The Planck mass can be derived from gravitational potential behavior in compactifications.
problem Deriving the Planck mass from gravitational potential behavior in compactifications.
method Physical considerations and Weyl law application to gravitational potential behavior.
result The Planck mass can be reconstructed from the asymptotics of the masses of spin 2 Kaluza--Klein modes.
Diagonal Frog: High-order positivity-preserving FD schemes for anisotropic Fokker-Planck equations
problem Positivity-preserving discretizations for anisotropic Fokker-Planck equations
method Diagonal Frog discretization
result Second-order accuracy and mass conservation
New method infers dynamical systems from population data.
problem Inferring dynamical systems from population data.
method Deducing and estimating Fokker-Planck equation, projecting to test functions, sparse inference.
result Induces driving forces of dynamical systems.
We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equ…
This paper bridges the gap between ODE and SDE in diffusion models using Fokker-Planck equations.
problem Empirical evidence shows that ODE-based samples from score-based diffusion models are inferior to SDE-based samples.
method The paper rigorously describes dynamics and approximations in training score-based diffusion models, linking them to Fokker-Planck equations.
result Adding a regularisation term based on the Fokker-Planck residual can close the gap between ODE- and SDE-induced distributions.
Wealth redistribution through Fokker-Planck equation controls preserves Gini coefficient.
problem Preserving Gini coefficient through proportional wealth tax.
method Formulating optimal redistribution as a control problem for Fokker-Planck equation.
result Progressive taxes redistribute within policy-relevant timescales.
Study extends wealth tax neutrality framework to heterogeneous investors.
problem Analyzing wealth tax neutrality in populations with varying return-generating ability.
method Extended Fokker-Planck framework to heterogeneous investors, deriving extended Fokker-Planck equation.
result Proportional wealth tax no longer neutral due to varying return-generating ability, leading to different real incidence and wealth distribution changes.
ADDA-KR uses KRnets for solving high-dimensional Fokker-Planck equations.
problem High-dimensional, unbounded Fokker-Planck equations.
method KRnet for density approximation, adaptive sampling for stochastic collocation.
result ADDA-KR efficiently approximates high-dimensional density functions.
We reformulate wealth taxation using Fokker-Planck equations to ensure tax neutrality.
problem Ensuring tax neutrality in wealth taxation frameworks.
method Reformulating the neutral wealth tax framework using stochastic dynamics and statistical physics, specifically Fokker-Planck equations.
result The framework clarifies when wealth taxation is a benign rescaling of dynamics and when it introduces new physics.
New method learns diffusion transition density for Bayesian inference.
problem Bayesian inference on diffusions with inaccessible boundaries.
method Neural Galerkin framework to solve FP equation with Dirac mass.
result Approximates likelihood function for efficient posterior sampling.
New method uses Fokker-Planck equation for sampling and inference.
problem Intractability of evaluating probability density in practical applications.
method Reformulates Fokker-Planck equation as a particle flow method, using velocity field.
result Turns intractable density evaluation into an advantage for variational inference, kernel mean embeddings, and sequential Monte Carlo.
The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker-Planck equation we describe a construction of exact …
FlowKac solves high-dimensional Fokker-Planck equations efficiently.
problem Intractability of Fokker-Planck equation solutions in high dimensions.
method Reformulates Fokker-Planck using Feynman-Kac, adaptive stochastic sampling, and normalizing flows.
result Significant computational efficiency and accuracy improvements over existing methods.
New gauge fields modify Fokker-Planck dynamics without changing the stationary state.
problem Understanding and modifying nonreversible dynamics in Fokker-Planck models.
method Formulate nonreversible perturbations as gauge fields, mapping to supersymmetric Hamiltonians, and learning finite forces.
result Learned finite forces can recover the optimal Lyapunov-equation solution in nonconvex landscapes.
Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a comp…
We consider here a Fokker--Planck equation with variable coefficient of diffusion which appears in the modeling of the wealth distribution in a multi-agent society. At difference with previous studies, to describe a society in which agents can have debts, we allow the wealth variable to be negative. It is shown that, e…
Develops a local Fokker--Planck geometric framework for more accurate score estimation.
problem Inaccurate estimation of score function in non-linear, state-dependent drifts.
method Local Fokker--Planck geometric framework, time change to cumulative-variance coordinate, heat-ball mean-value representations, exact high-dimensional sampling.
result Exact local mean-value representations for the score and density, improved accuracy in low-density regions.
