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

A stochastic differential equation evolves random trajectories, but the same dynamics also determines a nonrandom evolution equation for their probability law. For the Itô equation

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

the bridge is Itô’s formula. If $\mu_t$ is the law of $X_t$, then, for a suitable test function $f$,

$$\frac{\mathrm d}{\mathrm dt} \int f(x)\,\mu_t(\mathrm dx) =\int (L_t f)(x)\,\mu_t(\mathrm dx), \qquad L_t=b^i\partial_i+\frac12a^{ij}\partial_i\partial_j.$$

This weak equation is the general trajectory-to-law statement used here. Only when $\mu_t$ has a sufficiently regular density $p(t,x)$ and the integrations by parts are justified does it become the Fokker–Planck equation

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

The distinction matters: an SDE can define a perfectly good probability law even when no smooth density exists. This page develops the bridge carefully, including stochastic-integral conventions, boundary flux, stationary laws, and one regulated field mode for which every step can be checked explicitly.

Required background. Stochastic Processes and Correlation Functions, especially its distinctions among finite-dimensional laws, sample paths, stationarity, and correlation functions.

Itô processes and the trajectory-to-law method

Work on a filtered probability space $(\Omega,\mathcal F,(\mathcal F_t)_{t\geq0},\mathbb P)$ satisfying the usual conditions. Unless stated otherwise, $W_t=(W_t^1,\ldots,W_t^m)$ is standard $m$-dimensional Brownian motion, the state $X_t$ lies in $\mathbb R^d$, and repeated state and noise indices are summed. The stochastic integral is Itô’s integral. The finite-dimensional setting keeps the analytic assumptions visible; functional evolution for fields is only indicated at the end.

There are three logically separate steps:

1. Brownian motion and an integration convention define the random trajectory equation.
2. Itô’s formula determines how observables of that trajectory evolve.
3. Taking expectations and, when allowed, integrating by parts transfers the evolution to the probability law.

Pavliotis develops this path-to-density route in Pavliotis 2014, Chapters 3–4, especially pp. 49–66 and 77–80, PDF.

Brownian motion and quadratic variation

A one-dimensional standard Brownian motion $W_t$ starts at zero, has continuous paths, and has independent Gaussian increments

$$W_t-W_s\sim\mathcal N(0,t-s), \qquad 0\leq s<t.$$

For independent components, $\mathbb E[(W_t^a-W_s^a)(W_t^b-W_s^b)]=\delta^{ab}(t-s)$. Its paths are continuous, but they do not supply an ordinary time derivative. The useful replacement is quadratic variation. For any deterministic sequence of partitions $\Pi_n$ of $[0,t]$ whose mesh tends to zero,

$$\sum_k (W_{t_{k+1}}^a-W_{t_k}^a) (W_{t_{k+1}}^b-W_{t_k}^b) \longrightarrow \delta^{ab}t$$

in $L^2$ and hence in probability. The limit can be checked directly. For equal components, the sum has mean $t$ and variance $2\sum_k(\Delta t_k)^2\leq2t\lVert\Pi_n\rVert$; for distinct components, it has mean zero and variance at most $t\lVert\Pi_n\rVert$. The mnemonic

$$\mathrm dW_t^a\,\mathrm dW_t^b=\delta^{ab}\,\mathrm dt$$

records this limiting rule; it is not ordinary algebra with infinitesimals. Pavliotis introduces Brownian motion and its basic path properties in Pavliotis 2014, § 1.3, pp. 10–13, PDF and records the quadratic variation of an Itô integral in Pavliotis 2014, § 3.2, p. 54, PDF.

The formal notation $\xi^a(t)=\mathrm dW_t^a/\mathrm dt$ is Gaussian white noise. It has the distributional covariance

$$\mathbb E[\xi^a(t)\xi^b(s)] =\delta^{ab}\delta(t-s),$$

but $\xi(t)$ is not an ordinary random function evaluated pointwise. A Langevin equation written with $\xi$ must therefore be interpreted through an integral or SDE convention.

