Probability and Stochastic Processes

Use this chapter when a QFT calculation asks what makes a weight into a probability measure, how conditioning changes an expectation, which limit theorem supports an estimator, why Gaussian correlations reduce to pairings, how a stochastic trajectory induces evolution of probability laws, or how correlated samples change an uncertainty estimate. The common discipline is to name the probabilistic object and the hypotheses before manipulating its formula.

There are three independent entrances. The probability-foundations route starts with measures, random variables, conditional expectation, and modes of convergence. The Gaussian-fields route starts from characteristic and source-generating functions and proceeds to covariance and Wick structure. The stochastic-evolution route starts from processes and then branches to Brownian/Langevin dynamics or to Markov semigroups and sampling error. These routes meet, but they do not form one compulsory seven-page chain.

The chapter keeps several commonly conflated statements separate. A density is relative to a named reference measure; a random distribution is defined by its action on test functions rather than by point values; an invariant law need not imply reversibility; ergodicity alone does not supply a central limit theorem; and a Markov generator need not be self-adjoint. Those distinctions matter whenever formal path-integral notation, stochastic dynamics, and statistical inference occur in the same calculation.

The scope is reusable probability and stochastic analysis. Developed Euclidean field constructions, thermal or nonequilibrium interpretations, production Monte Carlo algorithms, and lattice-specific diagnostics belong to the later treatments linked below. A normalized nonnegative regulated Euclidean weight can define a probability measure. An oscillatory factor $e^{iS}$, a sign-changing weight, or an unconstructed continuum functional integral does not automatically do so.

Diagnose · Choose a route · Compare objects · Dependency map · Conventions · Page guide · Gaussian/OU thread · Review · Continue

Enter through the object you need

This overview has no prerequisite. Each target page does, however, state the measure-theoretic, algebraic, or earlier probabilistic structure used in its main argument. Use the checks below independently: uncertainty about stochastic calculus does not block the Gaussian route, and uncertainty about generating functionals does not block general process theory.

Observable readiness checks and exact repair routes

Check	Ready	Unsure	Repair
Can you distinguish a probability measure, the law of a random variable, and a density relative to a reference measure?	Enter probability foundations directly.	For one random variable, write its pushforward law and say whether a density exists and relative to which measure.	Repair Measures and Measurable Functions and Lebesgue Integration and Convergence Theorems, then use Probability Spaces, Random Variables, and Conditional Expectation.
Can you state whether a claimed limit is almost sure, in probability, in mean, or in law, and list the theorem's hypotheses?	Enter limit theorems after probability foundations.	Test which implications survive when expectations are not uniformly integrable or variances are infinite.	Use Probability Spaces; Limits, Completeness, and Modes of Convergence is recommended before Probabilistic Convergence and Limit Theorems.
Can you distinguish a characteristic function, a moment-generating function, a cumulant generator, and a Euclidean source functional?	Enter generating objects after probability foundations.	For each object, name its domain, normalization, and derivative interpretation.	Use Probability Spaces; Fourier Series, Fourier Transforms, and Plancherel Theory is recommended before Characteristic Functions, Moments, Cumulants, and Generating Functionals.
Can you say why a Gaussian law may be singular and why a continuum random field may exist only after smearing?	Enter Gaussian vectors and random distributions.	Check positivity of the covariance form, then ask separately about finite-dimensional consistency and path or distributional realization.	Use Generating Functionals and Bilinear and Hermitian Forms, Adjoints, and Isometries; Test-Function Spaces and Distributions is recommended before Gaussian Vectors, Processes, Random Distributions, and Wick Structure.
Can you distinguish finite-dimensional laws, sample paths, covariance functions, and spectral measures?	Enter stochastic processes after probability foundations.	Ask whether equality is in finite-dimensional law, is a modification, or is pathwise indistinguishability.	Use Probability Spaces, then Stochastic Processes and Correlation Functions.
Can you identify the stochastic-integral convention and map an SDE to its backward generator and forward law equation?	Enter Brownian and Langevin dynamics after processes.	Write the diffusion matrix, the factor of one half, and the boundary condition or probability current.	Use Stochastic Processes, then Brownian Motion, Stochastic Calculus, Langevin Equations, and Fokker–Planck Dynamics.
Can you separate invariance, ergodicity, reversibility, convergence to stationarity, a Markov-chain CLT, and an autocorrelation estimate?	Enter Markov semigroups and correlated-sample error.	For one chain, identify which statement is proved and which extra hypotheses the next statement needs.	Use both Stochastic Processes and Probabilistic Convergence and Limit Theorems before Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error.

The statistical and probability diagnostic gives item-level feedback. Its statistical ensembles and probability repair is a bounded review of ensembles, fluctuations, probability, estimators, and uncertainty. It does not by itself prepare Gaussian random distributions, stochastic calculus, or general Markov-chain limit theory.

Choose a route from the QFT question

In the table, requires marks preparation used in the target page’s main argument. Recommended preparation improves fluency but does not block entry. Continue points to the later page where the mathematical structure receives its developed physical or statistical role.

Goal-to-route choices through probability and stochastic analysis

Reader goal	Route and preparation	Observable result
Interpret a finite positive Euclidean integral probabilistically	Requires: measure and Lebesgue integration. Enter Probability Spaces, Random Variables, and Conditional Expectation, then continue to Euclidean Correlators and Schwinger Functions.	Name the normalized measure, construct laws as pushforwards, compute expectations and conditional expectations, and mark the boundary between a regulated measure and a formal continuum weight.
Justify an estimator or Gaussian fluctuation approximation	Requires: Probability Spaces. General convergence is recommended. Enter Probabilistic Convergence and Limit Theorems, then continue to Estimators, Covariance, and Resampling.	Match the desired conclusion to a convergence mode, state every LLN or CLT hypothesis, and distinguish an iid theorem from a correlated-sequence theorem.
Package full and connected correlations or organize a free Gaussian field	Requires: Probability Spaces, then Generating Functionals. Forms are required and distributions recommended before Gaussian and Wick Structure. Continue to Connected Correlators and Cumulants or Gaussian Fields and Sources.	Choose the correct generating object, recover connected blocks from its logarithm where derivatives exist, and state Gaussian pairing claims at the finite-dimensional or smeared level actually constructed.
Move between correlations, stochastic trajectories, and probability-law evolution	Requires: Probability Spaces, then Stochastic Processes. Continue through Brownian, Langevin, and Fokker–Planck Dynamics to Fokker–Planck Evolution and Stationary Measures.	Separate finite-dimensional laws from paths, declare Itô or Stratonovich, and derive backward-observable and forward-law evolution with the boundary data kept visible.
Quantify uncertainty from correlated samples without assuming reversible dynamics	Requires: both Stochastic Processes and Limit Theorems. Enter Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error, then continue to Autocorrelation Times and Effective Sample Size.	Act on observables and laws with the correct dual operators, test detailed balance only when relevant, and attach an autocorrelation time and variance-equivalent sample size to a named observable.

