Process Wiener

Note that the rank of Cg determines the number of independent Wiener processes that affect the system.99 The following two limiting cases are of practical interest. 

The uni-variate Wiener process W(t) produces fluctuations only in a one-dimensional sub-space. Moreover, since (ej ()) = (< . 11) = 0, the fluctuations will be zero outside the unit interval, and the allowable region for the joint composition PDF will remain onedimensional and bounded as required by (6.134). 

In both the Ito and Stratonovich formulations, the randomness in a set of SDEs is generated by an auxiliary set of statistically independent Wiener processes [12,16]. The solution of an SDE is defined by a hmiting process (which is different in different interpretations) that yields a unique solution to any stochastic initial value problem for each possible reahzation of this underlying set of Wiener processes. A Wiener process W t) is a Gaussian Markov diffusion process for which the change in value W t) — W(t ) between any two times t and t has a mean and variance... 

Wiener processes are continuous but not differentiable. To generate a particular time history for a set of L random variables, we use some set of M statistically independent Wiener processes W t),..., Wuit), which will herafter be labeled by indices m, = 1M. The number of Weiner processes M used to generate random processes for L coordinates need not always equal L. 

The Stratonovich interpretation of a generic set of L SDEs driven by M Wiener processes will be indicated in what follows by the notation... 

The interpretation of the Langevin equation presents conceptual difficulties that are not present in the Ito and Stratonovich interpretation. These difficulties are the result of the fact that the probability distribution for the random force rip(f) cannot be fully specihed a priori when the diffusivity and friction tensors are functions of the system coordinates. The resulting dependence of the statistical properties of the random forces on the system s trajectories is not present in the Ito and Stratonovich interpretations, in which the randomness is generated by standard Wiener processes Wm(f) whose complete probability distribution is known a priori. 

On substitution of Eqs. (18) and (19), Kn is obtained in closed form for the Uhlenbeck-Ornstein and Wiener processes, respectively. For example, for... 

A similar procedure gives the corresponding result for the Wiener process the first term of which was obtained by Feynman and Hibbs4 using a path integral approach. 

Higher terms in the expansion for Z(p) can be straightforwardly calculated using the theory of Gaussian Markov processes. For example in the case of the Wiener process we obtain for the first term inside the summation of Eq. (35)... 

If one chooses Pi(Vi, 0) = fi ) a non-stationary Markov process is defined, called the Wiener process or Wiener-Levy process. ) It is usually considered for f >0 alone and was originally invented for describing the stochastic behavior of the position of a Brownian particle (see VIII.3). The probability density for t > 0 is according to (2.2)... 

Exercise. Also prove for the Wiener process, when 0 < U [Pg.81]

Exercise. The transition probability P iy, t y0,0) of the Wiener process obeys (1.5) when G is the operator given by the kernel... 

Exercise. It has been remarked in 1 that a Markov process with time reversal is again a Markov process. Construct the hierarchy of distribution functions for the reversed Wiener process and verify that its transition probability obeys the Chapman-Kolmogorov equation. 

Exercise. In the Ornstein-Uhlenbeck process rescale the variables y = ay, t = fit and show that in a suitably chosen limit of a and ft the P reduces to that of the Wiener process. 

These processes are non-stationary because the condition singled out a certain time t0. Yet their transition probability depends on the time interval alone as it is the same as the transition probability of the underlying stationary process. Non-stationary Markov processes whose transition probability depends on the time difference alone are called homogeneous processes. 10 They usually occur as subensembles of stationary Markov processes in the way described here. However, the Wiener process defined in 2 is an example of a homogeneous process that cannot be embedded in a stationary Markov process. 

Exercise. From the Wiener process extract the subensemble corresponding to Y(t0) = y0. Find the evolution of <7(f)> for t > t0. Also find the variance 7(t)2 in this ensemble. 

Exercise. Let Y0 be the Wiener process and Yt, Y2,..., Yr random walks with different step sizes and transition probabilities. Show that Y0 + Yx + Y2 + + Yr is a process with independent increments, see (IV.4.7), and find its transition probability. 

Consider an ensemble of Brownian particles which at t = 0 are all at X = 0. Their positions at t O constitute a stochastic process X(t), which is Markovian by assumption and whose transition probability is determined by (3.1). That is, just the Wiener process defined in IV.2. Their density at t > 0 is given by the solution of (3.1) with initial condition P(X, 0) = 8(X), which is given by (IV.2.5) ... 

The process X(t) is now fully specified since it is Gaussian and the first two moments are known. But it is not the same as the Wiener process determined by (3.1), because the autocorrelation function is more complicated than (IV.2.7a). In fact, X(t) is not even Markovian, owing to the fact that it is still described on the fine time scale belonging to the Rayleigh particle. On the coarse time scale only time differences much larger than the damping time l/y of the velocity are admitted,... 

Exercise. Show that the Wiener process can be obtained as the limiting case of a compound Poisson process when the jumps become infinitely small, but infinitely dense on the time axis. Compare the Remark in 5. 

This white noise perturbance can be derived from a Wiener process W, and X then satisfies the stochastic differential equation... 

In Figs. 4.1 and 4.2, the broken lines do not represent the sample paths of the process X(t), but join the outcoming states of the system observed at a discrete set of times f, t2,.. . , tn. To understand the behavior of X(t), it is necessary to know the transition probability. In Fig. 4.3 are given numerical simulations of a Wiener process W(t) (Brownian motion) and a Cauchy process C(t), both supposed one dimensional, stationary, and homogeneous. Their transitions functions are defined... 

In contrast to the diffusion processes, the fluctuating force 3F t) can no longer be derived from the Wiener process, and its spectral density defined as 2kBT 0 times the Fourier transform of the memory function C(f) is frequency limited and has no more the white noise characteristics. 

In a pioneer work, Marcus established the link between some usual time-varying forms of h ( ) and / (a) in a single compartment [300]. For instance in h(t) = (f +/ ), a = 1 leads to A Gam(A,/3) and 1 < a < 2 defines the standard extreme stable-law density with exponent a. In the case of a = 1.5, the obtained distribution is known as the retention-time distribution of a Wiener process with drift. 

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