The Itô integral and a well-posed SDE

For an adapted step process $H_t$ and a partition $0=t_0<\cdots<t_n=T$, the Itô sum uses the left endpoint:

$$\int_0^T H_t\,\mathrm dW_t =\sum_{k=0}^{n-1} H_{t_k}(W_{t_{k+1}}-W_{t_k}).$$

For predictable $H$ with $\mathbb E\int_0^T\lVert H_t\rVert^2\mathrm dt<\infty$, completion in $L^2$ defines the integral and gives the Itô isometry

$$\mathbb E\left[ \left|\int_0^T H_t\,\mathrm dW_t\right|^2 \right] =\mathbb E\int_0^T\lVert H_t\rVert^2\,\mathrm dt.$$

Under the same square-integrability conditions its expectation is zero. The left-endpoint choice makes the integrand depend only on information available before the next Brownian increment.

An Itô SDE is shorthand for the integral equation

$$X_t=X_0 +\int_0^t b(s,X_s)\,\mathrm ds +\int_0^t\sigma(s,X_s)\,\mathrm dW_s.$$

A useful sufficient theorem is concrete. If $b$ and $\sigma$ are measurable in time, globally Lipschitz in the state uniformly on each finite time interval, and satisfy a linear-growth bound, then a square-integrable, $\mathcal F_0$-measurable initial condition produces a unique, nonexplosive strong solution when $W$ is Brownian with respect to the stated filtration. These assumptions are sufficient rather than necessary; outside them, well-posedness and nonexplosion require separate arguments and should not be silently inferred merely because an SDE has been written. Pavliotis states the global Lipschitz and linear-growth result in Pavliotis 2014, § 3.3, p. 57, PDF.

Itô’s formula and the local differential operator

Let $f\in C^{1,2}([0,T]\times\mathbb R^d)$ and assume the required integrability. The stochastic chain rule is

$$\begin{aligned} \mathrm df(t,X_t) ={}&\left( \partial_t f +b^i\partial_i f +\frac12a^{ij}\partial_i\partial_j f \right)(t,X_t)\,\mathrm dt\\ &+\partial_i f(t,X_t)\, \sigma^i{}_a(t,X_t)\,\mathrm dW_t^a. \end{aligned}$$

The second-derivative term is precisely the effect of Brownian quadratic variation. For a time-independent observable, define the local differential operator

$$(L_t f)(x) =b^i(t,x)\partial_i f(x) +\frac12a^{ij}(t,x)\partial_i\partial_j f(x).$$

This is the backward operator acting on observables. Its adjoint will act on laws. General Markov kernels, semigroups, domains of generators, and their long-time theory belong to the next page; the differential expression here is introduced only to derive the law equation.

A quick consistency check is $f(x)=x^2$ and $X_t=W_t$:

$$\mathrm d(W_t^2)=2W_t\,\mathrm dW_t+\mathrm dt.$$

Taking expectations gives $\mathbb E[W_t^2]=t$, the correct Brownian variance. The extra $\mathrm dt$ is what the ordinary chain rule would miss.

Itô and Stratonovich are different inputs

The Stratonovich integral is defined by symmetric rather than left-endpoint sums and obeys the classical-looking chain rule. For smooth coefficients,

$$\mathrm dX_t^i =\beta^i(t,X_t)\,\mathrm dt +\sigma^i{}_a(t,X_t)\circ\mathrm dW_t^a$$

describes the same process as the Itô equation with drift

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

Thus identical written drift and diffusion functions under the two symbols do not generally define identical laws. The correction is also not, in general, $\tfrac12\partial_j a^{ij}$: differentiating $a^{ij}$ produces an additional term involving $\partial_j\sigma^j{}_a$. If $\sigma$ is state independent, the correction vanishes and the two conventions agree. Pavliotis compares the conventions and their conversion in Pavliotis 2014, § 3.2, pp. 52–56, PDF.

A Langevin shorthand such as

$$\dot X=b(X)+\sqrt{2D}\,\xi$$

still requires the white-noise covariance normalization. Because the displayed amplitude is constant, Itô and Stratonovich give the same law. With standard white noise, the diffusion coefficient is $a=2D$, so the Fokker–Planck diffusion term is $D\,\partial_x^2p$. This factor of two is a common source of convention errors. If the amplitude instead depends on $X$, the stochastic convention is an additional necessary input.

