Brownian motion stochastic differential equation

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

The theory of Brownian motion for a constrained system is more subtle than that for an unconstrained system of pointlike particles, and has given rise to a substantial, and sometimes confusing, literamre. Some aspects of the problem, involving equilibrium statistical mechanics and the diffusion equation, have been understood for decades [1-8]. Other aspects, particularly those involving the relationships among various possible interpretations of the corresponding stochastic differential equations [9-13], remain less thoroughly understood. This chapter attempts to provide a self-contained account of the entire theory. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

There are many systems that can fluctuate randomly in space and time and cannot be described by deterministic equations. For example. Brownian motion of small particles occurs randomly because of random collisions with molecules of the medium in which the particles are suspended. It is useful to model such systems with what are known as stochastic differential equations. Stochastic differential equations feature noise terms representing the behavior of random elements in the system. Other examples of stochastic behavior arise in chemical reaction systems involving a small number of molecules, such as in a living cell or in the formation of particles in emulsion drops, and so on. A useful reference on stochastic methods is Gardiner (2003). 

The markets assume that the state variables evolve through a geometric Brownian motion, or Weiner process. It is therefore possible to model their evolution using a stochastic differential equation. The market also assumes that the cash flow stream of assets such as bonds and equities is a function of the state variables. 

This definition shows that Brownian motion is closely linked to the Gaussian/normal distribution. The formalism of Weiner processes opens stochastic processes to rigorous mathematical analysis and has enabled the use of Weiner processes in the field of stochastic differential equations. Stochastic differential equations are analogs of classical differential equations where... 

Consider relative motion by Brownian motion as well as by gravitational settling. We neglect interparticle force and correlated random motion, although their inclusion in the manner in which it was done in Section 3.3.5.3 is straightforward. In fact, the issue of interest in Section 3.3.5.3 does not differ from that in this section. The relative motion of particles of volume x as viewed from a particle of volume x is described by the stochastic differential equation... 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

The theory of stochastic processes began in the nineteenth century when physicists were trying to show that heat in a medium is essentially a random motion of the constituent molecules. At the end of that century, some researches began to adopt more direct mathematical models of random disturbances instead of considering random motion as due to collisions between objects having a random distribution of initial positions and velocities. In this context several physicists, among which Fok-ker (1914) and Planck (1915), developed partial differential equations, which were versions of what was subsequently called the Fokker-Planck equation, to study the theory of Brownian motion. 

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