The most important stochastic process is the Brownian motion or Wiener process. It was first discussed by Louis Bachelier (1900), who was interested in modeling fluctuations in prices in financial markets, and by Albert Einstein (1905), who gave a mathematical model for the irregular motion of colloidal particles first observed by the Scottish botanist Robert Brown in 1827. The first mathematically rigorous treatment of this model was given by Wiener (1923). Einstein’s results led to an early, dramatic confirmation of the molecular theory of matter in the French physicist Jean Perrin’s experiments to determine Avogadro’s number, for which Perrin was awarded a Nobel Prize in 1926. Today somewhat different models for physical Brownian motion are deemed more appropriate than Einstein’s, but the original mathematical model continues to play a central role in the theory and application of stochastic processes.

Let B(t) denote the displacement (in one dimension for simplicity) of a colloidally suspended particle, which is buffeted by the numerous much smaller molecules of the medium in which it is suspended. This displacement will be obtained as a limit of a random walk occurring in discrete time as the number of steps becomes infinitely large and the size of each individual step infinitesimally small. Assume that at times kδ, k = 1, 2,…, the colloidal particle is displaced a distance hX_k, where X₁, X₂,… are +1 or −1 according as the outcomes of tossing a fair coin are heads or tails. By time t the particle has taken m steps, where m is the largest integer ≤ t/δ, and its displacement from its original position is B_m(t) = h(X₁ +⋯+ X_m). The expected value of B_m(t) is 0, and its variance is h²m, or approximately h²t/δ. Now suppose that δ → 0, and at the same time h → 0 in such a way that the variance of B_m(1) converges to some positive constant, σ². This means that m becomes infinitely large, and h is approximately σ(t/m)^1/2. It follows from the central limit theorem (equation 12) that lim P{B_m(t) ≤ x} = G(x/σt^1/2), where G(x) is the standard normal cumulative distribution function defined just below equation (12). The Brownian motion process B(t) can be defined to be the limit in a certain technical sense of the B_m(t) as δ → 0 and h → 0 with h²/δ → σ².

The process B(t) has many other properties, which in principle are all inherited from the approximating random walk B_m(t). For example, if (s₁, t₁) and (s₂, t₂) are disjoint intervals, the increments B(t₁) − B(s₁) and B(t₂) − B(s₂) are independent random variables that are normally distributed with expectation 0 and variances equal to σ²(t₁ − s₁) and σ²(t₂ − s₂), respectively.

Einstein took a different approach and derived various properties of the process B(t) by showing that its probability density function, g(x, t), satisfies the diffusion equation ∂g/∂t = D∂²g/∂x², where D = σ²/2. The important implication of Einstein’s theory for subsequent experimental research was that he identified the diffusion constant D in terms of certain measurable properties of the particle (its radius) and of the medium (its viscosity and temperature), which allowed one to make predictions and hence to confirm or reject the hypothesized existence of the unseen molecules that were assumed to be the cause of the irregular Brownian motion. Because of the beautiful blend of mathematical and physical reasoning involved, a brief summary of the successor to Einstein’s model is given below.

Unlike the Poisson process, it is impossible to “draw” a picture of the path of a particle undergoing mathematical Brownian motion. Wiener (1923) showed that the functions B(t) are continuous, as one expects, but nowhere differentiable. Thus, a particle undergoing mathematical Brownian motion does not have a well-defined velocity, and the curve y = B(t) does not have a well-defined tangent at any value of t. To see why this might be so, recall that the derivative of B(t), if it exists, is the limit as h → 0 of the ratio [B(t + h) − B(t)]/h. Since B(t + h) − B(t) is normally distributed with mean 0 and standard deviation h^1/2σ, in very rough terms B(t + h) − B(t) can be expected to equal some multiple (positive or negative) of h^1/2. But the limit as h → 0 of h^1/2/h = 1/h^1/2 is infinite. A related fact that illustrates the extreme irregularity of B(t) is that in every interval of time, no matter how small, a particle undergoing mathematical Brownian motion travels an infinite distance. Although these properties contradict the commonsense idea of a function—and indeed it is quite difficult to write down explicitly a single example of a continuous, nowhere-differentiable function—they turn out to be typical of a large class of stochastic processes, called diffusion processes, of which Brownian motion is the most prominent member. Especially notable contributions to the mathematical theory of Brownian motion and diffusion processes were made by Paul Lévy and William Feller during the years 1930–60.

A more sophisticated description of physical Brownian motion can be built on a simple application of Newton’s second law: F = ma. Let V(t) denote the velocity of a colloidal particle of mass m. It is assumed that

The quantity f retarding the movement of the particle is due to friction caused by the surrounding medium. The term dA(t) is the contribution of the very frequent collisions of the particle with unseen molecules of the medium. It is assumed that f can be determined by classical fluid mechanics, in which the molecules making up the surrounding medium are so many and so small that the medium can be considered smooth and homogeneous. Then by Stokes’s law, for a spherical particle in a gas, f = 6πaη, where a is the radius of the particle and η the coefficient of viscosity of the medium. Hypotheses concerning A(t) are less specific, because the molecules making up the surrounding medium cannot be observed directly. For example, it is assumed that, for t ≠ s, the infinitesimal random increments dA(t) = A(t + dt) − A(t) and A(s + ds) − A(s) caused by collisions of the particle with molecules of the surrounding medium are independent random variables having distributions with mean 0 and unknown variances σ² dt and σ² ds and that dA(t) is independent of dV(s) for s < t.