From trajectory evolution to law evolution

Let $f\in C_c^\infty(\mathbb R^d)$ and let $\mu_t=\operatorname{Law}(X_t)$. Integrating Itô’s formula from $0$ to $t$ and taking expectations removes the martingale term:

$$\int f\,\mathrm d\mu_t -\int f\,\mathrm d\mu_0 =\int_0^t\!\int L_s f(x)\, \mu_s(\mathrm dx)\,\mathrm ds.$$

Equivalently, for almost every $t$,

$$\frac{\mathrm d}{\mathrm dt} \int f\,\mathrm d\mu_t =\int L_t f\,\mathrm d\mu_t.$$

This is a deterministic equation for probability measures, expressed weakly against test functions. It remains meaningful for a deterministic motion whose law is a moving delta measure and for degenerate diffusions supported on a lower-dimensional set—cases in which an ordinary density on $\mathbb R^d$ may not exist.

Now add hypotheses. Suppose $\mu_t(\mathrm dx)=p(t,x)\,\mathrm d^dx$, the coefficients and density have enough regularity, and the products are integrable so that the following integrations by parts are valid. Then

$$\begin{aligned} \int (L_t f)p\,\mathrm d^dx ={}&\int b^i(\partial_i f)p\,\mathrm d^dx\\ &+\frac12\int a^{ij} (\partial_i\partial_j f)p\,\mathrm d^dx\\ ={}&\int f\left[ -\partial_i(b^i p) +\frac12\partial_i\partial_j(a^{ij}p) \right]\mathrm d^dx. \end{aligned}$$

Because this holds for every compactly supported test function,

$$\partial_t p =-\partial_i(b^i p) +\frac12\partial_i\partial_j(a^{ij}p)$$

in the distributional sense, and classically when the displayed derivatives exist. This is the forward Kolmogorov, or Fokker–Planck, equation. It evolves the law; $L_t$ evolves observables. No sample-dependent noise appears after the expectation is taken. Pavliotis derives the classical density equation by adjoint integration by parts under smoothness assumptions in Pavliotis 2014, § 3.4, pp. 60–63, PDF.

Probability current and boundary behavior

Write the density equation as a continuity equation,

$$\partial_t p+\partial_iJ^i=0, \qquad J^i=b^i p-\frac12\partial_j(a^{ij}p).$$

For a domain $D$ with outward unit normal $n$,

$$\frac{\mathrm d}{\mathrm dt}\int_D p\,\mathrm d^dx =-\int_{\partial D}n_iJ^i\,\mathrm dS.$$

Probability is conserved only when the net boundary flux vanishes. On all of $\mathbb R^d$, sufficient decay can remove the flux at infinity. Periodic faces cancel in pairs. A reflecting boundary imposes $n_iJ^i=0$; at the path level, reflection is an additional boundary rule, not a consequence of the unconstrained interior SDE. An absorbing or killed boundary can carry outward flux, so the density remaining inside $D$ is subnormalized unless the absorbed state is included separately. Pavliotis compares absorbing, reflecting, and periodic Fokker–Planck boundary conditions in Pavliotis 2014, § 4.1, pp. 77–80, PDF.

These boundary conditions are part of the stochastic model. The same formal differential expression with reflecting and absorbing boundaries describes different law evolutions.

Stationary densities require more than a formal solution

Now suppose the coefficients are time independent and the domain and boundary rule are fixed. A stationary density $p_*$ must satisfy

$$L^\dagger p_*=0, \qquad p_*\geq0, \qquad \int_Dp_*\,\mathrm d^dx=1,$$

together with the domain’s boundary conditions. Equivalently, $\partial_iJ_*^i=0$. Vanishing current, $J_*=0$, is stronger than stationarity: divergence-free circulating currents can support a stationary nonequilibrium law. In an overdamped diffusion without variables that reverse under time reversal, zero stationary current is the usual reversible, detailed-balance condition. Pavliotis proves the equivalence between reversibility and zero stationary current in this diffusion setting in Pavliotis 2014, § 4.6, Proposition 4.13, pp. 104–105, PDF.

