08.14.01 · stat-mech / statistical-field-theory

Brownian motion, the Wiener measure, and the path integral

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Anchor (Master): Itzykson-Drouffe Vol. 1 §1.2; Glimm-Jaffe Ch. 3; Simon — Functional Integration §3

Intuition Beginner

Imagine a speck of pollen floating on water. Every instant, water molecules bump it from random directions, and it jitters in a jagged, unpredictable line. If you mark its position once a second and connect the dots, you get a zigzag path that never settles down. This is Brownian motion: motion built entirely out of tiny random kicks.

Now ask a different question. The speck starts here, and a moment later you find it over there. How likely was that? There is no single road it took. It could have wandered any of countless jagged routes. The honest answer adds up the chances of every possible route at once.

That habit of thinking — a particle does not pick one path, it explores all of them, and you sum over the whole bundle — is the seed of the path integral. The same bundle of random paths that describes a jittering speck also describes a free quantum particle, once you make one substitution about how time enters the weights.

Visual Beginner

The left panel shows one Brownian path: a single random walk refined until the steps are too small to see, leaving a continuous but infinitely wiggly curve. The right panel overlays many such paths sharing the same start and end points. The path integral is the weighted sum over this entire fan of routes.

Darker regions in the fan are where many paths overlap; those are the routes that contribute most to the total weight.

Worked example Beginner

Take a walker on a number line who starts at $0$ . Each step they flip a fair coin: heads moves $+ 1$ , tails moves $- 1$ . After $4$ steps, where might they be?

Count the routes. There is $1$ way to reach $+ 4$ (four heads) and $1$ way to reach $- 4$ . There are $4$ ways to reach $+ 2$ and $4$ ways to reach $- 2$ . There are $6$ ways to land back at $0$ . The totals are $1, 4, 6, 4, 1$ , adding to $16 = 2^{4}$ , every possible route.

So the chance of returning to $0$ is $6/16 = 0.375$ , while reaching the far edge $+ 4$ has chance $1/16 = 0.0625$ . The walker most often ends near the middle.

What this tells us: the probability of an endpoint is found by counting all paths that reach it, weighted equally. As steps shrink and multiply, this counting becomes the continuous sum over paths.

Check your understanding Beginner

Formal definition Intermediate+

A simple random walk in one dimension takes $N$ independent steps $ξ_{1}, \dots, ξ_{N}$ , each $\pm a$ with equal probability, after which the position is $S_{N} = ξ_{1} + \dots + ξ_{N}$ . With time step $τ$ , set $t = N τ$ and rescale so that $a^{2} / τ \to 2 D$ as $a, τ \to 0$ . The central limit theorem then makes $S_{N}$ converge in distribution to a Gaussian, and the probability density of the endpoint solves the diffusion equation

\frac{\partial P}{\partial t} = D \frac{\partial ^{2} P}{\partial x ^{2}}, P (x, t ∣ x_{0}) = \frac{1}{4 π D t} e^{- (x - x_{0})^{2} /4 D t} .

This continuum limit is the random walk turned into a process, following Itzykson-Drouffe ^{[Itzykson-Drouffe]}. The limiting object is Brownian motion $(B_{t})_{t \geq 0}$ : a real-valued process with $B_{0} = 0$ , independent increments, $B_{t} - B_{s} \sim N (0, t - s)$ for $s < t$ , and almost surely continuous sample paths.

The law of this process is a probability measure on the space $C ([0, t])$ of continuous paths. This is the Wiener measure $d W$ ^{[Wiener 1923]}. It is the unique Gaussian measure on path space with mean zero and covariance $E [B_{s} B_{t}] = min (s, t)$ . Heuristically one writes the unnormalised Wiener density as $exp (- \frac{1}{2} \int_{0}^{t} \overset{x}{˙} (s)^{2} d s)$ times a formal flat measure $D x$ , but the dotted derivative does not exist pointwise; the measure itself, defined through its finite-dimensional Gaussian marginals, is the rigorous object.

The free Euclidean scalar field of 08.06.01 is the multi-time generalisation of this construction: a Gaussian measure whose covariance is the lattice or continuum propagator. The two-point function of that field is exactly the random-walk generating function summing weighted paths between two points, which is the bridge to the field-theoretic 08.07.01.