The differential equation (18) has the solutionwhere β = f/m. From this equation and the assumed properties of A(t), it follows that E[V²(t)] → σ²/(2mf) as t → ∞. Now assume that, in accordance with the principle of equipartition of energy, the steady-state average kinetic energy of the particle, m lim_{t → ∞}E[V²(t)]/2, equals the average kinetic energy of the molecules of the medium. According to the kinetic theory of an ideal gas, this is RT/2N, where R is the ideal gas constant, T is the temperature of the gas in kelvins, and N is Avogadro’s number, the number of molecules in one gram molecular weight of the gas. It follows that the unknown value of σ² can be determined: σ² = 2RTf/N.

If one also assumes that the functions V(t) are continuous, which is certainly reasonable from physical considerations, it follows by mathematical analysis that A(t) is a Brownian motion process as defined above. This conclusion poses questions about the meaning of the initial equation (18), because for mathematical Brownian motion the term dA(t) does not exist in the usual sense of a derivative. Some additional mathematical analysis shows that the stochastic differential equation (18) and its solution equation (19) have a precise mathematical interpretation. The process V(t) is called the Ornstein-Uhlenbeck process, after the physicists Leonard Salomon Ornstein and George Eugene Uhlenbeck. The logical outgrowth of these attempts to differentiate and integrate with respect to a Brownian motion process is the Ito (named for the Japanese mathematician Itō Kiyosi) stochastic calculus, which plays an important role in the modern theory of stochastic processes.

The displacement at time t of the particle whose velocity is given by equation (19) is

For t large compared with β, the first and third terms in this expression are small compared with the second. Hence, X(t) − X(0) is approximately equal to A(t)/f, and the mean square displacement, E{[X(t) − X(0)]²}, is approximately σ²/f ² = RT/(3πaηN). These final conclusions are consistent with Einstein’s model, although here they arise as an approximation to the model obtained from equation (19). Since it is primarily the conclusions that have observational consequences, there are essentially no new experimental implications. However, the analysis arising directly out of Newton’s second law, which yields a process having a well-defined velocity at each point, seems more satisfactory theoretically than Einstein’s original model.

Stationary processes

The mathematical theory of stochastic processes attempts to define classes of processes for which a unified theory can be developed. The most important classes are stationary processes and Markov processes. A stochastic process is called stationary if, for all n, t₁ < t₂ <⋯< t_n, and h > 0, the joint distribution of X(t₁ + h),…, X(t_n + h) does not depend on h. This means that in effect there is no origin on the time axis; the stochastic behaviour of a stationary process is the same no matter when the process is observed. A sequence of independent identically distributed random variables is an example of a stationary process. A rather different example is defined as follows: U(0) is uniformly distributed on [0, 1]; for each t = 1, 2,…, U(t) = 2U(t − 1) if U(t − 1) ≤ 1/2, and U(t) = 2U(t − 1) − 1 if U(t − 1) > 1/2. The marginal distributions of U(t), t = 0, 1,… are uniformly distributed on [0, 1], but, in contrast to the case of independent identically distributed random variables, the entire sequence can be predicted from knowledge of U(0). A third example of a stationary process iswhere the Ys and Zs are independent normally distributed random variables with mean 0 and unit variance, and the cs and θs are constants. Processes of this kind can be useful in modeling seasonal or approximately periodic phenomena.

A remarkable generalization of the strong law of large numbers is the ergodic theorem: if X(t), t = 0, 1,… for the discrete case or 0 ≤ t < ∞ for the continuous case, is a stationary process such that E[X(0)] is finite, then with probability 1 the averageif t is continuous, converges to a limit as s → ∞. In the special case that t is discrete and the Xs are independent and identically distributed, the strong law of large numbers is also applicable and shows that the limit must equal E{X(0)}. However, the example that X(0) is an arbitrary random variable and X(t) ≡ X(0) for all t > 0 shows that this cannot be true in general. The limit does equal E{X(0)} under an additional rather technical assumption to the effect that there is no subset of the state space, having probability strictly between 0 and 1, in which the process can get stuck and never escape. This assumption is not fulfilled by the example X(t) ≡ X(0) for all t, which gets stuck immediately at its initial value. It is satisfied by the sequence U(t) defined above, so by the ergodic theorem the average of these variables converges to 1/2 with probability 1. The ergodic theorem was first conjectured by the American chemist J. Willard Gibbs in the early 1900s in the context of statistical mechanics and was proved in a corrected, abstract formulation by the American mathematician George David Birkhoff in 1931.