Stochastic Processes and Correlation Functions

A stochastic process is a family of random variables indexed by time, space, or another parameter, all defined on one probability space. Its complete finite-observation content is the collection of joint laws at every finite set of indices. When those laws satisfy the appropriate consistency conditions, they determine a probability law on the product, or cylinder, sigma-algebra.

Correlation functions are useful summaries of that law, not substitutes for it. A mean and a two-point covariance determine every finite-dimensional law only in the Gaussian case. They do not by themselves provide continuous sample paths, joint measurability in the index and outcome, stationarity, a Markov property, or a rule for replacing ensemble averages by one long trajectory. These distinctions are what make the same language usable for stochastic time series and regulated random fields without confusing either with a formal path integral or a generic quantum correlator.

Required background. Probability Spaces, Random Variables, and Conditional Expectation supplies measurable random variables, pushforward laws, expectation, independence, and almost-sure equality.

Processes, index sets, and laws

Throughout the mathematical core, $(\Omega,\mathcal F,\mathbb P)$ is a probability space, $I$ is an index set, and $(S,\mathcal S)$ is a measurable state space. The main formulas use real-valued processes and explicitly state their moment assumptions. The QFT-facing example is a classical thermal stochastic field with a finite spatial regulator and one real mode retained. It is not a claim about an arbitrary Lorentzian quantum field.

For the spectral discussion, angular frequency follows the site’s Fourier convention:

$$\begin{aligned} \widetilde C(\omega) &=\int_{\mathbb R}\mathrm d\tau\, e^{+i\omega\tau}C(\tau),\\ C(\tau) &=\int_{\mathbb R}\frac{\mathrm d\omega}{2\pi}\, e^{-i\omega\tau}\widetilde C(\omega), \end{aligned}$$

whenever the ordinary transforms exist. This convention is restated because several probability texts use cycles per unit time or reverse the signs.

A process is a family and a random path

Definition. An $S$-valued stochastic process indexed by $I$ is a family

$$X=(X_i)_{i\in I}, \qquad X_i:(\Omega,\mathcal F) \longrightarrow(S,\mathcal S),$$

in which every coordinate $X_i$ is measurable. Typical choices are $I=\mathbb Z$ for a discrete time series, $I=\mathbb R$ for continuous time, and $I=\mathbb R^d$ for a random field. The same object has two complementary views:

• Fix $i$. Then $\omega\mapsto X_i(\omega)$ is a random variable.
• Fix $\omega$. Then $i\mapsto X_i(\omega)$ is a sample path, realization, or field configuration.

Equivalently, define

$$X:\Omega\longrightarrow S^I, \qquad X(\omega)(i)=X_i(\omega).$$

Equip $S^I$ with the product sigma-algebra $\mathcal S^{\otimes I}$ generated by sets that inspect only finitely many coordinates. Coordinatewise measurability then makes $X$ a random element of this product space. Pavliotis gives the family, path, and finite-dimensional viewpoints in Pavliotis 2014, § 1.1, pp. 1–3, PDF.

There is an important missing conclusion. If $I$ itself has a measurable or topological structure, coordinatewise measurability does not automatically make

$$(i,\omega)\longmapsto X_i(\omega)$$

jointly measurable. Nor does it make sample paths continuous, differentiable, or càdlàg. Such properties require additional hypotheses and usually a carefully chosen version of the process.

Finite-dimensional laws specify the cylinder law

For a finite subset $J=\{i_1,\ldots,i_n\}\subset I$, let

$$X_J=(X_{i_1},\ldots,X_{i_n})$$

and define its law by pushforward:

$$\mu_J=(X_J)_{\#}\mathbb P.$$

The collection $(\mu_J)_{J\subset I,\,J\text{ finite}}$ is the family of finite-dimensional distributions. If coordinates are written as ordered tuples, permuting the indices must permute the law, and repeated indices must represent repeated copies of the same random variable. In subset notation, the central compatibility condition is marginal consistency. For $K\subset J$, with coordinate projection $\pi_{J,K}:S^J\to S^K$,

$$(\pi_{J,K})_{\#}\mu_J=\mu_K.$$

This equation says that forgetting observations cannot change the joint law of the observations retained.

Kolmogorov consistency theorem, in the scope used here. Suppose $S$ is a Borel subset of a finite-dimensional Euclidean space. Every projectively consistent family $(\mu_J)$ of finite-dimensional probability measures extends uniquely to a probability measure $Q$ on $(S^I,\mathcal S^{\otimes I})$. On that canonical probability space, the coordinate maps

$$X_i(x)=x(i)$$

form a process with exactly the prescribed finite-dimensional laws. Kevei states the arbitrary-index theorem and its canonical construction in Kevei 2026, § 4.2, Theorems 4.3 and 4.8, pp. 43–46, PDF.