Keep the probabilistic objects distinct

The same integral or covariance notation can represent different structures. The comparison below records what each structure licenses and the extra step needed before a stronger conclusion is valid.

Probabilistic structures, warranted operations, and stop rules

Structure	Operation and warranted result	What does not follow automatically
Probability measure on an outcome space	A normalized nonnegative countably additive measure assigns probabilities to events and expectations to integrable random variables.	An oscillatory, complex, sign-changing, unnormalized, or merely formal functional weight is not thereby a probability measure.
Law and density	The law is the pushforward measure of a random variable. A density is its Radon–Nikodym derivative relative to a named reference measure.	A law need not possess a density, and changing the reference measure changes the density without changing the law.
Conditional expectation	Conditioning on a σ-algebra produces an almost-sure equivalence class satisfying the defining integral identities and projection properties.	A pointwise expression conditioned on Y = y needs a regular conditional law and a chosen version; values on null sets are not canonical.
Probabilistic convergence and a limit theorem	Almost-sure, probability, mean, and distributional convergence answer different questions. An LLN or CLT transfers only under its stated dependence, moment, and regularity hypotheses.	Convergence in law does not generally move expectations, and stationarity or ergodicity alone does not guarantee a CLT.
Characteristic, moment, cumulant, and source generators	A characteristic function always exists and determines a finite-dimensional law. Moment and source derivatives encode moments or responses only on a suitable domain; a local logarithm encodes cumulants.	A moment-generating function need not exist away from zero, a logarithm need not be global, and a finite list of vanishing higher cumulants does not prove Gaussianity.
Gaussian vector, process, or random distribution	Within the Gaussian class, mean and positive-semidefinite covariance determine every finite-dimensional law, and centered moments reduce to pairings.	Covariance positivity alone does not provide sample-path regularity, point values, local products, or reflection positivity. For jointly Gaussian variables, zero covariance implies independence; outside that class it does not in general.
Stochastic process and correlation function	A process is a jointly defined indexed family. A projectively consistent family of finite-dimensional laws with Borel–Euclidean coordinate state spaces specifies its cylinder law, while correlations summarize selected moments; a continuous stationary covariance has a spectral measure.	Correlations do not determine a non-Gaussian law, weak stationarity need not imply strict stationarity or ergodicity, and a spectral measure need not have a density.
SDE, generator, and forward law evolution	A well-posed SDE in a declared stochastic-integral convention defines a process. Its generator acts backward on observables, while its adjoint evolves laws forward.	A formal differential expression without a domain and boundary behavior is not a complete generator; a weak law need not possess a smooth density satisfying a classical PDE.
Markov semigroup and invariant law	Transition kernels compose by Chapman–Kolmogorov and induce a semigroup. An invariant law is fixed by the forward action.	Invariance is not uniqueness, convergence, ergodicity, reversibility, or a CLT. Detailed balance and self-adjointness describe a special reversible case, not Markov dynamics in general.
Autocorrelation time and effective sample size	When an observable's covariance series is absolutely summable, its integrated autocorrelation time gives the leading sample-mean variance relative to iid sampling. A Gaussian error law separately requires a valid CLT.	The value is not a property of the chain alone, a finite run does not reveal an infinite covariance sum exactly, and effective sample size is defined only for a positive asymptotic variance ratio and is variance-equivalent rather than a literal count.

Probability spaces, laws, independence, and conditional expectation are developed in Durrett 2019, §§ 1.1–1.3, 1.6, 2.1, and 4.1, printed pp. 1–18, 28–34, 43–49, and 205–215, open author-hosted PDF. The same source supports the convergence, LLN, CLT, and characteristic-function distinctions in §§ 2.2–2.5, 3.2–3.4, and 3.10, printed pp. 56–87, 116–152, and 199–201. Gaussian-process specification and the separation of mean-square from sample-path continuity are treated in Rasmussen and Williams 2006, §§ 2.2, 4.1.1, and 4.2.1, printed pp. 13–14 and 81–82. Generalized stochastic processes and characteristic functionals on test-function spaces are treated in Fageot, Amini, and Unser 2014, §§ 2.1–2.2, printed pp. 4–5, arXiv PDF. The projective-consistency hypotheses for a cylinder law and the separate path-regularity problem are developed in Kevei 2026, §§ 4.1–4.2, printed pp. 41–46, and the path-space discussion beginning on printed p. 49, open author-hosted PDF.

The process, Markov, SDE, forward-equation, current, and reversibility distinctions are developed in Pavliotis 2014, §§ 1.1–1.3, 2.2–2.4, 3.1–3.5, 4.1, and 4.6, printed pp. 1–13, 30–39, 49–66, 77–80, and 104–105, author-manuscript PDF. Conditions for general-state Markov-chain convergence, laws of large numbers, CLTs, and Poisson-equation methods are separated in Roberts and Rosenthal 2004, printed pp. 4–5, §§ 3.2 and 5, printed pp. 14–18 and 42–47, author version PDF. The covariance-sum error formula, integrated-autocorrelation convention, and finite-window cautions are treated in Wolff 2004/2006, § 2, printed pp. 4–6, and §§ 3.1–3.3, printed pp. 7–13, arXiv PDF.

How the seven pages fit together

The hard-preparation graph contains ten direct edges: seven within this chapter and three from earlier chapters. The recommended graph contains three additional external edges. Chapter order is a reference order, not a demand to read every page in sequence.

1. Establish the root. Probability Spaces, Random Variables, and Conditional Expectation requires measures and Lebesgue integration.
2. Split into three branches. Probability foundations lead independently to Limit Theorems, Generating Functionals, and Stochastic Processes.
3. Continue the Gaussian branch. Generating Functionals and bilinear or Hermitian forms are both required for Gaussian Vectors, Processes, Random Distributions, and Wick Structure. General stochastic processes do not require this Gaussian page.
4. Split the process branch. Stochastic Processes leads separately to Brownian, Langevin, and Fokker–Planck Dynamics and to the Markov page. Brownian motion is an important model, not a hard prerequisite for general Markov semigroups.
5. Join dependence with asymptotics. Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error requires both Stochastic Processes and Limit Theorems because it combines dependent evolution with asymptotic inference.

The three hard external preparations are Measures and Measurable Functions, Lebesgue Integration and Convergence Theorems, and Bilinear and Hermitian Forms, Adjoints, and Isometries. The three recommended preparations are Limits, Completeness, and Modes of Convergence, Fourier Series, Fourier Transforms, and Plancherel Theory, and Test-Function Spaces, Distributions, Support, and Convergence.

