Defining Standard Brownian Motion

Why Define the Noise Precisely?

In a pharmacokinetic model, a stochastic driver represents variation in a state between observation times, not merely scatter in the measured concentrations. To specify a Brownian driver, it is not enough to say that it is “random” or “normally distributed.” Its starting value, the distribution and dependence of its increments, and the regularity of its paths are all part of the definition. These conditions determine which probability calculations are justified.

The term standard Brownian motion describes a stochastic process with a particular law and path behavior. A specific process is defined on a probability space and has particular sample paths; its law describes the distribution of those paths. Keeping these two meanings separate is useful when comparing models, simulations, or mathematical constructions.

Definition of Standard Brownian Motion

The variance parameter in \(N(0,t-s)\) is the elapsed time, not its square. The independent-increment condition means that increments on disjoint intervals are independent random variables, even if the intervals are not adjacent. Equivalently, each increment after time \(s\) is independent of the process history up to \(s\), represented by \(\sigma(W_u:0\leq u\leq s)\).

The continuity condition is almost sure: there may be a probability-zero set of outcomes on which the path is not continuous. The definition does not assert that every possible outcome has a continuous path. Nor does it say that a path is deterministic or differentiable. Continuity concerns the realized function of time; the Gaussian and independence conditions concern probability distributions.

Consequences for Means and Covariances

The definition gives the distribution of increments directly. It also determines the first and second moments of the process at individual times. The following result is useful when checking calculations involving Brownian values.

Proof. Since \(W_t-W_0\sim N(0,t)\) and \(W_0=0\) almost surely, \(W_t\sim N(0,t)\). Therefore \(E[W_t]=0\) and \(\operatorname{Var}(W_t)=t\). To obtain the covariance, first suppose \(s\leq t\) and write \[ W_t=W_s+(W_t-W_s). \] The increment \(W_t-W_s\) is independent of \(W_s\), has mean zero, and has finite variance. Thus \[ \operatorname{Cov}(W_s,W_t) =\operatorname{Cov}(W_s,W_s)+\operatorname{Cov}(W_s,W_t-W_s) =s+0=s. \] If \(t<s\), the same argument with the times reversed gives covariance \(t\). In either case the covariance is \(\min(s,t)\). \(\square\)

In particular, Brownian values at two different times are generally not independent: their covariance is positive when both times are positive. It is increments on disjoint intervals that are independent. For example, \(W_s\) and \(W_t-W_s\) are independent for \(0\leq s<t\), while \(W_s\) and \(W_t\) are not independent when \(s>0\).

A Process and Its Wiener Law

A process \(W\) is a collection of random variables on a probability space. At a particular outcome \(\omega\), it produces a path \(t\mapsto W_t(\omega)\). The law of the process describes the probabilities assigned to possible paths; it does not identify which path occurred on a particular outcome.

The defining conditions determine a unique Wiener measure. The key reason is that the increment distributions and independence determine every finite-dimensional distribution, while continuous paths are determined by their values at rational times. Here is the precise argument.

Proof. Choose any finite set of times \(0\leq t_1<\cdots<t_m\). The vector of Brownian values \((W_{t_1},\ldots,W_{t_m})\) is obtained by adding the increments over the successive intervals \([0,t_1], [t_1,t_2],\ldots,[t_{m-1},t_m]\). Those increments are independent, and each has a normal distribution with variance equal to its interval length. Consequently, the joint distribution of the vector is determined by the definition and is the same for every standard Brownian motion.

It remains to pass from finite-dimensional distributions to measures on path space. Evaluations \(x\mapsto x(q)\) at nonnegative rational times \(q\) generate the Borel sigma-field of \(C_0[0,\infty)\). To see why continuity matters here, for any continuous path the supremum of its distance from another continuous path over a bounded interval equals the supremum over rational times in that interval. Thus open sets for uniform convergence on bounded intervals can be described using countably many rational-time evaluations. The two path laws agree on every finite collection of these evaluations by the preceding argument. The corresponding cylinder sets form a generating class closed under finite intersections, so uniqueness of probability measures implies that the laws agree on the Borel sigma-field. \(\square\)

Model Checks, Alternatives, and Limitations

When Brownian motion is used as a model assumption, data can be checked against its implications, but finite observations cannot establish the full definition. For observed increments over intervals of known lengths, useful diagnostics include:

• Mean and variance: increments should be centered, with variance proportional to elapsed time.
• Distribution: standardized increments should be broadly compatible with a normal distribution, allowing for sampling uncertainty.
• Dependence: increments over non-overlapping intervals should not show systematic dependence.
• Path behavior: obvious jumps are inconsistent with the almost-sure continuity assumption, although sparse measurements cannot verify continuity between sampling times.

These checks are especially important when observations are sparse: a smooth-looking sequence of measured concentrations does not demonstrate that the latent process has continuous paths, and irregular measurements do not by themselves identify which source of variability produced them. A model’s observation process and its latent process must be distinguished when interpreting such patterns.

Standard Brownian motion is a convenient model for continuous random variation with independent, time-homogeneous Gaussian increments. It is not a universal description of randomness. A deterministic smooth input has no random increments; a jump process allows discontinuities; and a mean-reverting Gaussian process generally has dependent increments. These alternatives may be more appropriate when the scientific mechanism or observed behavior contradicts Brownian assumptions. Choosing among them is a modeling decision, not something resolved by calling all unexplained variability “noise.”

A common mistake is to infer independence of \(W_s\) and \(W_t\) from the fact that Brownian motion has independent increments. The definition applies to increments on disjoint intervals, not to the process values themselves. For \(0<s<t\), the values share the earlier part of the path, and their covariance is \(s\), not zero. The correction is to express the quantities of interest as increments first, then check whether their time intervals are disjoint.