The theorem constructs a cylinder law. It does not say that $Q$ is concentrated on continuous or càdlàg paths. In the canonical space $\mathbb R^{[0,\infty)}$, for example, the subset $C[0,\infty)$ is not an event in the cylinder sigma-algebra: cylinder events depend on at most countably many coordinates, whereas continuity cannot be decided from arbitrary values on a fixed countable subset. Kevei makes this obstruction explicit on Kevei 2026, p. 49, PDF. One must separately construct a regular modification or work directly on a chosen path space with its own Borel sigma-algebra.

Four levels of agreement between processes

Several statements that sound like “the same process” have different strengths. In this section, $X$ and $Y$ are real-valued processes, so their equality events are measurable.

Same finite-dimensional laws. Processes $X$ and $Y$, possibly on different probability spaces, agree in finite-dimensional distribution if

$$(X_{i_1},\ldots,X_{i_n}) \overset{\mathrm d}{=} (Y_{i_1},\ldots,Y_{i_n})$$

for every finite index tuple. They then define the same cylinder law.

Modification. If $X$ and $Y$ live on the same probability space, $Y$ is a modification of $X$ when

$$\mathbb P(X_i=Y_i)=1 \qquad \text{for every fixed }i\in I.$$

The exceptional null set may depend on $i$.

Indistinguishability. The processes are indistinguishable when

$$\begin{gathered} \text{there is an }N\in\mathcal F\text{ with }\mathbb P(N)=0 \text{ such that}\qquad\\ X_i(\omega)=Y_i(\omega) \quad\text{for every }i\in I \text{ and every }\omega\notin N. \end{gathered}$$

Thus one measurable null set works simultaneously for all indices; no measurability of an uncountable intersection is being assumed.

Agreement of selected correlations. This means only that some moments, often the mean and covariance, agree. It is weaker than agreement of all finite-dimensional laws.

The implications are

$$\text{indistinguishable} \Longrightarrow \text{modifications} \Longrightarrow \text{same finite-dimensional laws},$$

and neither converse is automatic. To see the first failure, let $U\sim\operatorname{Uniform}[0,1]$ and define on $I=[0,1]$

$$X_t=0, \qquad Y_t=\mathbf 1_{\{U=t\}}.$$

For every fixed $t$, $\mathbb P(U=t)=0$, so $Y$ is a modification of $X$ and the two processes have identical finite-dimensional laws. For every outcome, however, the path $Y$ has a spike at $t=U$, so the processes are not indistinguishable. This example and the definitions appear in Kevei Kevei 2026, § 3.1, p. 33, PDF.

Regularity can close this gap when the index space is separable. For example, if $I$ is a separable metric space and two real processes have continuous paths almost surely and are modifications of each other, equality on a countable dense subset of $I$ holds outside one null set and continuity extends it to every index. The two versions are then indistinguishable.

Correlation functions summarize moments

Assume the products below are integrable. The raw $n$-point correlation function is

$$G_n(i_1,\ldots,i_n) =\mathbb E[X_{i_1}\cdots X_{i_n}].$$

At first and second order, write

$$\begin{aligned} m(i)&=\mathbb E[X_i],\\ G_2(i,j)&=\mathbb E[X_iX_j],\\ C(i,j) &=\operatorname{Cov}(X_i,X_j)\\ &=G_2(i,j)-m(i)m(j). \end{aligned}$$

Thus the raw two-point function and the covariance coincide only for a centered process. When $C(i,i)>0$ and $C(j,j)>0$, the normalized correlation coefficient is

$$\rho(i,j) =\frac{C(i,j)}{ \sqrt{C(i,i)C(j,j)}}.$$

It is undefined when either variance vanishes. Cauchy–Schwarz gives $|\rho(i,j)|\leq1$.

Every covariance kernel is symmetric and positive semidefinite. Indeed, for finite indices $i_1,\ldots,i_n$ and real coefficients $a_1,\ldots,a_n$,

$$\begin{aligned} \sum_{r,s=1}^n a_ra_s C(i_r,i_s) &=\mathbb E\!\left[ \left( \sum_{r=1}^n a_r(X_{i_r}-m(i_r)) \right)^2 \right]\\ &\geq0. \end{aligned}$$

This is a necessary condition for a proposed covariance. In the Gaussian case it is also the key existence condition, and mean plus covariance determine all finite-dimensional laws; see Gaussian Vectors, Processes, Random Distributions, and Wick Structure. Outside the Gaussian class it is far from sufficient. An independent standard-normal sequence and an independent Rademacher sequence taking values $\pm1$ both have

$$m(n)=0, \qquad C(n,k)=\delta_{nk},$$

but their one-time laws, fourth moments, and finite-dimensional laws differ.