Shared conventions and stop rules

The pages use one convention package so that a formula can be moved from probability to field theory or stochastic dynamics without silently changing signs or normalizations.

Measures, laws, and conditioning

A probability measure $\mathbb P$ is normalized and nonnegative, $\mathbb P(\Omega)=1$. For a random variable $X:\Omega\to E$, its law is the pushforward

$$\mu_X(A)=\mathbb P(X\in A)=\mathbb P\!\left(X^{-1}(A)\right).$$

If $\mu_X$ is absolutely continuous relative to a declared reference measure $\nu$, then its density is $p_X=\mathrm d\mu_X/\mathrm d\nu$. The notation $p_X(x)\,\mathrm dx$ therefore includes a choice of reference measure; it is not a replacement for the measure-theoretic statement.

Equalities between random variables and conditional expectations are understood almost surely unless a pointwise version has been constructed. Writing $\mathbb E[X\mid Y=y]$ requires a regular conditional law (or an equivalent construction) and a version; its values at exceptional $y$ are not fixed by the almost-sure equivalence class.

Characteristic, source, and spectral signs

For $X\in\mathbb R^d$, the characteristic-function convention is

$$\varphi_X(t)=\mathbb E\!\left[e^{+it^{\mathsf T}X}\right], \qquad p_X(x)=\int_{\mathbb R^d}\frac{\mathrm d^d t}{(2\pi)^d}\, e^{-it^{\mathsf T}x}\varphi_X(t)$$

when an inversion theorem applies. The characteristic function always exists, but its derivatives encode moments only under the relevant integrability conditions.

For a normalized positive Euclidean measure and sources $J$ for which the exponential is integrable, the source convention is

$$Z_E[J]=\mathbb E\!\left[e^{\langle J,\Phi\rangle}\right], \qquad W_E[J]=\log Z_E[J]\quad\text{when }0<Z_E[J]<\infty, \qquad Z_E[0]=1.$$

This is a moment-generating convention, not a characteristic functional and not a Lorentzian path integral. In particular, Lorentzian conventions such as $Z_L[J]=\int e^{iS+i\int J\phi}$ and $W_L=-i\log Z_L$ carry factors of $i$ that must not be imported into $Z_E$.

For a second-order stationary process, the angular-frequency convention is

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

If the positive-definite stationary covariance $C$ is continuous, Bochner’s theorem supplies a unique finite positive spectral measure. For a covariance, continuity at the origin already implies continuity everywhere. The displayed density $\widetilde C$ is available only when that spectral measure is absolutely continuous.

Itô dynamics and the two operator directions

Itô is the default convention. For

$$\mathrm dX_t^i=b^i(X_t)\,\mathrm dt +\sigma^i{}_a(X_t)\,\mathrm dW_t^a, \qquad a^{ij}=\sigma^i{}_a\sigma^j{}_a,$$

the backward generator on an appropriate domain of observables is

$$Lf=b^i\partial_i f+\frac12a^{ij}\partial_i\partial_j f.$$

When the law has a sufficiently regular density, the forward equation is

$$\partial_t p=L^\dagger p =-\partial_i(b^ip) +\frac12\partial_i\partial_j(a^{ij}p).$$

The function space, operator domain, boundary behavior, and weak meaning of this equation are part of the statement. For multiplicative noise, a Stratonovich drift $\beta$ represents the same process as the Itô drift

$$b^i=\beta^i+\frac12\sigma^j{}_a\partial_j\sigma^i{}_a.$$

Thus changing the stochastic-integral convention without changing the drift changes the model.

For a Markov process, the transition semigroup acts backward on observables,

$$P_tf(x)=\mathbb E_x[f(X_t)],$$

while probability laws evolve by the dual forward action. The mnemonic $P_t=e^{tL}$ is semigroup notation, not permission to ignore the generator’s domain. Detailed balance may make $P_t$ self-adjoint in a weighted $L^2$ space, but neither property is required in the definition of a Markov process.

Autocorrelation and uncertainty

For a stationary sampled observable $f_n=f(X_n)$, set

$$\gamma_f(k)=\operatorname{Cov}(f_0,f_k), \qquad \rho_f(k)=\frac{\gamma_f(k)}{\gamma_f(0)}, \qquad \gamma_f(0)>0.$$

When the covariance series is absolutely summable, this chapter uses Wolff’s factor-of-two convention

$$\tau_{\mathrm{int},f} =\frac12+\sum_{k=1}^{\infty}\rho_f(k).$$

Then $2\gamma_f(0)\tau_{\mathrm{int},f}$ is the leading $N^{-1}$ sample-mean variance coefficient. When $2\tau_{\mathrm{int},f}>0$, define the variance-equivalent count

$$N_{\mathrm{eff},f}=\frac{N}{2\tau_{\mathrm{int},f}}.$$

If the ratio vanishes, report the degenerate $N^{-1}$ coefficient rather than a literal infinite effective count. A Gaussian error law additionally needs a valid CLT. Both quantities are observable-specific. A finite-run window is an estimator with its own bias–variance tradeoff, not the infinite sum itself.

Exact page guide

The seven pages below appear in chapter order. The questions are independent entry tests, and the preparation statements identify the actual hard and recommended edges.

Probability Spaces, Random Variables, and Conditional Expectation

Question. What structure defines probability, expectation, conditioning, and independence?

This page begins with a measurable space and a normalized nonnegative measure. It constructs random-variable laws by pushforward, distinguishes independence from zero covariance, defines conditional expectation relative to a sub-$\sigma$-algebra, and explains almost-sure uniqueness and regular conditional laws. Measures and measurable functions and Lebesgue integration are both required.

The finite-dimensional QFT application is a positive regulated Euclidean Gaussian integral. It is deliberately bounded: it does not turn an oscillatory Lorentzian factor or an unconstructed continuum path integral into a probability measure. Continue to Euclidean Correlators and Schwinger Functions.

Probabilistic Convergence, Laws of Large Numbers, and Central Limit Theorems

Question. Which convergence notion and limit theorem justify an estimator or fluctuation approximation?

This page separates almost-sure, probability, mean, and distributional convergence; records the valid implications and missing arrows; and then matches laws of large numbers and central limit theorems to explicit moment and dependence assumptions. Probability Spaces is required. General limit and completeness language is recommended.

The main stop rule is that a conclusion determines the theorem: consistency, expectation convergence, asymptotic normality, and a finite-sample error bound are different requests. The iid CLT does not apply unchanged to one correlated Markov-chain trajectory. Continue to Estimators, Covariance, and Resampling.

Characteristic Functions, Moments, Cumulants, and Generating Functionals

Question. How do generating objects package distributions, moments, connected correlations, and responses?

