# Introduction To Stochastic Processes

# Introduction To Stochastic Processes

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view of the fact that the physical equations of motion are deterministic and this issue was discussed in Section 1.5.1. Random functions, ordered sequences of random variable, were discussed in Section 1.5.3. The focus of this chapter is a particular class of random functions, stochastic processes, for which the ordering parameter is time. Time is a continuous ordering parameter, however in many practical situations observations of the random function z(t) are made at discrete time 0 < t1 < t2, . . . ,< tn < T. In this case the sequence {z(ti)} is a discrete sample of the stochastic process z(t). Let us start with an example. Consider a stretch of highway between two intersections, and let the variable of interest be the number of cars within this road segment at any given time, N(t). This number is obviously a random function of time whose properties can be deduced from observation and also from experience and intuition. First, this function takes positive integer values but this is of no significance: we could redefine N → N − (N) and the new variable will assume both positive and negative values. Second and more significantly, this function is characterized by several timescales: 1. Let τ1 is the average time it takes a car to go through this road segment, for example 1 min, and compare N(t) and N(t +∆ t) for ∆ t << τ1 and ∆ t τ1. Obviously N(t) ≈ N(t + ∆ t) for ∆ t << τ1 while in the opposite case the random numbers N(t) and N(t + ∆ t) will be almost uncorrelated. Figure 7.1 shows a typical result of one observation of this kind. The apparent lack of correlations between successive points in this data set expresses the fact that numbers sampled at intervals equal to or longer than the time it takes to traverse the given distance are not correlated. 2. The apparent lack of systematic component in the time series displayed here reflects only a relatively short-time behavior.

*Keywords:*
Gaussian distribution, Markovian stochastic processes, Stirling formula, Wiener–Khintchine theorem, average strengths of Fourier components, binomial distribution, correlation time, harmonic analysis, memory time, power spectrum

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