Higher connected correlations are joint cumulants, not ordinary moments. The conversion between moments, cumulants, and source derivatives is developed in Characteristic Functions, Moments, Cumulants, and Generating Functionals.

Strict and second-order stationarity

Let the index set be $\mathbb R$ or $\mathbb Z$ so that shifts are defined. A process is strictly stationary when every finite-dimensional law is translation invariant:

$$(X_{t_1+h},\ldots,X_{t_n+h}) \overset{\mathrm d}{=} (X_{t_1},\ldots,X_{t_n})$$

for every $n$, every allowed shift $h$, and every index tuple.

A real process is second-order stationary, also called weakly or wide-sense stationary, when every $X_t\in L^2$, the mean is constant, and the covariance depends only on the lag:

$$m(t)=\mu, \qquad C(t,s)=C(t-s).$$

Strict stationarity plus finite second moments implies second-order stationarity. The converse fails. Let $A$ and $B$ be independent Rademacher variables and, for $n\in\mathbb Z$, set

$$X_n =A\cos\!\left(\frac{n\pi}{4}\right) +B\sin\!\left(\frac{n\pi}{4}\right).$$

Then

$$\mathbb E[X_n]=0, \qquad \mathbb E[X_nX_m] =\cos\!\left(\frac{(n-m)\pi}{4}\right),$$

so the process is second-order stationary. But $X_0=A$ takes only the values $\pm1$, whereas

$$X_1=\frac{A+B}{\sqrt2}$$

also takes the value $0$. Even the one-time law changes with $n$, so the process is not strictly stationary. For a Gaussian process, by contrast, mean and covariance determine every finite-dimensional law, so second-order stationarity is equivalent to strict stationarity. Pavliotis develops these two notions and the Gaussian equivalence in Pavliotis 2014, § 1.2, pp. 3–4, PDF.

For a spatial random field, translation invariance is often called homogeneity. Isotropy is a separate rotational invariance. A homogeneous field can select a preferred direction, and an isotropic field need not be translation invariant. For distribution-valued fields, these statements are made through smeared variables and their joint laws rather than undefined point values.

Stationarity also does not authorize replacing an ensemble expectation by a time average along one realization. The extra hypotheses behind that step, together with Markov evolution and correlated-sample uncertainty, are developed in Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error.

From stationary covariance to spectral measure

For a real second-order stationary process indexed by $I=\mathbb R$, write

$$C(\tau) =\operatorname{Cov}(X_{t+\tau},X_t).$$

The covariance is even and positive semidefinite. If $C$ is continuous, Bochner’s theorem supplies a unique finite positive measure $\nu$ such that

$$C(\tau) =\int_{\mathbb R}e^{-i\omega\tau}\,\nu(\mathrm d\omega), \qquad \nu(\mathbb R)=C(0).$$

The measure $\nu$ is the spectral measure. It may contain atoms, so a spectral density need not exist. If it is absolutely continuous in the normalization

$$\nu(\mathrm d\omega) =\frac{\widetilde C(\omega)}{2\pi}\, \mathrm d\omega,$$

then $\widetilde C(\omega)\geq0$ almost everywhere and

$$\begin{aligned} C(\tau) &=\int_{\mathbb R}\frac{\mathrm d\omega}{2\pi}\, e^{-i\omega\tau}\widetilde C(\omega),\\ C(0) &=\int_{\mathbb R}\frac{\mathrm d\omega}{2\pi}\, \widetilde C(\omega). \end{aligned}$$

When $C\in L^1(\mathbb R)$, the density is its ordinary forward transform shown in the local convention above. Rasmussen and Williams state Bochner’s theorem and the Wiener–Khinchin pair in Rasmussen and Williams 2006, § 4.2.1, p. 82, PDF. Their frequency $s$ counts cycles and appears as $e^{2\pi i s\tau}$. For the real covariance used here, the spectrum is even; after writing angular frequency $\omega=2\pi s$, the equivalent forward-plus, inverse-minus pair is the one used above.