Key theorem with proof Intermediate+

Theorem (diffusion limit of the random walk). Let $S_{N}$ be the symmetric $\pm a$ random walk after $N$ steps, with $t = N τ$ held fixed and $a^{2} / τ \to 2 D$ . Then the rescaled endpoint $S_{N}$ converges in distribution to a Gaussian of variance $2 D t$ , whose density solves the diffusion equation above.

Proof. The single step has mean $0$ and variance $a^{2}$ , so $S_{N}$ has mean $0$ and variance $N a^{2} = (t / τ) a^{2} = t (a^{2} / τ) \to 2 D t$ . The characteristic function of one step is $E [e^{ik ξ}] = cos (k a) = 1 - \frac{1}{2} k^{2} a^{2} + O (a^{4})$ . By independence,

E [e^{ik S_{N}}] = (cos k a)^{N} = (1 - \frac{1}{2} k^{2} a^{2} + O (a^{4}))^{N} .

Write $N = t / τ$ and $a^{2} = (a^{2} / τ) τ \to 2 D τ$ as the spacing shrinks. Then $\frac{1}{2} k^{2} a^{2} = k^{2} D τ + o (τ)$ , and

(1 - k^{2} D τ + o (τ))^{t / τ} ⟶ e^{- k^{2} D t} .

This is the characteristic function of $N (0, 2 D t)$ , whose density is the heat kernel $P (x, t ∣ 0) = (4 π D t)^{- 1/2} e^{- x^{2} /4 D t}$ . Direct differentiation confirms $\partial_{t} P = D \partial_{x}^{2} P$ . The endpoint law of the limiting process is therefore Gaussian with the stated covariance, and a Kolmogorov consistency check on these marginals produces the Wiener measure on $C ([0, t])$ . $□$

Bridge. This computation builds toward the Master-tier Feynman-Kac formula: the heat kernel $e^{- t H_{0}}$ for the free Hamiltonian $H_{0} = - D \partial_{x}^{2}$ is precisely the propagator just derived, and adding a potential $V$ inserts the weight $e^{- \int V d s}$ along each path. The foundational reason the formal functional integral 08.07.01 computes anything is that this is exactly the Wiener measure dressed by a potential; the central insight is that the Gaussian path measure already carries the kinetic term $\frac{1}{2} \int \overset{x}{˙}^{2}$ , so the integral over paths is a genuine probability average rather than a divergent symbol. This pattern generalises directly: the free field 08.06.01 is the same construction with the random walk replaced by random surfaces, and putting these together gives the random-walk representation of every Euclidean propagator. The bridge is that summing weighted paths and exponentiating a generator are one operation seen from two sides, which appears again in the Wick-rotated quantum correspondence 08.09.01.

Exercises Intermediate+

Exercise 6 (hard, short-answer).

Argue heuristically why the discretised kinetic weight $\prod_{n} exp (- \frac{( x _{n + 1} - x _{n} ) ^{2}}{4 D τ})$ becomes the Wiener measure as $τ \to 0$ , and identify where the flat measure $D x$ and its normalisation come from.

Hint

Each factor is a one-step heat kernel; the product of $N$ of them over $N$ Gaussian integrals is the time-sliced propagator.

Answer

Each factor is the one-step heat kernel $(4 π D τ)^{- 1/2} exp (- (x_{n + 1} - x_{n})^{2} /4 D τ)$ . The product over $N$ slices, integrated against $\prod_{n} d x_{n}$ , reconstructs the full propagator by repeated convolution. The per-slice normalisation $(4 π D τ)^{- 1/2}$ is absorbed into the symbol $D x$ ; in the limit the increments scale as $(x_{n + 1} - x_{n})^{2} \sim 2 D τ$ , so the exponent is finite while the formal $\int \overset{x}{˙}^{2}$ is not, which is why the measure rather than the density is the real object.

Rubric: full credit for time-slicing, the role of the normalisation, and the $τ$ scaling of increments.

Exercise 7 (hard, short-answer).