This page distinguishes characteristic functions, moment-generating functions, cumulant generators, and Euclidean or Lorentzian source functionals. It tracks domains, logarithm branches, derivative hypotheses, normalization, and factors of $i$. Probability Spaces is required; Fourier theory is recommended.

The logarithm organizes connected set partitions where the relevant derivatives exist, but a formal power series is not automatically a convergent generating function. Continue to Connected Correlators and Cumulants.

Gaussian Vectors, Processes, Random Distributions, and Wick Structure

Question. Why are Gaussian correlations fixed by mean and covariance, and which infinite-dimensional claims need extra care?

This page allows singular finite-dimensional Gaussian laws, proves the Wick–Isserlis pairing structure, then separates consistent finite-dimensional Gaussian laws from path regularity and random-distribution realization. Generating Functionals and bilinear or Hermitian forms are required. Test-function spaces and distributions are recommended.

For continuum fields, the basic variables are smeared quantities $\Phi(f)$. Point values, products at coincident points, and local Wick powers need additional construction. Classical commuting Gaussian pairing is also not the same theorem as operator-ordered or fermionic Wick expansion. Continue to Gaussian Fields and Sources.

Stochastic Processes and Correlation Functions

Question. How are time- or space-indexed random variables specified and compared through finite-dimensional laws and correlations?

This page distinguishes an indexed random family from its sample paths, cylinder law, modifications, and finite-dimensional distributions. It then separates strict from second-order stationarity and covariance functions from spectral measures. Probability Spaces is required; Gaussian structure is not.

The controlled application is one regulated Gaussian relaxation mode. Its correlation and spectrum are classical stochastic objects, not generic time-ordered, Wightman, retarded, or Schwinger correlators. Continue to Langevin Field Equations and Noise.

Brownian Motion, Stochastic Calculus, Langevin Equations, and Fokker–Planck Dynamics

Question. How does stochastic trajectory evolution induce deterministic evolution of probability laws?

This page develops Brownian quadratic variation, Itô integration, Itô’s formula, the Itô– Stratonovich conversion, generators, forward equations, currents, and stationary densities. Stochastic Processes is required.

Existence and uniqueness hypotheses precede formal manipulation of an SDE, and Brownian paths are not ordinary differentiable forcing functions. A density-level Fokker–Planck equation also requires more regularity than weak forward evolution of measures. Continue to Fokker–Planck Evolution and Stationary Measures.

Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error

Question. How do transition kernels and generators encode Markov evolution, and how do ergodicity and autocorrelation affect uncertainty?

This page keeps the backward action on observables distinct from the forward action on laws, includes the generator domain, and separates invariance, stationarity, reversibility, ergodicity, convergence, LLNs, CLTs, and covariance-sum error formulas. Stochastic Processes and Limit Theorems are both required.

Detailed balance is a useful special case rather than a definition, and a nonreversible semigroup can still have an invariant probability measure. Autocorrelation time and effective sample size must name the observable and the normalization convention. Continue to Autocorrelation Times and Effective Sample Size.

One Gaussian and Ornstein–Uhlenbeck mode across the chapter

A single normalized finite-volume mode makes the joins among the seven pages visible without pretending to construct an interacting continuum field. Let

$$\kappa>0, \qquad \Gamma>0, \qquad T>0, \qquad \lambda=\Gamma\kappa, \qquad D=\Gamma T, \qquad v=\frac{D}{\lambda}=\frac{T}{\kappa}.$$

Here $\kappa$ is the positive quadratic kernel of the retained mode, $\Gamma$ is its relaxation coefficient, $T$ sets the noise strength, $\lambda$ is its relaxation rate, $D$ is its diffusion coefficient, and $v$ is its stationary variance. This is the one-mode form of the Gaussian relaxational example in Täuber 2006, § 1.1, especially equations (20)–(26), printed pp. 6–8, arXiv PDF; the continuum momentum delta function is absent because only one normalized regulated mode is retained.

The same regulated mode viewed from each page

Page lens	Object used	Diagnostic equality or distinction
Probability and conditioning	A centered Gaussian law π with variance v, and a stationary pair separated by Δ.	E[XΔ | X0] = αX0 and conditional variance v(1 − α²), where α = e−λΔ.
Limit theorems	Independent replicas from π versus successive values on one trajectory.	The iid average has variance v/N exactly; a single trajectory has covariance contributions and needs a dependent-sequence theorem.
Generating objects	Characteristic function, Euclidean source generator, and two-time connected generator.	The logarithm is quadratic, so cumulants above order two vanish for this Gaussian law.
Gaussian and Wick structure	The covariance kernel K(s,t) = v e−λ|t−s|.	The four-point moment is a sum over three pairings; path properties are a separate question.
Processes and correlations	The stationary covariance C(τ) and its Lorentzian-shaped spectral density.	The equal-time spectral integral returns v; mean and covariance determine the full law only because the process is Gaussian.
Langevin and Fokker–Planck dynamics	An Itô OU equation, its backward generator, forward density equation, and probability current.	The normalized Gaussian π is stationary because its current vanishes in this scalar reversible model.
Markov semigroup and sampling error	The Gaussian transition kernel and its sampled AR(1) chain.	For the linear observable, ρ(k) = αk; the centered square has ρ(k) = α2k, so uncertainty is observable-specific.

From a law to conditioning

The stationary one-mode law is

$$\pi(\mathrm dx) =\frac{1}{\sqrt{2\pi v}} \exp\!\left(-\frac{x^2}{2v}\right)\mathrm dx.$$

At this stage it is simply a normalized probability measure. To foreshadow time dependence without assuming a full process, define a centered jointly Gaussian pair $(X_0,X_\Delta)$ with covariance

$$\operatorname{Cov} \begin{pmatrix}X_0\\X_\Delta\end{pmatrix} =v \begin{pmatrix} 1 & \alpha\\ \alpha & 1 \end{pmatrix}, \qquad \Delta>0, \qquad \alpha=e^{-\lambda\Delta}\in(0,1).$$

Gaussian conditioning then gives, almost surely,

$$\mathbb E[X_\Delta\mid X_0]=\alpha X_0, \qquad \operatorname{Var}(X_\Delta\mid X_0)=v(1-\alpha^2).$$

These two-variable statements do not yet construct compatible laws for every time or establish path regularity.

Independent replicas and dependent observations

For independent replicas $X_1,\ldots,X_N\sim\pi$,

$$\overline X_N=\frac1N\sum_{j=1}^N X_j \sim\mathcal N\!\left(0,\frac{v}{N}\right), \qquad \frac1v\sum_{j=1}^N X_j^2\sim\chi_N^2.$$

The normality of $\overline X_N$ here is an exact finite-$N$ consequence of Gaussian closure, not an approximation supplied by the CLT. For a general iid law with finite variance, the CLT instead supplies an asymptotic distribution after centering and scaling. Values sampled successively from the OU trajectory below are not independent replicas, so their mean has a different variance even though every one-time marginal is $\pi$.