Covariance continuity controls only mean-square continuity. For a centered stationary process,

$$\mathbb E[(X_{t+h}-X_t)^2] =2\bigl(C(0)-C(h)\bigr).$$

Thus continuity of $C$ at the origin is equivalent to $X_{t+h}\to X_t$ in $L^2$. It does not imply almost-sure continuity of sample paths. This distinction is summarized in Rasmussen and Williams Rasmussen and Williams 2006, § 4.1.1, p. 81, PDF.

Controlled QFT-facing example: one regulated Gaussian mode

Consider a classical, nonconserved order-parameter field in a finite spatial box with a momentum cutoff. Select one real normal-mode coordinate labelled by wavevector $\mathbf q$ and give it positive Gaussian stiffness

$$\kappa_{\mathbf q}=r+|\mathbf q|^2>0.$$

Let $T>0$ be the thermal temperature and $\Gamma>0$ a relaxation coefficient. With $k_{\mathrm B}=1$, the stationary Gaussian mode produced by linear relaxational dynamics has covariance

$$\boxed{ C_{\mathbf q}(\tau) =\frac{T}{\kappa_{\mathbf q}} e^{-\Gamma\kappa_{\mathbf q}|\tau|}. }$$

This formula alone specifies all finite-dimensional laws of the centered Gaussian mode. Its spectral density in the convention of this page is

$$\boxed{ \widetilde C_{\mathbf q}(\omega) =\frac{2T\Gamma}{ \omega^2+\Gamma^2\kappa_{\mathbf q}^2}. }$$

Three checks are immediate.

Equal-time normalization. The inverse transform gives

$$\begin{aligned} \int_{\mathbb R}\frac{\mathrm d\omega}{2\pi}\, \widetilde C_{\mathbf q}(\omega) &=\frac{2T\Gamma}{2\Gamma\kappa_{\mathbf q}}\\ &=\frac{T}{\kappa_{\mathbf q}} =C_{\mathbf q}(0). \end{aligned}$$

Positivity. The Lorentzian spectral density is nonnegative, as Bochner’s theorem requires.

Infrared stop rule. If the regulated zero-mode stiffness tends to zero, then $C_{\mathbf 0}(0)=T/\kappa_{\mathbf 0}$ diverges. The proposed stationary Gaussian probability law is no longer normalizable in that mode. A positive stiffness, a nonlinear stabilizing potential, a boundary condition or constraint eliminating the zero mode, or another genuine infrared regulator is needed.

Täuber obtains this correlation from the linear equilibrium model-A dynamics in Täuber 2006, § 1.1, equations (20)–(26), p. 8. His coefficient $D$ is denoted $\Gamma$ here, his $k_{\mathrm B}$ is set to one, and the continuum momentum delta function has been removed by retaining one normalized finite-volume mode. The stochastic equation, noise normalization, and fluctuation–dissipation derivation are intentionally not repeated. Continue to Brownian Motion, Stochastic Calculus, Langevin Equations, and Fokker–Planck Dynamics for the mathematical evolution machinery and to Langevin Field Equations and Noise for the developed thermal-field treatment.

Classical correlations are not generic quantum correlators

For real classical random variables, multiplication commutes, so

$$G_2(i,j)=G_2(j,i),$$

and covariance matrices are positive semidefinite. A Lorentzian Wightman function $\langle\phi(x)\phi(y)\rangle$, a time-ordered propagator, a retarded response, and a closed-time-path correlator carry operator ordering, state, and boundary-value information. They need not be symmetric real covariance kernels and must not be declared classical stochastic correlations merely because they have two arguments.

Positive Euclidean field measures can supply genuine random variables, often only after smearing. Reflection positivity and analytic continuation are additional structures, not consequences of ordinary covariance positivity. The regulated thermal mode above is explicitly a classical stochastic application and makes no stronger claim.

Common pitfalls

Specifying only one-time marginals. The laws of individual $X_i$ do not specify how different indices are coupled. A process requires all consistent finite-dimensional joint laws.

Treating covariance as the law. Mean and covariance determine a Gaussian process, not a general process. Equal covariance also does not imply independence outside jointly Gaussian families.

Equating cylinder-law existence with regular paths. Kolmogorov consistency constructs a product-space measure. Joint measurability, continuity, and a chosen path topology require separate arguments.

Using weak stationarity as strict stationarity. Lag-dependent covariance controls only second moments. Non-Gaussian higher-dimensional laws may still change under translations.

Reading an ordered quantum two-point function as a covariance. Ordering, complex phases, causal prescriptions, and positivity conditions distinguish quantum correlators from classical moment functions.