The quadratic variation of Brownian motion satisfies $\sum_{k} (B_{t_{k + 1}} - B_{t_{k}})^{2} \to t$ as the partition of $[0, t]$ refines (with $D = 1/2$ ). Explain why this forces ordinary calculus to fail and signals the need for Itô's correction.

Hint

For a smooth path the sum of squared increments tends to zero; here it tends to $t$ .

Answer

For a differentiable path the squared increments scale as $(Δ t)^{2}$ and their sum vanishes. For Brownian motion each squared increment averages $Δ t$ , so the sum converges to $t > 0$ . The non-vanishing quadratic variation means $(d B)^{2}$ behaves like $d t$ , so a Taylor expansion of $f (B_{t})$ must keep the second-order term, producing the extra $\frac{1}{2} f^{''} d t$ of Itô calculus.

Rubric: full credit for contrasting smooth and Brownian scaling and naming the Itô second-order term.

Advanced results Master

Fix the free Hamiltonian $H_{0} = - \frac{1}{2} Δ$ on $L^{2} (R^{d})$ and a potential $V$ bounded below. The semigroup $e^{- t H_{0}}$ has the heat kernel as integral kernel, and the Wiener measure $d W_{x}$ is the probability law on $C ([0, t]; R^{d})$ of Brownian paths started at $x$ . The Feynman-Kac formula states

(e^{- t H} f) (x) = \int_{C ([0, t])} e^{- \int_{0}^{t} V (ω (s)) d s} f (ω (t)) d W_{x} (ω), H = - \frac{1}{2} Δ + V,

which one writes informally as $\int e^{- \int_{0}^{t} (\frac{1}{2} \overset{x}{˙}^{2} + V (x)) d s} D x$ with the kinetic term carried by the Wiener measure rather than the integrand ^{[Kac 1949]}. The kernel of $e^{- t H}$ between fixed endpoints is the same integral over the bridge measure of paths pinned at $x$ and $y$ .

This is the rigorous prequel of the formal path integral 08.07.01. Where the physics derivation writes a divergent product of free measures and a complex phase $e^{i S}$ , the Euclidean version replaces the phase by a positive Boltzmann-like weight and the flat measure by the genuine Wiener probability measure. The Wick rotation $t \mapsto - i t$ that converts $e^{- t H}$ into the unitary $e^{- i t H}$ is exactly the analytic continuation recorded in 08.09.01; the Euclidean (imaginary-time) integral is the one that converges, and the real-time Feynman integral ^{[Feynman 1948]} is its boundary value.

The free Gaussian field 08.06.01 is the field-theoretic Feynman-Kac: its measure is Gaussian with covariance $(- Δ + m^{2})^{- 1}$ , and the random-walk representation expresses that propagator as $\int_{0}^{\infty} d T e^{- m^{2} T} (heat kernel at time T)$ — a sum over random paths of every length. Interactions multiply the Gaussian field measure by $e^{- \int V (ϕ)}$ , the field analogue of the Feynman-Kac potential factor, and perturbation theory expands that exponential into Wick contractions.

Synthesis. The Feynman-Kac formula is the foundational reason the path integral 08.07.01 is more than formal: the central insight is that the divergent kinetic exponent and the divergent flat measure are not two separate objects to be tamed but a single convergent Wiener probability measure, so summing over paths is a genuine average and the foundational reason every Euclidean propagator admits a random-walk representation. This is exactly the construction that the free field 08.06.01 generalises from random walks to random surfaces, and it is dual to the operator semigroup picture, since $e^{- t H}$ on Hilbert space and the path-space integral compute the same kernel. Putting these together, the Trotter product formula is the bridge: time-slicing $e^{- t H} = lim (e^{- t H_{0} / N} e^{- t V / N})^{N}$ interleaves free diffusion with the potential weight, and in the limit the free steps assemble into the Wiener measure while the potential steps assemble into $e^{- \int V}$ . The same identification appears again in the Wick-rotated correspondence 08.09.01, where imaginary-time statistical mechanics and real-time quantum dynamics are two readings of one analytic object, and it generalises to the field-theoretic functional integral whose convergence rests entirely on this Euclidean, probabilistic footing.