Existence does not imply uniqueness, attraction from every initial law, or ergodicity. Those are additional long-time questions. A particularly simple failed candidate is free Brownian motion on $\mathbb R$: the stationary equation admits a formal constant solution, but no nonzero constant is normalizable on the line. Formal annihilation by $L^\dagger$ is therefore not enough.

Controlled example: one regulated Ornstein–Uhlenbeck mode

Retain one normalized real mode $\phi_{\mathbf q}$ of a scalar field in finite spatial volume and set

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

The linear relaxational Langevin equation is

$$\mathrm d\phi_{\mathbf q}(s) =-\Gamma\kappa_{\mathbf q}\phi_{\mathbf q}(s)\,\mathrm ds +\sqrt{2\Gamma T}\,\mathrm dW_s.$$

This is an Ornstein–Uhlenbeck process. Its noise is additive, so Itô and Stratonovich interpretations coincide. Multiplication by the integrating factor $e^{\Gamma\kappa_{\mathbf q}s}$ gives

$$\begin{aligned} \phi_{\mathbf q}(s) ={}&e^{-\Gamma\kappa_{\mathbf q}s}\phi_{\mathbf q}(0)\\ &+\sqrt{2\Gamma T} \int_0^s e^{-\Gamma\kappa_{\mathbf q}(s-u)}\,\mathrm dW_u. \end{aligned}$$

Conditional on $\phi_{\mathbf q}(0)=\phi_0$, the result is Gaussian with

$$\begin{aligned} m_s&=e^{-\Gamma\kappa_{\mathbf q}s}\phi_0,\\ v_s&=\frac{T}{\kappa_{\mathbf q}} \left(1-e^{-2\Gamma\kappa_{\mathbf q}s}\right). \end{aligned}$$

For $s>0$, its transition density is therefore

$$p(\phi,s\mid\phi_0) =\frac{1}{\sqrt{2\pi v_s}} \exp\!\left[-\frac{(\phi-m_s)^2}{2v_s}\right].$$

The trajectory coefficients are $b(\phi)=-\Gamma\kappa_{\mathbf q}\phi$ and $a=2\Gamma T$, so the induced density equation and current are

$$\begin{aligned} \partial_s p &=\Gamma\kappa_{\mathbf q}\partial_\phi(\phi p) +\Gamma T\partial_\phi^2p,\\ J &=-\Gamma\kappa_{\mathbf q}\phi p -\Gamma T\partial_\phi p. \end{aligned}$$

Solving $J_*=0$ and normalizing gives

$$p_*(\phi) =\sqrt{\frac{\kappa_{\mathbf q}}{2\pi T}} \exp\!\left[-\frac{\kappa_{\mathbf q}\phi^2}{2T}\right].$$

Several checks agree:

• Direct differentiation gives $J_*=0$, hence $L^\dagger p_*=0$.

• Multiplying the Fokker–Planck equation by $\phi$ and $\phi^2$ and integrating gives

  $$\begin{aligned} \frac{\mathrm d}{\mathrm ds}\langle\phi\rangle &=-\Gamma\kappa_{\mathbf q}\langle\phi\rangle,\\ \frac{\mathrm d}{\mathrm ds}\langle\phi^2\rangle &=-2\Gamma\kappa_{\mathbf q}\langle\phi^2\rangle +2\Gamma T. \end{aligned}$$

• The stationary variance is $T/\kappa_{\mathbf q}$, and the stationary two-time covariance is

  $$\left\langle \phi_{\mathbf q}(s+\tau)\phi_{\mathbf q}(s) \right\rangle_* =\frac{T}{\kappa_{\mathbf q}} e^{-\Gamma\kappa_{\mathbf q}|\tau|},$$

  matching the correlation calculation on the preceding page.