Characteristic, source, and Wick data

For the same one-time law,

$$\varphi_X(k)=e^{-vk^2/2}, \qquad Z_E(J)=e^{vJ^2/2}, \qquad W_E(J)=\frac{vJ^2}{2}.$$

At two stationary times,

$$W_E(J_0,J_t) =\frac v2\left( J_0^2+2e^{-\lambda t}J_0J_t+J_t^2 \right), \qquad t\geq0.$$

Every cumulant above second order vanishes. The moment–cumulant partition relation and the finite-dimensional Wick–Isserlis formula used here are developed in McCullagh 2018, Chapter 2, §§ 2.2–2.7, printed pp. 29–54, and § 3.9.1, printed pp. 85–86, official author-hosted chapter directory. Equivalently, the Gaussian covariance kernel

$$K(s,t)=v e^{-\lambda|t-s|}$$

fixes all finite-dimensional Gaussian laws. Wick pairing gives the useful observable check

$$\mathbb E[X_s^2X_t^2] =v^2+2v^2e^{-2\lambda|t-s|}, \qquad \operatorname{Cov}(X_s^2,X_t^2) =2v^2e^{-2\lambda|t-s|}.$$

This covariance is positive semidefinite and consistent with the process constructed below, but covariance positivity in a general problem must not be silently upgraded to continuous sample paths or pointwise continuum fields.

Correlation and spectrum

In stationarity the centered autocovariance is

$$C(\tau)=v e^{-\lambda|\tau|},$$

so the declared Fourier convention gives

$$\widetilde C(\omega) =\frac{2v\lambda}{\omega^2+\lambda^2} =\frac{2\Gamma T}{\omega^2+\Gamma^2\kappa^2}.$$

The normalization check is

$$\int_{\mathbb R}\frac{\mathrm d\omega}{2\pi}\, \widetilde C(\omega)=C(0)=v.$$

Outside the Gaussian class, the same two-point function would not specify the full finite-dimensional laws.

Trajectory evolution and law evolution

The scalar Itô Ornstein–Uhlenbeck equation is

$$\mathrm dX_s=-\lambda X_s\,\mathrm ds+\sqrt{2D}\,\mathrm dW_s.$$

For a deterministic starting value $X_0=x$, its solution is Gaussian with

$$\mathbb E_x[X_s]=e^{-\lambda s}x, \qquad \operatorname{Var}_x(X_s)=v\left(1-e^{-2\lambda s}\right).$$

The noise amplitude is $\sqrt{2D}$, so the factor of one half in the general generator leaves diffusion coefficient $D$:

$$Lf(x)=-\lambda x f'(x)+D f''(x).$$

The forward density equation and current are

$$\partial_s p =\lambda\,\partial_x(xp)+D\,\partial_x^2p =-\partial_x J, \qquad J=-\lambda xp-D\,\partial_xp.$$

The stationary density

$$\pi(x)=\sqrt{\frac{\lambda}{2\pi D}} \exp\!\left(-\frac{\lambda x^2}{2D}\right)$$

has $J=0$. Zero current makes this scalar example reversible under its usual boundary behavior; it is not part of the definition of a Markov process. The generator, forward equation, current, and invariant density follow the convention in Pavliotis 2014, §§ 3.1–3.5 and 4.1, printed pp. 49–66 and 77–80, open author-manuscript PDF.

The parameter $s$ can represent model time, algorithmic time, or an auxiliary stochastic-quantization time depending on the application. Nothing in this one-mode SDE identifies it automatically with physical Lorentzian time or identifies $T$ automatically with a physical temperature.

Semigroup and observable-specific error

The backward transition semigroup has the explicit kernel

$$P_sf(x) =\int_{\mathbb R}f(y)\, \mathcal N\!\left( e^{-\lambda s}x, v(1-e^{-2\lambda s}) \right)(\mathrm dy).$$

For $\Delta>0$, sampling gives the recursion

$$X_{n+1} =\alpha X_n+\sqrt{v(1-\alpha^2)}\,\xi_n, \qquad \xi_n\overset{\mathrm{iid}}{\sim}\mathcal N(0,1).$$

It is stationary when $X_0\sim\pi$ and $X_0$ is independent of the iid innovations. The transition-semigroup interpretation follows Pavliotis 2014, §§ 2.2–2.4, printed pp. 30–39, open author-manuscript PDF.

For the linear observable $f(x)=x$,

$$\rho_f(k)=\alpha^k, \qquad \tau_{\mathrm{int},f} =\frac{1+\alpha}{2(1-\alpha)}, \qquad N_{\mathrm{eff},f} =N\frac{1-\alpha}{1+\alpha}.$$

Before taking a large-$N$ limit, the exact stationary sample-mean variance is

$$\operatorname{Var}(\overline X_N) =\frac vN\left[ 1+2\sum_{k=1}^{N-1} \left(1-\frac{k}{N}\right)\alpha^k \right].$$

Because the stationary chain is jointly Gaussian, this sample mean is exactly Gaussian for every $N$. That exact fact is special. A general Markov-chain CLT needs its own hypotheses. The covariance-sum normalization follows Wolff 2004/2006, § 2, printed pp. 4–6, arXiv PDF.

For the centered quadratic observable $q(x)=x^2-v$, Wick pairing gives

$$\rho_q(k)=\alpha^{2k}, \qquad \tau_{\mathrm{int},q} =\frac{1+\alpha^2}{2(1-\alpha^2)}.$$

Thus the same chain has different correlation times for different observables. Quoting a single chain-wide “effective sample size” without an observable is incomplete.

Reversibility is optional

A two-dimensional extension makes the logical point sharp. Let

$$R= \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix}, \qquad \mathrm d\mathbf X_s =\left(-\lambda I+\Omega R\right)\mathbf X_s\,\mathrm ds +\sqrt{2D}\,\mathrm d\mathbf W_s.$$

The centered isotropic Gaussian with covariance $vI$ remains invariant, but its stationary current is

$$\mathbf J(\mathbf x)=\Omega R\mathbf x\,\pi(\mathbf x).$$

This current is divergence-free because $\operatorname{tr}R=0$ and $(R\mathbf x)\mathbin{\cdot}\mathbf x=0$, yet it is nonzero when $\Omega\neq0$. When initialized from $\pi$, the process is Markov and stationary but not reversible. For other initial laws, the dynamics is Markov and has $\pi$ as a nonreversible invariant law. Its semigroup need not be self-adjoint, and different observables can see damped oscillatory correlations. This finite-dimensional check blocks the mistaken identification of Markov evolution, detailed balance, and self-adjoint dynamics.