Full proof set Master

Proposition (Feynman-Kac via the Trotter product formula). Let $H_{0} = - \frac{1}{2} Δ$ generate the Brownian semigroup and let $V$ be bounded and continuous. Then for $f \in L^{2} (R^{d})$ ,

(e^{- t H} f) (x) = \int_{C ([0, t])} e^{- \int_{0}^{t} V (ω (s)) d s} f (ω (t)) d W_{x} (ω) .

Proof. By the Trotter product formula for the bounded perturbation $V$ of the self-adjoint generator $H_{0}$ ,

e^{- t H} = N \to \infty lim (e^{- \frac{t}{N} H_{0}} e^{- \frac{t}{N} V})^{N},

with convergence in the strong operator topology. Each free factor $e^{- \frac{t}{N} H_{0}}$ acts by convolution against the heat kernel $p_{t / N} (x, y) = (2 π t / N)^{- d /2} exp (- ∣ x - y ∣^{2} N /2 t)$ , and each potential factor is multiplication by $e^{- \frac{t}{N} V}$ . Writing the $N$ -fold product out with intermediate points $x = x_{0}, x_{1}, \dots, x_{N}$ gives

(e^{- t H} f) (x) = N \to \infty lim \int k = 0 \prod N - 1 p_{t / N} (x_{k}, x_{k + 1}) e^{- \frac{t}{N} V (x_{k})} f (x_{N}) k = 1 \prod N d x_{k} .

The product of heat kernels $\prod_{k} p_{t / N} (x_{k}, x_{k + 1}) \prod_{k} d x_{k}$ is, by the Kolmogorov consistency theorem, the finite-dimensional marginal of the Wiener measure $d W_{x}$ on the time grid ${k t / N}$ . As $N \to \infty$ the grid refines; because Brownian paths are almost surely continuous and $V$ is continuous and bounded, the Riemann sum $\frac{t}{N} \sum_{k} V (x_{k})$ converges $W_{x}$ -almost surely to $\int_{0}^{t} V (ω (s)) d s$ , and the integrand is uniformly bounded by $e^{t s u p ∣ V ∣} ∣ f ∣$ . Dominated convergence over path space exchanges the limit with the integral, giving the stated identity. $□$

Proposition (the diffusion semigroup is positivity preserving and contractive). The free Brownian semigroup $e^{- t H_{0}}$ satisfies $e^{- t H_{0}} f \geq 0$ whenever $f \geq 0$ , and $∥ e^{- t H_{0}} f ∥_{L^{\infty}} \leq ∥ f ∥_{L^{\infty}}$ .

Proof. The kernel $p_{t} (x, y) = (2 π t)^{- d /2} exp (- ∣ x - y ∣^{2} /2 t)$ is strictly positive, so $(e^{- t H_{0}} f) (x) = \int p_{t} (x, y) f (y) d y \geq 0$ for $f \geq 0$ . Since $\int p_{t} (x, y) d y = 1$ for every $x$ , the kernel is a probability density, and $∣ (e^{- t H_{0}} f) (x) ∣ \leq \int p_{t} (x, y) ∣ f (y) ∣ d y \leq ∥ f ∥_{L^{\infty}}$ . Positivity is the analytic shadow of the fact that $d W_{x}$ is a probability measure: the semigroup transports mass without creating negative weight, which is why the Euclidean integral, unlike the oscillatory real-time one, is a bona fide average. $□$

Connections Master

This unit is the rigorous foundation that 08.07.01 is derived from, not merely related to: the formal functional integral with weight $e^{- S}$ is the Feynman-Kac integral with the kinetic term reabsorbed into the Wiener measure, so every manipulation of the path integral in 08.07.01 inherits its meaning from the construction here.
The free Gaussian field 08.06.01 is the multi-dimensional generalisation of the Wiener measure: its covariance is the resolvent $(- Δ + m^{2})^{- 1}$ , and the random-walk representation of that propagator as a Laplace transform of heat kernels is the field-theoretic version of the path sum derived in this unit.
The Wick rotation 08.09.01 is the analytic continuation that turns the convergent Euclidean Feynman-Kac integral $e^{- t H}$ into the oscillatory real-time amplitude $e^{- i t H}$ ; the Euclidean integral built here is the well-defined object from which the Minkowski path integral is recovered as a boundary value.
Forward to 08.14.02: the stochastic-calculus reading of the same paths — Itô's formula, the non-vanishing quadratic variation flagged in Exercise 7, and the generator $- \frac{1}{2} Δ + V$ as the drift-diffusion operator — develops the differential side of the integral constructed here.
Forward to 08.14.03: the field-theoretic functional integral and its perturbative (Feynman-diagram) expansion build on this unit's identification of the Gaussian path measure with a propagator, replacing single random walks by interacting random surfaces.