Täuber derives this relaxational Gaussian field dynamics and its thermal-noise normalization in Täuber 2006, § 1.1, pp. 6–8. Here finite volume and a single real coordinate avoid delta-function normalization and continuum existence questions.

The limits reveal the assumptions. At fixed initial condition, $T\downarrow0$ removes the noise and leaves deterministic relaxation. At fixed $s$, $\kappa_{\mathbf q}\downarrow0$ gives $v_s\to2\Gamma Ts$, the Brownian variance. But the stationary Gaussian then ceases to be normalizable: the zero-restoring-force limit has no stationary probability density on $\mathbb R$.

Stochastic-quantization orientation

The same finite-dimensional algebra suggests the basic stochastic- quantization construction, but its noise scale should not be confused with the physical temperature of the thermal model. Introduce a separate positive scale $\Theta$ and define the dimensionless regulated Euclidean action for this mode by

$$S_E^{(\mathbf q)}(\phi) =\frac{\kappa_{\mathbf q}\phi^2}{2\Theta}.$$

Then the drift can be written

$$-\Gamma\kappa_{\mathbf q}\phi =-\Gamma\Theta\,\partial_\phi S_E^{(\mathbf q)},$$

and fictitious noise amplitude $\sqrt{2\Gamma\Theta}$ gives a zero-current stationary density proportional to $e^{-S_E^{(\mathbf q)}}$. In the preceding thermal example one sets $\Theta=T$. In stochastic quantization, $\Theta$ instead represents the auxiliary noise/action normalization, conventionally related to $\hbar$ and often set to one; it is not a physical temperature. The parameter $s$ is reinterpreted as an auxiliary, fictitious time used to sample a regulated Euclidean measure. It is not the Lorentzian time of the quantum theory, nor an extra Euclidean spacetime coordinate. Damgaard and Hüffel present the fictitious-time Langevin equation, its formal functional Fokker–Planck equation, and the $e^{-S_E}$ stationary candidate in Damgaard and Hüffel 1987, § 3.1, pp. 236–239, PDF.

This one-mode calculation is an orientation, not a continuum theorem. It does not establish existence or uniqueness of an interacting field measure, convergence in fictitious time, gauge fixing, or equivalence to a desired quantum theory. Those issues require regulators, functional analysis, and the field-specific dynamics developed later. For functional Fokker–Planck evolution and physical stationary measures, continue to Fokker–Planck Evolution and Stationary Measures.

Common pitfalls

Differentiating Brownian paths. White noise is a generalized random field, not an ordinary time function. Interpret $\dot W$ through a stochastic integral and state the convention.

Using the ordinary chain rule in Itô form. Quadratic variation supplies the second-derivative term. Omitting it changes both observable evolution and the resulting law equation.

Changing notation without changing the drift. Itô and Stratonovich coefficients are convention dependent when the noise is multiplicative. Convert the drift before comparing two equations.

Assuming every law has a smooth density. The weak measure equation comes first. Degenerate noise, deterministic directions, or singular initial laws can prevent a full-dimensional smooth density.

Solving only the stationary differential equation. A stationary probability law must also be nonnegative, normalizable, and compatible with the boundary conditions. Zero current is stronger than stationarity.

Ignoring the boundary. Reflecting, periodic, and absorbing boundaries produce different flux balances even when the interior operator has the same formula.

Check your understanding

1. Recover Brownian variance from Itô’s formula

Apply Itô’s formula to $f(x)=x^2$ for $X_t=W_t$ and derive $\mathbb E[W_t^2]=t$.

Solution

Since $f'(x)=2x$ and $f''(x)=2$,

$$\mathrm d(W_t^2)=2W_t\,\mathrm dW_t+\mathrm dt.$$

Integrating from $0$ to $t$ and taking expectations removes the Itô integral, provided its square-integrability condition holds. Since $W_0=0$,

$$\mathbb E[W_t^2]=t.$$

The $\mathrm dt$ term encodes Brownian quadratic variation.

2. Convert a multiplicative-noise equation

Convert

$$\mathrm dX_t =\alpha X_t\,\mathrm dt +\beta X_t\circ\mathrm dW_t$$

from Stratonovich to Itô form. What error results from keeping the same drift?