The soft-mode stop rule

If $\kappa\to0^+$ with $\Gamma$ and $T$ fixed, then $\lambda=\Gamma\kappa\to0$ and the stationary variance $v=T/\kappa$ diverges. Starting instead from $X_0=0$ at a fixed finite time,

$$v\left(1-e^{-2\lambda s}\right) =\frac{D}{\lambda}\left(1-e^{-2\lambda s}\right) \longrightarrow 2Ds.$$

The finite-time law tends to Brownian diffusion, while the stationary probability law is lost. These are compatible statements about different limits. The one-mode regulator must therefore remain visible whenever the stationary Gaussian is used as a model for a field mode.

Synthesis

The chapter’s workflow can be compressed into six questions.

1. What is random? Name the outcome space, measurable variables, and the normalized nonnegative probability measure—or state explicitly that the expression is only formal.
2. What information is being retained? A law, density, conditional law, generating function, covariance, path, or transition kernel supports a different set of operations.
3. What limit is claimed? State the convergence mode and every moment, dependence, regularity, or domain hypothesis used by the theorem.
4. Which evolution direction is meant? The Markov semigroup acts backward on observables; its dual acts forward on laws. A density PDE is an additional regularity statement.
5. Which long-time property is proved? Invariance, uniqueness, convergence, ergodicity, reversibility, and a CLT must be checked separately.
6. Which observable carries the uncertainty? State the autocorrelation convention, finite-window procedure, and observable before quoting an effective sample size.

The Gaussian/OU thread is valuable precisely because every layer is explicit: a normalized law, a quadratic generator, a covariance kernel, a well-posed Itô process, a Markov semigroup, and an exactly checkable correlated-sample variance. Its simplicity is also its limit. It does not establish continuum field construction, interacting stochastic dynamics, or the validity of a production inference workflow.

Review the chapter

Each check gives concise success criteria and points back to the page that develops the relevant capability.

1. Separate the probabilistic objects

Review modes: Retrieval and comparison.
Tested capability: Distinguish a probability measure, a law, a density, a random distribution, and a formal path integral.
Needed pages: Probability Spaces, Random Variables, and Conditional Expectation and Gaussian Vectors, Processes, Random Distributions, and Wick Structure.
Expected response: A five-entry glossary that states the ambient space and the relation between adjacent objects.
Verification: Every density is named relative to a reference measure; smearing a random distribution produces a random variable; no formal functional integral is silently treated as a measure.
Repair if objects are conflated: Return to Measures and Measurable Functions.

For a measurable map $X:\Omega\to E$, classify $\mathbb P$, $\mu_X$, $p$, $\Phi$, and $\int\mathcal D\phi\,e^{-S[\phi]}$. State which object is pushed forward, which requires a reference measure, and which may exist only after smearing.

Check

$\mathbb P$ is a probability measure on $(\Omega,\mathcal F)$, and $\mu_X=X_\ast\mathbb P$ is the law of $X$ on $E$. A density $p=\mathrm d\mu_X/\mathrm d\nu$ exists only relative to a named measure $\nu$. A random distribution $\Phi$ need not have pointwise values, but $\Phi(f)$ is a random variable for each allowed test function $f$. A formal functional integral is not a probability measure until its measurable space, reference or constructed measure, normalization, and convergence have been established.

2. Translate moments into connected cumulants

Review modes: Explanation and representation change.
Tested capability: Choose the correct generating object and explain why its logarithm generates connected cumulants.
Needed page: Characteristic Functions, Moments, Cumulants, and Generating Functionals.
Expected response: A scalar-to-functional dictionary plus one explicit second-derivative identity.
Verification: $\varphi_X$ is distinguished from $M_X$; $Z_E[0]=1$ and $W_E[0]=0$; derivatives are asserted only where the needed moments and differentiation interchange are justified.
Repair if moments and cumulants are interchanged: Revisit Characteristic Functions, Moments, Cumulants, and Generating Functionals.

Write the characteristic function $\varphi_X$, moment-generating function $M_X$, cumulant generator $K_X$, Euclidean source functional $Z_E[J]$, and connected generator $W_E[J]$. Explain how the scalar relation $K_X=\log M_X$ becomes $W_E=\log Z_E$ and show explicitly what the first two derivatives of $W_E$ recover at zero source.

Check

For real $t$, $\varphi_X(t)=\mathbb E[e^{itX}]$ always exists, whereas $M_X(t)=\mathbb E[e^{tX}]$ is defined only where the exponential is integrable. Where $M_X$ is finite and nonzero, $K_X=\log M_X$. The field counterparts are $Z_E[J]=\mathbb E[e^{\langle J,\Phi\rangle}]$ and $W_E[J]=\log Z_E[J]$ on their source domain. At zero source, the first derivative gives the mean and the second gives the connected two-point function:

$$\left.\delta^2W_E\right|_0 =\left.\delta^2Z_E\right|_0 -\left.\delta Z_E\right|_0\left.\delta Z_E\right|_0,$$

because $Z_E[0]=1$. Taking a logarithm organizes connected parts; it does not create moments that fail to exist.

3. Check the Gaussian-to-Wick derivation

Review mode: Derivation or proof check.
Tested capability: Derive Gaussian moments and connected cumulants from a source functional while stating the covariance and smearing assumptions.
Needed pages: Characteristic Functions, Moments, Cumulants, and Generating Functionals and Gaussian Vectors, Processes, Random Distributions, and Wick Structure.
Expected response: A short source-derivative derivation of the two- and four-point functions, followed by the Gaussian cumulant conclusion.
Verification: The fourth moment has exactly three pairings, every test function occurs once in each pairing, and connected cumulants above order two vanish.
Repair if Wick pairing is invoked without Gaussianity or smearing: Return to Gaussian Vectors, Processes, Random Distributions, and Wick Structure.

Let a centered Gaussian random distribution have covariance form $C$ and source functional $Z_E[J]=\exp\!\bigl(\tfrac12 C(J,J)\bigr)$. Put $J=\sum_{i=1}^4t_if_i$. Differentiate with respect to the $t_i$ to recover the two- and four-point functions. Identify every assumption used and state what changes for a non-Gaussian law.

Check

The covariance form must be positive semidefinite, and each $\Phi(f_i)$ must be a defined smeared random variable. Since $W_E[J]=\tfrac12C(J,J)$, its connected derivatives above order two vanish. The derivatives of $Z_E$ give

$$\begin{aligned} \mathbb E[\Phi(f_1)\Phi(f_2)] &=C(f_1,f_2),\\ \mathbb E\!\left[\prod_{i=1}^4\Phi(f_i)\right] &=C(f_1,f_2)C(f_3,f_4) +C(f_1,f_3)C(f_2,f_4)\\ &\quad+C(f_1,f_4)C(f_2,f_3). \end{aligned}$$

Odd centered Gaussian moments vanish. For a non-Gaussian law, higher connected cumulants need not vanish and Wick pairing is not available.