Historical & philosophical context Master

Einstein's 1905 account of Brownian motion gave the diffusion law $⟨ x^{2} ⟩ = 2 D t$ and tied $D$ to molecular fluctuations, but it described only the endpoint distribution, not the paths. Norbert Wiener supplied the missing object in 1923, constructing a countably additive probability measure on the space of continuous functions and proving that its sample paths are almost surely continuous and almost surely nowhere differentiable ^{[Wiener 1923]}. This was the first rigorous integration theory on an infinite-dimensional path space, predating Kolmogorov's axiomatisation of probability.

Richard Feynman's 1948 reformulation of quantum mechanics expressed the propagator as a sum over histories weighted by $e^{i S /ℏ}$ , an integral whose oscillatory measure resisted rigorous definition ^{[Feynman 1948]}. Mark Kac, who had heard Feynman present the idea, observed that replacing real time by imaginary time converts the oscillatory weight into the positive Wiener weight, and in 1949 proved the formula now bearing both names: the imaginary-time propagator $e^{- t H}$ is a genuine Wiener integral with the potential entering as $e^{- \int V}$ ^{[Kac 1949]}. Glimm and Jaffe later made this the technical backbone of constructive quantum field theory, where Euclidean functional integrals are the rigorous definitions and Minkowski amplitudes are their analytic continuations.

Bibliography Master

@article{Wiener1923DifferentialSpace,
  author  = {Wiener, Norbert},
  title   = {Differential Space},
  journal = {Journal of Mathematics and Physics},
  volume  = {2},
  year    = {1923},
  pages   = {131--174}
}

@article{Kac1949WienerFunctionals,
  author  = {Kac, Mark},
  title   = {On Distributions of Certain Wiener Functionals},
  journal = {Transactions of the American Mathematical Society},
  volume  = {65},
  year    = {1949},
  pages   = {1--13}
}

@article{Feynman1948SpaceTime,
  author  = {Feynman, Richard P.},
  title   = {Space-Time Approach to Non-Relativistic Quantum Mechanics},
  journal = {Reviews of Modern Physics},
  volume  = {20},
  year    = {1948},
  pages   = {367--387}
}

@book{ItzyksonDrouffe1989,
  author    = {Itzykson, Claude and Drouffe, Jean-Michel},
  title     = {Statistical Field Theory, Volume 1},
  publisher = {Cambridge University Press},
  year      = {1989}
}

@book{Simon1979FunctionalIntegration,
  author    = {Simon, Barry},
  title     = {Functional Integration and Quantum Physics},
  publisher = {Academic Press},
  year      = {1979}
}

Prerequisites

08.06.01
08.07.01

Tier anchors

beginner: Einstein 1905 random-walk picture; Berg — Random Walks in Biology Ch. 1
intermediate: Itzykson-Drouffe — Statistical Field Theory Vol. 1 §1.1-1.2
master: Itzykson-Drouffe Vol. 1 §1.2; Glimm-Jaffe Ch. 3; Simon — Functional Integration §3

References

raw/pdfs/kardar-1-3-laws-entropy-potentials.pdf · Kardar statistical physics lecture notes (diffusion and Gaussian conventions)
quantum-well · statistical physics index
Itzykson and Drouffe — Statistical Field Theory Vol. 1 · Ch. 1 §1.1-1.2, random walks and the continuum limit
Wiener — Differential Space · J. Math. Phys. (MIT) 2 (1923) 131-174
Kac — On Distributions of Certain Wiener Functionals · Trans. Amer. Math. Soc. 65 (1949) 1-13
Feynman — Space-Time Approach to Non-Relativistic Quantum Mechanics · Rev. Mod. Phys. 20 (1948) 367-387

Estimated time

beginner: 18m
intermediate: 45m
master: 85m