Solution

Here $\sigma(x)=\beta x$, so $\tfrac12\sigma\sigma'=\tfrac12\beta^2x$. The equivalent Itô equation is

$$\mathrm dX_t =\left(\alpha+\frac12\beta^2\right)X_t\,\mathrm dt +\beta X_t\,\mathrm dW_t.$$

Keeping the Itô drift equal to $\alpha X_t$ would instead describe a process whose logarithmic drift differs by $\beta^2/2$; the two equations would not have the same law.

3. Read mass change from the current

Let $p$ satisfy $\partial_t p+\partial_iJ^i=0$ on a bounded domain $D$. Compare reflecting and absorbing boundaries.

Solution

The divergence theorem gives

$$\frac{\mathrm d}{\mathrm dt}\int_Dp\,\mathrm d^dx =-\int_{\partial D}n_iJ^i\,\mathrm dS.$$

For a reflecting boundary, $n_iJ^i=0$ pointwise, so the mass in $D$ is constant. At an absorbing boundary, outward current can be positive. The mass of trajectories still inside $D$ then decreases; it becomes a survival probability rather than a normalized interior law.

4. Diagnose the soft-mode limit

Derive the stationary density of the regulated mode from $J_*=0$ and explain why it fails when $\kappa_{\mathbf q}=0$.

Solution

The equation

$$-\Gamma\kappa_{\mathbf q}\phi p_* -\Gamma T\partial_\phi p_*=0$$

implies

$$\partial_\phi\log p_* =-\frac{\kappa_{\mathbf q}}{T}\phi.$$

Therefore

$$p_*(\phi)=C \exp\!\left[-\frac{\kappa_{\mathbf q}\phi^2}{2T}\right],$$

and Gaussian normalization gives $C=\sqrt{\kappa_{\mathbf q}/(2\pi T)}$ when $\kappa_{\mathbf q}>0$. At $\kappa_{\mathbf q}=0$, the formal solution is constant and cannot be normalized on $\mathbb R$. Finite-time Brownian evolution still exists, but a stationary probability density does not.

Synthesis and continuations

Brownian quadratic variation modifies the chain rule. Itô’s formula converts that modification into a local operator on observables, and expectation gives a deterministic weak evolution of probability measures. A density-level Fokker–Planck equation follows only with additional regularity and boundary control. Its current exposes conservation, boundary loss, and the distinction between stationarity and detailed balance.

The regulated Ornstein–Uhlenbeck mode closes the circle: the SDE, transition law, Fokker–Planck equation, moments, stationary Gaussian, and two-time covariance all agree, while the soft-mode limit shows exactly where the stationary construction fails. For kernels, semigroups, invariant measures, ergodicity, and correlated-sample uncertainty, continue to Markov Generators, Semigroups, Ergodicity, and Correlated-Sample Error.

References

• Poul H. Damgaard and Helmuth Hüffel, “Stochastic Quantization”, Physics Reports 152 (1987), 227–398, Open PDF, doi:10.1016/0370-1573(87)90144-X. § 3.1, pp. 236–239, motivates fictitious-time Langevin evolution and the formal functional Fokker–Planck stationary measure.

• Grigorios A. Pavliotis, Stochastic Processes and Applications, PDF, Springer, 2014; linked author manuscript dated November 11, 2015. § 1.3, pp. 10–13, develops Brownian motion; §§ 3.1–3.5, pp. 49–66, cover SDEs, Itô and Stratonovich integrals, existence, Itô’s formula, the Fokker–Planck connection, and the Ornstein–Uhlenbeck process; § 4.1, pp. 77–80, develops the forward equation, current, and boundary conditions; § 4.6, pp. 104–105, relates reversibility to vanishing stationary current.

• Uwe C. Täuber, “Field Theory Approaches to Nonequilibrium Dynamics”, arXiv:cond-mat/0511743v2, 2006. § 1.1, pp. 6–8, supplies the QFT-facing model-A Langevin equation, thermal noise normalization, and Gaussian mode correlations.