4. Move from trajectories to laws and correlations

Review modes: Representation change and explanation.
Tested capability: Translate an Itô stochastic equation into backward observable evolution, forward law evolution, stationary covariance, and spectral data.
Needed pages: Stochastic Processes and Correlation Functions, Brownian Motion, Stochastic Calculus, Langevin Equations, and Fokker–Planck Dynamics, and Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error.
Expected response: One connected chain of equations plus a sentence distinguishing the domains of the backward and forward operators.
Verification: The factor $\tfrac12$ in the general diffusion generator leaves coefficient $D$; the stationary variance is $D/\lambda$; the spectral density integrates to the equal-time variance.
Repair if backward and forward evolution are confused: Revisit Brownian Motion, Stochastic Calculus, Langevin Equations, and Fokker–Planck Dynamics.

For $\lambda,D>0$, start from

$$\mathrm dX_s=-\lambda X_s\,\mathrm ds+\sqrt{2D}\,\mathrm dW_s.$$

Write the backward generator, forward density equation and probability current, stationary law, stationary covariance, spectral density, and the one-step coefficient after sampling every $\Delta>0$.

Check

The backward generator acts on observables and the adjoint acts on laws:

$$Lf=-\lambda xf'+Df'', \qquad \partial_s p =\lambda\partial_x(xp)+D\partial_x^2p =-\partial_xJ, \qquad J=-\lambda xp-D\partial_xp.$$

With decaying boundary behavior, the zero-current stationary Gaussian has variance $v=D/\lambda$. Its covariance and spectral density are

$$C(s)=v e^{-\lambda|s|}, \qquad \widetilde C(\omega) =\frac{2v\lambda}{\omega^2+\lambda^2}, \qquad \int_{\mathbb R}\frac{\mathrm d\omega}{2\pi}\, \widetilde C(\omega)=v.$$

Sampling at spacing $\Delta$ gives $\alpha=e^{-\lambda\Delta}\in(0,1)$.

5. Transfer the error analysis to two time scales

Review mode: Transfer to a fresh example.
Tested capability: Compute observable-specific autocorrelation and uncertainty when more than one relaxation scale contributes.
Needed pages: Gaussian Vectors, Processes, Random Distributions, and Wick Structure, Stochastic Processes and Correlation Functions, Probabilistic Convergence, Laws of Large Numbers, and Central Limit Theorems, and Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error.
Expected response: The law, covariance, normalized autocorrelation, integrated autocorrelation time, and variance-equivalent sample size of one named observable.
Verification: $\rho_Y(0)=1$; equal relaxation rates reduce to the one-mode formula; changing the observable weights generally changes $\tau_{\mathrm{int}}$.
Repair if effective sample size is treated as chain-wide: Return to Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error.

Let $X^{(1)}$ and $X^{(2)}$ be independent stationary OU modes with variances $v_i$, sampled at spacing $\Delta$, and let $\alpha_i=e^{-\lambda_i\Delta}$. For $Y_n=aX^{(1)}_{n\Delta}+bX^{(2)}_{n\Delta}$, compute the requested quantities and state the conditions under which the covariance sum is meaningful.

Check

For integer lag $k\geq0$, write $V=a^2v_1+b^2v_2>0$ and $C_Y(k)=\operatorname{Cov}(Y_0,Y_k)$. Then

$$Y_n\sim\mathcal N(0,V), \qquad C_Y(k)=a^2v_1\alpha_1^k+b^2v_2\alpha_2^k,$$

and therefore

$$\rho_Y(k)=\frac{C_Y(k)}{V}, \qquad \tau_{\mathrm{int},Y} =\frac12+ \frac{ a^2v_1\alpha_1/(1-\alpha_1) +b^2v_2\alpha_2/(1-\alpha_2) }{V}.$$

When the covariance sum converges and $2\tau_{\mathrm{int},Y}>0$, $N_{\mathrm{eff},Y}=N/(2\tau_{\mathrm{int},Y})$. If $\alpha_1=\alpha_2=\alpha$, this reduces to $(1+\alpha)/(2(1-\alpha))$. The dependence on $a$ and $b$ shows why the summary belongs to the observable, not to the chain alone.

6. Diagnose an overclaimed simulation report

Review mode: Failure diagnosis.
Tested capability: Locate missing hypotheses in claims about Euclidean measures, reversibility, limit theorems, and correlated-sample uncertainty.
Needed pages: Probability Spaces, Random Variables, and Conditional Expectation, Probabilistic Convergence, Laws of Large Numbers, and Central Limit Theorems, and Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error.
Expected response: A numbered list pairing each unsupported claim with its missing hypothesis or a counterexample and a corrected claim.
Verification: The response identifies a measure-construction gap, separates invariance from reversibility, refuses to infer a CLT from stationarity alone, and names the observable attached to $N_{\mathrm{eff}}$.
Repair if the first error is missed: Revisit Probability Spaces, Random Variables, and Conditional Expectation.
Repair if the remaining errors are missed: Revisit Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error.

A report states: “Because $e^{-S_E}$ is nonnegative, the continuum path integral is a probability density. Its Gaussian invariant law makes the sampling process reversible and its semigroup self-adjoint. Stationarity then guarantees a central limit theorem, so the chain has a single effective sample size.” Diagnose every unsupported step.

Check

Nonnegativity does not supply a measurable space, reference or constructed measure, finite normalization, or continuum limit. An invariant Gaussian law does not imply detailed balance: a stationary process may carry a nonzero divergence-free probability current and have a non-self-adjoint semigroup. Stationarity alone does not imply a CLT; an appropriate dependence or operator hypothesis and a finite asymptotic variance are still needed. Finally, autocorrelation time and effective sample size depend on the named observable and on the convention and estimator used.

7. Reconstruct the chapter-scale route

Review mode: Synthesis.
Tested capability: Choose the shortest valid route from a QFT need, distinguish hard dependencies from optional representations, and stop at the chapter boundary.
Needed pages: Probability Spaces, Limit Theorems, Generating Functionals, Gaussian and Wick Structure, Stochastic Processes, Brownian and Langevin Dynamics, and Markov Semigroups and Correlated-Sample Error.
Expected response: A typed dependency list followed by a short explanation of the mathematical transitions and stopping boundary.
Verification: Probability Spaces is the common foundation; Generating Functionals precedes Gaussian/Wick; Stochastic Processes and Limit Theorems both feed the Markov route; Brownian/Langevin is optional for an abstract Markov treatment; reversibility is never assumed.
Repair if the dependency types are unclear: Return to the exact page guide.

