Related papers: Probabilistic Analysis of Replicator-Mutator Equat…
In this paper, we develop an approach to recursively estimate the quadratic risk for matrix recovery problems regularized with spectral functions. Toward this end, in the spirit of the SURE theory, a key step is to compute the (weak)…
A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The H-theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck…
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations)…
This chapter is an overview of foundational results in the mathematical theory of replicator systems. Its primary aim is to provide a unified framework for the mathematical formalisation of evolutionary processes in the spirit of…
Inspired on the continued-fraction technique to solve the classical Fokker--Planck equation, we develop continued-fraction methods to solve quantum master equations in phase space (Wigner representation of the density matrix). The approach…
We study the relaxation to equilibrium for a class linear one-dimensional Fokker-Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker-Planck equations is that, for…
We derive the Fokker-Planck equation on the parametric space. It is the Wasserstein gradient flow of relative entropy on the statistical manifold. We pull back the PDE to a finite dimensional ODE on parameter space. Some analytical example…
Many theorems about Kolmogorov complexity rely on existence of combinatorial objects with specific properties. Usually the probabilistic method gives such objects with better parameters than explicit constructions do. But the probabilistic…
Based on a general discrete model for a semiflexible polymer chain, we introduce a formal derivation of a kinetic equation for semiflexible polymers in the half-plane via a continuum limit. It turns out that the resulting equation is the…
These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares…
We propose a novel method to solve a chemical diffusion master equation of birth and death type. This is an infinite system of Fokker-Planck equations where the different components are coupled by reaction dynamics similar in form to a…
We study the Cauchy problem for Fokker--Planck--Kolmogorov equations with unbounded and degenerate coefficients. Sufficient conditions for the existence and uniqueness of solutions are indicated.
A multi-species Fokker-Planck model for simulating particle collisions in a plasma is presented. The model includes various parameters that must be tuned. Under reasonable assumptions on these parameters, the model satisfies appropriate…
Polymer quantum mechanics has been studied as a simplified picture that reflects some of the key properties of Loop Quantum Gravity; however, while the fate of relativistic symmetries in Loop Quantum Gravity is still not established, it is…
Stochastic differential equations play an important role in various applications when modeling systems that have either random perturbations or chaotic dynamics at faster time scales. The time evolution of the probability distribution of a…
We introduce a multi-dimensional variant of the kinetic Cucker-Smale model with singular and matrix-valued communication weight, which reduces to the singular kinetic Cucker-Smale equation in the one-dimensional case. We propose an…
In this paper, we explicitly find all solutions of the title Diophantine equation, using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker-Davenport reduction method in…
In a recent paper Calogero and Alcantara derived a Lorentz-invariant Fokker-Planck equation, which corresponds to the evolution of a particle distribution associated with relativistic Brownian Motion. We study the "one and one-half"…
We show that there is a PDE formulation in terms of Fokker-Planck equations for weak optimal transport problems. The main novelty is that we introduce a minimization problem involving Fokker-Planck equations in the extended sense of…
In this paper, we investigate the fundamental solution of the fractional Fokker-Planck equation. Utilizing the Littlewood-Paley decomposition technology, we present a concise proof of the pointwise estimate for the fundamental solution.