Improved solver maintains positivity and accuracy across all time steps.
problem Linear second-order schemes for Fokker-Planck equation cannot preserve positivity.
method Flux-Corrected Diagonal Frog (FCDF) framework using nonlinear extension and iterative limiter.
result FCDF schemes are unconditionally positive across all time steps and maintain second-order accuracy.
Study measures inequality in social-economic systems using Fokker-Planck equations and Lotka-Volterra dynamics.
problem Measuring inequality in oscillatory social-economic systems described by Fokker-Planck equations and Lotka-Volterra dynamics.
method Used Fokker-Planck equations and Lotka-Volterra dynamics to model inequality, focusing on coefficient of variation as a measure.
result Inequality initially tends to decrease in oscillatory systems, contrary to steady-state models.
A new numerical scheme approximates nonlinear filtering densities for noisy and partial measurements.
problem Approximating nonlinear filtering densities for noisy and partial measurements.
method Deep splitting scheme applied to the Fokker--Planck equation followed by Bayes' formula.
result Convergence rate established for the numerical scheme under parabolic Hörmander condition.
MF-PID uses interacting samples to efficiently transport probability mass.
problem Efficiently transporting probability mass in generative models.
method Introducing Mean-Field Path-Integral Diffusion (MF-PID) where samples become interacting agents.
result MF-PID achieves 19-24% reductions in control energy for demand-response control of energy systems.
In this paper, we obtain a necessary and sufficient condition for L∞-uniqueness of Sturm-Liouville operator a(x)dx2d2+b(x)dxd−V on an open interval of $\rr$, which is equivalent to the L1-uniqueness of the associated Fokker-Planck equation. For a general elliptic operator $\LL^V:=Δ…
Distributions derived from non-extensive Tsallis statistics are closely connected with dynamics described by a nonlinear Fokker-Planck equation. The combination shows promise in describing stochastic processes with power-law distributions and superdiffusive dynamics. We investigate intra-day price changes in the S&P500…
The so-called "Yard-Sale Model" of wealth distribution posits that wealth is transferred between economic agents as a result of transactions whose size is proportional to the wealth of the less wealthy agent. In recent work [B.M. Boghosian, "Kinetics of Wealth and the Pareto Law," {\it Phys. Rev. E} {\bf 89} (2014) 042…
A new MFG framework for evolving clusters from Gaussian mixtures.
problem Evolutionary clustering of time-dependent Gaussian mixtures.
method Control-theoretic framework based on Mean Field Games (MFG) with coupled HJB and Fokker-Planck systems.
result MFG dynamics recover classical EM algorithm trajectories with mass conservation.
Over the moduli space of rank n semi-stable lattices is a universal family of tori. Along the fibers, there are natural differential operators and differential equations, particularly, the heat equations and the Fokker-Planck equations in statistical mechanics. In this paper, we explain why, by taking averages over t…
Proves subelliptic estimates for geometric Kramers-Fokker-Planck operators on closed manifolds.
problem Proving subelliptic estimates for a specific class of operators on closed manifolds.
method Significantly different method from previous works, using dyadic partition and local analysis in position variable.
result Maximal subelliptic estimates with control of constants in high and low friction regimes.
Interpolates mean shift and spectral clustering on graphs.
problem Data clustering algorithms.
method Fokker-Planck equations on data graphs.
result New theoretical insights on diffusion maps and mean shift dynamics.
The usual derivation of the Fokker-Planck partial differential eqn. assumes the Chapman-Kolmogorov equation for a Markov process. Starting instead with an Ito stochastic differential equation we argue that finitely many states of memory are allowed in Kolmogorov's two pdes, K1 (the backward time pde) and K2 (the Fokker…
New method speeds up SDE inference by matching moments to FPK equation.
problem Efficiency of sampling schemes in high-dimensional SDEs.
method Direct approximation of Fokker-Planck-Kolmogorov equation by matching moments.
result Fast, scalable inference in high-dimensional latent spaces.