A regulated free Euclidean field has a genuine centered Gaussian measure. A stationary, possibly nonreversible Markov chain samples field configurations, and the target is an uncertainty estimate for a two-point observable. Give the minimal route through this chapter. Mark hard prerequisites, identify the optional continuous-time SDE branch, and state which further work belongs to Foundations, Thermal and Nonequilibrium QFT, Lattice and Hamiltonian QFT, or a rigorous mathematical treatment.

Check

The measure-theoretic foundation leads to two branches. The generating branch runs from Probability Spaces to Generating Functionals and then to Gaussian/Wick structure; it packages moments and connected correlations but does not construct a formal measure. The dynamical branch runs from Probability Spaces to Stochastic Processes. Markov uncertainty additionally requires Limit Theorems. Brownian/Langevin dynamics is needed only when an SDE representation is part of the problem, not for abstract Markov evolution.

The two branches are used together in the stated application rather than being joined by an invented hard edge. The estimator must name its observable, CLT assumptions, autocorrelation convention, and finite-run estimator. Developed correlator physics belongs to Foundations, stochastic field interpretation to Thermal and Nonequilibrium QFT, production sampling diagnostics to Lattice and Hamiltonian QFT, and theorem-first continuum measure construction to the rigorous treatment.

For broader foundational uncertainty, use the item-level diagnostic or the bounded probability repair. Neither route certifies readiness for random-distribution construction, stochastic calculus, or dependent-sample asymptotics.

Continue from here

• For positive Euclidean measures and their relation to QFT correlators, continue to Euclidean Correlators and Schwinger Functions and Gaussian Fields and Sources.
• For full versus connected correlations, continue to Connected Correlators and Cumulants.
• For stochastic field evolution and its physical interpretation, continue to Langevin Field Equations and Noise and Fokker–Planck Evolution and Stationary Measures.
• For estimators and correlated-output diagnostics in lattice work, continue to Estimators, Covariance, and Resampling and Autocorrelation Times and Effective Sample Size.
• For a constructed infinite-dimensional Euclidean Gaussian measure, continue to Gaussian Euclidean Fields as Measures.

References

• Rick Durrett, Probability: Theory and Examples, fifth edition, Cambridge University Press, 2019. Open author-hosted PDF. §§ 1.1–1.3 and 1.6, printed pp. 1–18 and 28–34, cover probability spaces, random variables, laws, and expectation; § 2.1, pp. 43–49, covers independence; § 4.1, pp. 205–215, covers conditional expectation; §§ 2.2–2.5, pp. 56–87, cover probabilistic convergence and laws of large numbers; §§ 3.2, 3.4.1–3.4.2, and 3.10, pp. 116–124, 143–152, and 199–201, cover weak convergence and central limit theorems; and §§ 3.3.1–3.3.3 and 3.3.5, pp. 125–136 and 140–143, cover characteristic functions and their moment limits. The linked file is an author-hosted copy of a copyrighted book.
• Julien Fageot, Arash Amini, and Michael Unser, “On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling”, arXiv:1401.6850v2, 2014. §§ 2.1–2.2, printed pp. 4–5, cover generalized stochastic processes, characteristic functionals, and the Bochner–Minlos construction. Open preprint.
• Péter Kevei, Stochastic Processes, notes dated May 21, 2026. Open author-hosted PDF. §§ 4.1–4.2, printed pp. 41–46, develop arbitrary products, projective consistency, and the Kolmogorov consistency theorem; the discussion beginning on printed p. 49 separates construction of finite-dimensional laws from regular sample paths.
• Peter McCullagh, Tensor Methods in Statistics, Chapman and Hall, 1987; Dover edition, 2018. Official author-hosted chapter directory. Chapter 2, §§ 2.2–2.7, printed pp. 29–54, cover moment–cumulant partition relations and Gaussian cumulants; § 3.9.1, printed pp. 85–86, covers the Wick–Isserlis pairing formula. The linked files are first-edition chapter PDFs; the source page identifies the 2018 Dover edition.
• Grigorios A. Pavliotis, Stochastic Processes and Applications, Springer, 2014; linked author manuscript dated November 11, 2015. Open author-manuscript PDF. §§ 1.1–1.3, printed pp. 1–13, cover processes, stationarity, covariance, and Brownian motion; §§ 2.2–2.4, pp. 30–39, cover transition kernels, Markov semigroups, generators, forward evolution, and invariant measures; §§ 3.1–3.5, pp. 49–66, cover SDEs, Itô and Stratonovich conventions, existence, Itô’s formula, Fokker–Planck evolution, and the Ornstein– Uhlenbeck process; §§ 4.1 and 4.6, pp. 77–80 and 104–105, cover currents, boundary behavior, and reversibility. The linked file is an author manuscript of a copyrighted book.
• Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, MIT Press, 2006. Official chapter directory. § 2.2, printed pp. 13–14, covers Gaussian processes, mean and covariance, and consistent finite-dimensional laws; § 4.1.1, printed p. 81, distinguishes mean-square from sample-path continuity; § 4.2.1, printed p. 82, covers the spectral representation. The directory provides official open chapter PDFs of a copyrighted book.
• Gareth O. Roberts and Jeffrey S. Rosenthal, “General State Space Markov Chains and MCMC Algorithms,” Probability Surveys 1 (2004), 20–71; linked author version corrected through 2023. Open author-version PDF. Printed pp. 4–5 cover detailed balance; § 3.2, pp. 14–18, separates invariance, irreducibility, aperiodicity, convergence, and laws of large numbers; § 5, pp. 42–47, covers Markov-chain CLT conditions, covariance-sum variance, and Poisson-equation methods.
• Uwe C. Täuber, “Field Theory Approaches to Nonequilibrium Dynamics”, arXiv:cond-mat/0511743v2, 2006. § 1.1, especially equations (20)–(26), printed pp. 6–8, gives the QFT-facing linear Langevin mode, thermal-noise normalization, and Gaussian time- and frequency-domain correlations. Open preprint.
• Ulli Wolff, “Monte Carlo Errors with Less Errors”, Computer Physics Communications 156 (2004), 143–153; arXiv:hep-lat/0306017v4, 2006 revision. § 2, printed pp. 4–6, gives the covariance-sum uncertainty, the convention $\tau_{\mathrm{int}}=\tfrac12+\sum_{k\geq1}\rho(k)$, and $N/(2\tau_{\mathrm{int}})$; §§ 3.1–3.3, printed pp. 7–13, discuss finite-window bias, noise, and finite-run cautions. Open preprint.