On certain solutions of the diffusion equation for a two-dimensional Brownian particle density

We find approximate solutions of the diffusion equation for a two-dimensional Brownian particle density. We obtain expressions for the corrections connected with the fluctuations of the transfer coefficients, which can be applied in particular to describe the fluctuations of the concentration of Brownian particles in small volumes of the medium. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Coloured noise for dispersion of contaminants in shallow waters

In this article, we explore the application of a set of stochastic differential equations called particle model in simulating the advection and diffusion of pollutants in shallow waters. The Fokker–Planck equation associated with this set of stochastic differential equations is interpreted as an advection–diffusion equation. This enables us to derive an underlying particle model that is exactly consistent with the advection–diffusion equation. Still, neither the advection–diffusion equation nor the related traditional particle model accurately takes into account the short-term spreading behaviour of particles. To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this article. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection–diffusion equation.     

Coagulation, diffusion and the continuous Smoluchowski equation

The Smoluchowski equations are a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers or by positive reals, these corresponding to the discrete or the continuous form of the equations. For dimension d≥3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a method similar to that used to derive the discrete form of the equations in [A. Hammond, F. Rezakhanlou, The kinetic limit of a system of coagulating Brownian particles, Arch. Ration. Mech. Anal. 185 (2007) 1–67]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of the cited work. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.     

Scaling limits of interacting diffusions with arbitrary initial distributions

Summary It is remarked that for Brownian particles interacting with a smooth repulsive pair potential the nonlinear diffusion equation which S. Varadhan has derived under an entropy bound for initial densities is valied whatever initial distribution they start with.Research partially supported by Japan Society for the Promotion of Science     

Branching Brownian Motion with Decay of Mass and the Nonlocal Fisher-KPP Equation

In this work we study a nonlocal version of the Fisher-KPP equation, and its relation to a branching Brownian motion with decay of mass as introduced in Addario-Berry and Penington (2015) , i.e., a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in ℝ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighborhood around them (as measured by the function φ). We obtain two types of results. First, we study the behavior of solutions to the partial differential equation above. We show that, under suitable conditions on φ and u₀, the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik. Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the nonlocal Fisher-KPP equation. We then harness this to obtain several new results concerning the behavior of the particle system. © 2019 Wiley Periodicals, Inc.     

Attractiveness of Brownian queues in tandem

Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. The result is also interpreted in the related context of Brownian particles acting under one-sided reflection.

     

Exit time moments, boundary value problems, and the geometry of domains in Euclidean space

Let X _t be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using the moments of the exit time for particles driven by X _t , as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all critical points are maxima. Received: 23 January 1997 / Revised version: 21 January 1998     

Positivity of the self-diffusion matrix of interacting Brownian particles with hard core

We prove the positivity of the self-diffusion matrix of interacting Brownian particles with hard core when the dimension of the space is greater than or equal to 2. Here the self-diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle. We will do this for all activities, z>0, of Gibbs measures; in particular, for large z– the case of high density particles. A typical example of such a particle system is an infinite amount of hard core Brownian balls. Received: 22 September 1997 / Revised version: 15 January 1998     

A stochastic model of a chemical reaction with diffusion

Summary Stochastic systems of Brownian motions with multiple deletion of particles are introduced to model a chemical reaction with diffusion. Convergence to the solution of a deterministic nonlinear reaction-diffusion equation is proved without high-density assumptions.     