Projects Markovian processes from Itô semimartingales with jumps.
problem Modeling Itô semimartingales with jumps using Markovian projections.
method Construct Markovian projections for Itô semimartingales with jumps using non-local FPKEs.
result Markovian projections match the marginal laws of the original process.
Study of a risk-averse informed trader in a multi-asset market with non-Gaussian prices.
problem Existence of equilibrium in a multi-asset market with non-Gaussian prices and a risk-averse informed trader.
method Constructed equilibrium using Fokker-Planck equation and coupled partial differential equations with an optimal transport constraint.
result Equilibrium exists in a market with multiple assets and non-Gaussian prices.
The evolution of the probability distributions of Japan and US major market indices, NIKKEI 225 and NASDAQ composite index, and JPY/DEM and DEM/USD currency exchange rates is described by means of the Fokker-Planck equation (FPE). In order to distinguish and quantify the deterministic and random influences on these…
In recent work, Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy -- complete concentration of wealth by a single …
This paper derives the non-analytic solution to the Fokker-Planck equation of fractional Brownian motion using the method of Laplace transform. Sequentially, by considering the fundamental solution of the non-analytic solution, this paper obtains the transition probability density function of the random variable that i…
We derive a diffusion approximation for the kinetic Vlasov-Fokker-Planck equation in bounded spatial domains with specular reflection type boundary conditions. The method of proof involves the construction of a particular class of test functions to be chosen in the weak formulation of the kinetic model. This involves t…
A new kernel framework analyzes spatio-temporal data from dynamic equations.
problem Analyzing spatio-temporal data from dynamic equations with noisy measurements.
method Kernel-based framework with representer theorem for minimizing error with given samples.
result Minimizes error in solutions of dynamic equations with noisy spatio-temporal data.
The article models illiquid stocks using quantum calculus with asymptotic methods.
problem Modeling illiquid financial markets.
method Application of quantum stochastic calculus and asymptotic methods.
result Power series solutions can approximate quantum stochastic processes for longer time frames.
Solves specific mean-field game equations with ODEs.
problem Mean-field game equations in economic applications.
method Reduces coupled PDEs to a quadratically nonlinear system of ODEs.
result Shows specific data leads to solvable ODE system.
We recently showed that the S&P500 stock market index is well described by Tsallis non-extensive statistics and nonlinear Fokker-Planck time evolution. We argued that these results should be applicable to a broad range of markets and exchanges where anomalous diffusion and `heavy' tails of the distribution are present.…
Study examines USD exchange rate dynamics using Kramers-Moyal expansion.
problem Understanding and predicting exchange rate instability.
method Kramers-Moyal expansion and Fokker-Planck formalism applied to log-return data.
result Identifies a stabilizing linear drift and nonlinear diffusion term in exchange rate fluctuations.
We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov-Fokker-Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants o…
The paper proposes a method to learn evolving multivariate distributions from sample paths.
problem Learning the temporal evolution of multivariate densities from sample data.
method Normalizing flows to construct time-dependent mappings.
result The method can approximate evolving probability density functions from observed data.
Foreign exchange markets show that currency units (= accounting or nominal price units) are variables. Technical and economic progress evidence that the consumer baskets (= purchasing power units or real price units) are also variables. In contrast, all physical measurement units are constants and either defined in the…
Study on convergence of SDEs using entropy methods.
problem Analyzing convergence of stochastic differential equations.
method Applied Lyapunov method to Fokker-Planck equation with weighted relative Fisher information.
result Exponential convergence of probability density function to invariant distribution in L1 distance. We study the evolution of probability distribution functions of returns, from the tick data of the Korean treasury bond (KTB) futures and the S$&$P 500 stock index, which can be described by means of the Fokker-Planck equation. We show that the Fokker-Planck equation and the Langevin equation from the estimated Kramers…
Physics-informed neural networks approximate diffusion process pdfs efficiently.
problem Approximating the probability density function of diffusion processes.
method Physics-informed neural networks solving Fokker-Planck or integro-differential equations.
result Neural network solutions approximate target solutions for various types of differential equations.
New bounds for heavy-tailed SDEs without info-theory terms.
problem Understanding generalization of heavy-tailed stochastic optimization.
method Fractional Fokker-Planck equation to estimate entropy flows.
result High-probability bounds with better dimension dependence.