties of both formulations, including particle number, momentum, and energy, which are verified numerically on the following benchmark problems: manufactured solution, linear and nonlinear Landau damping, two-stream instability, bump-on-tail instability, and ion-acoustic wave.

Local clustering of relic neutrinos: comparison of kinetic field theory and the Vlasov equation

Gravitational clustering in our cosmic vicinity is expected to lead to an enhancement of the local density of relic neutrinos. We derive expressions for the neutrino density, using a perturbative approach to kinetic field theory and perturbative solutions of the Vlasov equation up to second order. Our work reveals that both formalisms give exactly the same results and can thus be considered equivalent. Numerical evaluation of the local relic neutrino density at first and second order provides some fundamental insights into the frequently applied approach of linear response to neutrino clustering (also known as the Gilbert equation). Against the naive expectation, including the second-order contribution does not lead to an improvement of the prediction for the local relic neutrino density but to a dramatic overestimation. This is because perturbation theory breaks down in a momentum-dependent fashion and in particular for densities well below unity.

Solving Vlasov Equation with Neural Networks

Solving Vlasov Equation with Neural Networks

Вихрова октупольна мода у кінетичній моделі колективних збуджень в ядрах

The nature of a new isoscalar octupole resonance found within a kinetic model based on the Vlasov equation for finite Fermi systems with moving surfaces is studied. It is shown that this octupole resonance is due to dynamic effects of the nuclear surface, like the low-energy isoscalar dipole resonance (vortex dipole mode) observed in heavy nuclei. It is found that the velocity field associated with the new octupole resonance has a vortex character in the surface region of the nuclear liquid and, moreover, the vortex motion of nucleons is fragmented into three areas near the nuclear surface. At the same time, the velocity field associated with the high-energy octupole resonance found within our kinetic model displays an octupole deformation form and includes a compression within the nuclear fluid, which is consistent with the corresponding quantum calculations in the random phase approximation.

Quantized tensor networks for solving the Vlasov–Maxwell equations

The Vlasov–Maxwell equations provide an ab initio description of collisionless plasmas, but solving them is often impractical because of the wide range of spatial and temporal scales that must be resolved and the high dimensionality of the problem. In this work, we present a quantum-inspired semi-implicit Vlasov–Maxwell solver that uses the quantized tensor network (QTN) framework. With this QTN solver, the cost of grid-based numerical simulation of size $N$ is reduced from $O(N)$ to $O(\text {poly}(D))$ , where $D$ is the ‘rank’ or ‘bond dimension’ of the QTN and is typically set to be much smaller than $N$ . We find that for the five-dimensional test problems considered here, a modest $D=64$ appears to be sufficient for capturing the expected physics despite the simulations using a total of $N=2^{36}$ grid points, which would require $D=2^{18}$ for full-rank calculations. Additionally, we observe that a QTN time evolution scheme based on the Dirac–Frenkel variational principle allows one to use somewhat larger time steps than prescribed by the Courant–Friedrichs–Lewy constraint. As such, this work demonstrates that the QTN format is a promising means of approximately solving the Vlasov–Maxwell equations with significantly reduced cost.

From many-body quantum dynamics to the Hartree–Fock and Vlasov equations with singular potentials

In a combined mean-field and semiclassical regime, we consider the time evolution of N fermions interacting through singular pair interaction potentials of the form \pm |x-y|^{-a} , which includes the Coulomb and gravitational interactions. We prove that the many-body dynamics of mixed states are well approximated by solutions of the Hartree–Fock and Vlasov equations in terms of Schatten norms. The errors in these approximations are expressed in terms of the expected number of particles, N , and the Planck constant, h . For cases where a\in(0,1/2) , we obtain local-in-time results when N^{-1/2}\ll h \leq N^{-1/3} . Notably, this leads to the derivation of the Vlasov equation with singular potentials. For cases where a\in[1/2,1] , our results hold only within a small time scale or require an N -dependent cut-off. A fundamental ingredient in our analysis is the propagation of regularity for solutions to the Hartree–Fock equation uniformly in the Planck constant, which holds for a\in(0,1] .

Violent relaxation in one-dimensional self-gravitating system: Deviation from the Vlasov limit due to finite-N effects.

We investigate the effect of a finite particle number N on the violent relaxation leading to the quasistationary state (QSS) in a one-dimensional self-gravitating system. From the theoretical point of view, we demonstrate that the local Poissonian fluctuations embedded in the initial state give rise to an additional term proportional to 1/N in the Vlasov equation. This term designates the strength of the local mean-field variations by fluctuations. Because it is of the mean-field origin, we interpret it differently from the known collision term in the way that it effects the violent relaxation stage. Its role is to deviate the distribution function from the Vlasov limit, in the collisionless manner, at a rate proportional to 1/N while the violent relaxation is progressing. This hypothesis is tested by inspecting the QSSs in simulations of various N. We observe that the core phase-space density can exceed the limiting density deduced from the Vlasov equationand its deviation degree is in accordance with the 1/N estimate. This indicates the deviation from the standard mean-field approximation of the violent relaxation process by that 1/N term. In conclusion, the finite-N effect has a significant contribution to the QSS apart from that it plays a role in the collisional stage that takes place long after. The conventional collisionless Vlasov equationmight not be able to describe the violent relaxation of a system of particles properly without the correction term of the local finite-N fluctuations.