Fokker-planck equation for brownian motion of rotating spherical particles : PMM vol. 40, n 6, 1976, pp. 1127–1131

The Langevin equation to derive the Fokker-Planck equation is used for the Brownian motion of particles in translational motion. The Fokker-Planck equation for the Brownian motion of particles which have, in addition to the translational velocity also an angular velocity, has not, so far, been derived. This can apparently be explained by the fact that in the case of the rotational motion, the Langevin equation for the translational motion velocity vector must be supplemented by a corresponding equation for an angular velocity vector. The latter equation must contain, in addition to the systematic moment of reaction linearly dependent on the angular velocity of rotation itself, a random moment rapidly varying with time. Moreover, to ensure the compatibility of two differential vector equations within the system, additional relations which must be introduced, must connect not only the coefficients of the systematic reactions, but also the. random vectors varying rapidly with time.In [1],the Boltzmann's equation for a mixture of two gases was used to derive a Fokker-Planck equation for a translational motion of Brownian particles. The same method can be applied to the Brownian motion of spherical particles which have, in addition to the translational velocities, angular velocities of self-rotations. In this case there is no need to introduce additional relations connecting the random rapidly varying vectors.In the present paper we derive the Fokker-Planck equations for a new model of rotating spherical molecules which was used in [2].     

Genealogy of extremal particles of branching Brownian motion

Branching Brownian motion describes a system of particles that diffuse in space and split into offspring according to a certain random mechanism. By virtue of the groundbreaking work by M. Bramson on the convergence of solutions of the Fisher‐KPP equation to traveling waves, the law of the rightmost particle in the limit of large times is rather well understood. In this work, we address the full statistics of the extremal particles (first‐, second‐, third‐largest, etc.). In particular, we prove that in the large t‐limit, such particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0, or within a distance of order 1 from time t. The approach relies on characterizing, up to a certain level of precision, the paths of the extremal particles. As a byproduct, a heuristic picture of branching Brownian motion “at the edge” emerges, which sheds light on the still unknown limiting extremal process. © 2011 Wiley Periodicals, Inc.     

The approximation of additive functionals of diffusion processes

Let us consider a diffusion process in R_d . Around each point x one may consider a ring of size ? and a process which counts the crossings over the ring. Integrating with respect to a measure μ(dx) and letting ?→ 0 one gets an additive functional. This is a natural generalization of the approximation theorem of the local time of one dimensional Brownian motion by means of “downcrossings”. For multidimensional Brownian motion the result was established by Bally. The present paper introduces a new method which allows us to handle general diffusions     

One approach to the problem of nonparametric estimation in statistics of random processes based on the method of ill-posed problem

We consider the problem of estimation of integrated volatility, i.e., of the integral of the diffusion coefficient squared, in a stochastic differential equation for a random process that corresponds to geometric Brownian motion. In additon to purely theoretical interest, this problem is of interest for applications since the problem of evaluation of integrated volatility for financial assets is an important part of financial engineering topics. In the present paper, we suggest a new approach to the above-mentioned problem. We derive an integral equation whose solution determines the value of integrated volatility. This integral equation is a typical ill-posed problem of mathematical physics. The main idea of the proposed reduction of the original problem to an ill-posed problem consists of making its solution robust with respect to anomalous values of statistical data which are generated, for example, by market microstructure effects, such as the bid-ask spread. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 117–128.     

On the potential in non‐Gaussian chain polymer models

In this paper, we investigate the potential for a class of non‐Gaussian processes so‐called generalized grey Brownian motion. We obtain a closed analytic form for the potential as an integral of the M‐Wright functions and the Green function. In particular, we recover the special cases of Brownian motion and fractional Brownian motion. In addition, we give the connection to a fractional partial differential equation and its the fundamental solution.     

Diffusion with interactions and collisions between coloured particles and the propagation of chaos

Summary One considers a system ofN particles on the real line which are of two different types (colours). Their dynamics is given by a stochastic differential equation with constant diffusion part; the drift felt by a particle of either type depends on the empirical measures of type 1 and 2 particles at every instant; further a reflection condition is imposed so that particles of different type are not allowed to cross each other. The article studies the Vlasov-McKean limit of the system asN: propagation of chaos and an evolution equation for the limiting empirical measures is established, from where in particular an equation for the separating front between the two types follos.To the Memory of the late Professor Gishiro Maruýama