Check your understanding

1. A modification that is visibly different

For the spike process $Y_t=\mathbf1_{\{U=t\}}$ and the zero process $X_t=0$, show directly that every finite-dimensional law agrees while the two paths differ for every outcome.

Solution

Fix $t_1,\ldots,t_n$. Since a finite set has zero probability under the uniform law,

$$\mathbb P\!\left(U\in\{t_1,\ldots,t_n\}\right)=0.$$

Therefore

$$(Y_{t_1},\ldots,Y_{t_n})=(0,\ldots,0)$$

almost surely, exactly as for $X$. But for every outcome $\omega$, choosing $t=U(\omega)$ gives $Y_t(\omega)=1$ while $X_t(\omega)=0$. Hence there is no outcome on which the paths agree at every time.

2. Covariance without strict stationarity

For the Rademacher construction

$$X_n=A\cos(n\pi/4)+B\sin(n\pi/4),$$

derive its covariance and identify the first obstruction to strict stationarity.

Solution

Independence and centering give $\mathbb E[A^2]=\mathbb E[B^2]=1$ and $\mathbb E[AB]=0$. Thus

$$\begin{aligned} \mathbb E[X_nX_m] ={}&\cos(n\pi/4)\cos(m\pi/4)\\ &+\sin(n\pi/4)\sin(m\pi/4)\\ =&\cos((n-m)\pi/4). \end{aligned}$$

The mean is zero and covariance depends only on the lag. Nevertheless, $X_0=A$ never vanishes, while $X_1=(A+B)/\sqrt2$ equals zero with probability $1/2$. The one-time marginal already fails translation invariance, so the process is not strictly stationary.

3. Check the regulated mode at equal time

Evaluate the inverse spectral integral for $\widetilde C_{\mathbf q}(\omega)$ and explain what fails as $\kappa_{\mathbf q}\downarrow0$.

Solution

Using

$$\int_{-\infty}^{\infty} \frac{\mathrm d\omega}{\omega^2+a^2} =\frac{\pi}{a} \qquad(a>0),$$

one finds

$$\begin{aligned} \int_{\mathbb R}\frac{\mathrm d\omega}{2\pi}\, \frac{2T\Gamma}{ \omega^2+\Gamma^2\kappa_{\mathbf q}^2} &=\frac{T}{\kappa_{\mathbf q}}\\ &=C_{\mathbf q}(0). \end{aligned}$$

As $\kappa_{\mathbf q}$ tends to zero, both the equal-time variance and the integrated spectral weight diverge. A centered Gaussian coordinate with infinite variance is not the normalized stationary mode assumed in the calculation.

Synthesis and continuations

A stochastic process is specified by compatible finite-dimensional joint laws, which determine its cylinder law. Sample-path regularity and joint measurability are extra. Equality of finite-dimensional laws, modification, indistinguishability, and agreement of correlations are successively different comparison statements. Correlation functions encode moments; under second-order stationarity their covariance depends on a lag and has a positive spectral measure, but only Gaussianity promotes mean and covariance to a full law.

For stochastic integrals, trajectory equations, and deterministic evolution of probability laws, continue to Brownian Motion, Stochastic Calculus, Langevin Equations, and Fokker–Planck Dynamics. For Markov kernels, semigroups, ergodicity, integrated autocorrelation, and estimator uncertainty, continue to Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error.

References

• Péter Kevei, Stochastic Processes, PDF, notes dated May 21, 2026. § 3.1, p. 33, distinguishes finite-dimensional agreement, modifications, and indistinguishability. §§ 4.1–4.2, pp. 41–46, develop cylinder products and Kolmogorov consistency, and p. 49 exposes the path-regularity gap.

• Grigorios A. Pavliotis, Stochastic Processes and Applications, PDF, Springer, 2014. § 1.1, pp. 1–3, supplies the teaching route through processes, paths, and finite-dimensional distributions; § 1.2, pp. 3–7, supports strict and second-order stationarity, covariance positivity, spectral measures, and exponential covariance.

• Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, Chapter 4, PDF, MIT Press, 2006. § 4.1.1, p. 81, separates mean-square from sample-path continuity; § 4.2.1, p. 82, states Bochner’s theorem and the Wiener–Khinchin Fourier pair.

• Uwe C. Täuber, “Field Theory Approaches to Nonequilibrium Dynamics”, arXiv:cond-mat/0511743v2, 2006. § 1.1, pp. 6–8, develops the regulated linear relaxational mode, its thermal noise convention, and its time- and frequency-domain Gaussian correlations.