Modified Lawson Methods for Vlasov Equations

Modified Lawson Methods for Vlasov Equations

Advanced fuel fusion, phase space engineering, and structure-preserving geometric algorithms

Non-thermal advanced fuel fusion trades the requirement of a large amount of recirculating tritium in the system for that of large recirculating power. Phase space engineering technologies utilizing externally injected electromagnetic fields can be applied to meet the challenge of maintaining non-thermal particle distributions at a reasonable cost. The physical processes of the phase space engineering are studied from a theoretical and algorithmic perspective. It is emphasized that the operational space of phase space engineering is limited by the underpinning symplectic dynamics of charged particles. The phase space incompressibility according to the Liouville theorem is just one of many constraints, and Gromov's non-squeezing theorem determines the minimum footprint of the charged particles on every conjugate phase space plane. In this sense and level of sophistication, the mathematical abstraction of phase space engineering is symplectic topology. To simulate the processes of phase space engineering, such as the Maxwell demon and electromagnetic energy extraction, and to accurately calculate the minimum footprints of charged particles, recently developed structure-preserving geometric algorithms can be used. The family of algorithms conserves exactly, on discretized spacetime, symplecticity and thus incompressibility, non-squeezability, and symplectic capacities. The algorithms apply to the dynamics of charged particles under the influence of external electromagnetic fields as well as the charged particle–electromagnetic field system governed by the Vlasov–Maxwell equations.

Robust and conservative dynamical low-rank methods for the Vlasov equation via a novel macro-micro decomposition

Dynamical low-rank (DLR) approximation has gained interest in recent years as a viable solution to the curse of dimensionality in the numerical solution of kinetic equations including the Boltzmann and Vlasov equations. These methods include the projector-splitting and Basis-update & Galerkin (BUG) DLR integrators, and have shown promise at greatly improving the computational efficiency of kinetic solutions. However, this often comes at the cost of conservation of charge, current and energy. In this work we show how a novel macro-micro decomposition may be used to separate the distribution function into two components, one of which carries the conserved quantities, and the other of which is orthogonal to them. We apply DLR approximation to the latter, and thereby achieve a clean and extensible approach to a conservative DLR scheme which retains the computational advantages of the base scheme. Moreover, our approach requires no change to the mechanics of the DLR approximation, so it is compatible with both the BUG family of integrators and the projector-splitting integrator which we use here. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as an integrator which exactly conserves charge and energy and exhibits second-order accuracy on our test problems. To highlight the flexibility of the proposed macro-micro decomposition, we implement a pair of velocity space discretizations, and verify the claimed accuracy and conservation properties on a suite of plasma benchmark problems.

Optimal stopping of conditional McKean–Vlasov jump diffusions

The purpose of this paper is to study the optimal stopping problem of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal.The key is that we combine the conditional McKean–Vlasov equation with the associated stochastic Fokker–Planck partial integro-differential equation for the conditional law of the state. This leads to a Markovian system which can be handled by using a version of a Dynkin formula.Our verification result is illustrated by finding the optimal time to sell in a market with common noise and jumps.

Discrete moments models for Vlasov equations with non constant strong magnetic limit

Discrete moments models for Vlasov equations with non constant strong magnetic limit

Classical and Quantum Mechanical Models of Many-Particle Systems

The workshop focused on the collective behavior of many-particle systems in various application fields: physics (gas dynamics, plasmas, quantum mechanics), mathematical biology (cell mobility, evolution of trait-structured species), and social sciences (wealth distribution). This includes famous models such as the Boltzmann equation of gas dynamics, Vlasov equation for plasmas, Fokker–Planck equations, Smoluchowski and related equations, Keller–Segel system of chemotaxis.

Nonlinear Fokker–Planck equations with fractional Laplacian and McKean–Vlasov SDEs with Lévy noise

This work is concerned with the existence of mild solutions to nonlinear Fokker–Planck equations with fractional Laplace operator (-Δ)s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(- \\Delta )^s$$\\end{document} for s∈12,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\in \\left( \\frac{1}{2},1\\right) $$\\end{document}. The uniqueness of Schwartz distributional solutions is also proved under suitable assumptions on diffusion and drift terms. As applications, weak existence and uniqueness of solutions to McKean–Vlasov equations with Lévy noise, as well as the Markov property for their laws are proved.