Chapman–Kolmogorov Equation

Every stochastic model that claims to be Markovian — memoryless, its future fixed by its present alone — must answer one consistency question: if I propagate the system from an early time to a late time in one step, do I get the same answer as propagating it through any intermediate time and summing over where it might have been? The Chapman–Kolmogorov (CK) equation is that consistency condition made exact. For the transition probability density $P(x_2,t_2\mid x_1,t_1)$ it states that inserting an intermediate time $t_2$ between $t_1$ and $t_3$ , then integrating over the unobserved intermediate state, must reproduce the direct kernel $P(x_3,t_3\mid x_1,t_1)$ . Two ingredients and nothing more go into it: the Markov property (the future given the present does not also need the past) and the law of total probability (sum over the intermediate state you did not measure). No stationarity, no detailed balance, no equilibrium is assumed. What emerges is a semigroup law — the transition kernels compose like probability-conserving evolution operators, $U(t_3,t_1)=U(t_3,t_2)\,U(t_2,t_1)$ — the stochastic cousin of the unitary composition law of quantum mechanics, except that information is lost rather than preserved, so there is no inverse. The reason CK sits at the head of the whole stochastic-methods chapter is that its infinitesimal-time limit generates the two great continuum equations of nonequilibrium statistical mechanics. Keep finite jump rates and the short-time expansion of CK produces the master equation; keep small Gaussian increments and it produces the Fokker–Planck equation. CK is their common parent. It also underlies why Markov-chain Monte Carlo converges (repeated composition of one stochastic matrix), why coarse-graining time is legitimate for Markov dynamics, and how one tests, on real single-molecule or turbulence data, whether a Markovian description is even admissible. Necessary but not sufficient for Markovianity, CK is wielded as a check, not a definition.

Key takeaways

The Chapman–Kolmogorov equation is the consistency law obeyed by the transition probabilities of any Markov process: $P(x_3,t_3\mid x_1,t_1)=\int\mathrm{d}x_2\,P(x_3,t_3\mid x_2,t_2)\,P(x_2,t_2\mid x_1,t_1)$ for $t_1<t_2<t_3$ .
It follows from just two facts — the Markov property and the law of total probability — with no stationarity or detailed balance assumed.
In discrete state spaces it is matrix multiplication of stochastic matrices, $\mathbf P(t_1,t_3)=\mathbf P(t_1,t_2)\,\mathbf P(t_2,t_3)$ : the semigroup law underlying Monte-Carlo dynamics.
Its infinitesimal-time limit generates the master equation (finite jump rates) and the Fokker–Planck equation (small Gaussian increments) — both derived in their own articles; CK is their parent.
The Gaussian free-diffusion kernel satisfies CK exactly because convolving two Gaussians adds their variances, $2D\tau_1+2D\tau_2=2D(\tau_1+\tau_2)$ — the semigroup made explicit.
Key idea: Markov + total probability ⇒ transition kernels compose. Inserting an intermediate time and summing over the unobserved state must reproduce the direct kernel; this single semigroup law is the seed from which the master and Fokker–Planck equations grow.

The core insight

Markov + total probability ⇒ transition kernels compose: inserting an intermediate time and summing over the (unobserved) intermediate state must reproduce the direct kernel. This single semigroup law is the seed from which the master and Fokker–Planck equations grow.

Where this lives today

Chapman–Kolmogorov is a working tool, not a museum piece. Every Markov-chain Monte Carlo sampler — from lattice-field-theory simulations to the diffusion models behind modern image generators — relies on the semigroup law: applying one engineered transition kernel $n$ times equals the $n$ -step kernel $\mathbf P^n$ , and its convergence to a target distribution is a statement about the spectral gap of that CK operator. In single-molecule biophysics and in turbulence analysis, experimentalists compute the two-time transition density directly and test whether $\int\mathrm{d}x_2\,P(x_3\mid x_2)P(x_2\mid x_1)$ reproduces the measured $P(x_3\mid x_1)$ ; a failure is the operational signature that the process has memory and needs a generalised (non-Markovian) description. Iterating CK over many infinitesimal intervals is also the discretised path integral for stochastic processes — the Onsager–Machlup and Martin–Siggia–Rose functional integrals — the stochastic analogue of Feynman's construction.

Origins · 1 minandrey kolmogorov and how this was discovered

andrey kolmogorov

1903–1987 · Soviet mathematician who axiomatised probability theory in 1933 and, with the analytic theory of Markov processes, put the transition-probability consistency equation on a rigorous footing. The equation carries his name jointly with the geophysicist Sydney Chapman, who reached the same relation in the kinetic theory of gases; an equivalent composition relation had been written earlier by Smoluchowski (1906). Kolmogorov's forward and backward equations are the differential descendants of the identity discussed here.

History

1906

— Andrey Markov introduces chains with the memoryless property that bears his name

1928

— Sydney Chapman derives the composition relation in the kinetic theory of diffusing gases

1931

— Kolmogorov's paper on analytic methods in probability establishes the forward and backward equations

1933

— Kolmogorov's Grundbegriffe axiomatises probability theory, grounding transition kernels rigorously

1945

— Wang and Uhlenbeck survey Brownian motion, popularising the CK framework in physics

The physical picture

Think of a relay race in which the baton — the system — must be held by some runner at every checkpoint. The probability of getting from runner A at the start to runner C at the finish is the sum, over every possible runner B at the midway checkpoint, of (probability A→B) × (probability B→C). The Markov property is precisely what lets you forget how the baton reached B once it is there: only the present matters. Chapman–Kolmogorov is nothing more than this bookkeeping, promoted to an exact functional identity for the transition kernel. Formally the propagators of a Markov process form a semigroup: $U(t_3,t_1)=U(t_3,t_2)\,U(t_2,t_1)$ . These operators are stochastic — non-negative, probability-conserving — rather than unitary, and there is no inverse because information about the starting state is being washed out. The word semigroup (group without inverses) captures the irreversibility of the coarse-grained description even though the underlying microscopic dynamics is time-reversible.

Figure

Chapman–Kolmogorov: sum over the intermediate state x₂

Because CK relates the same kernel at three times in a nonlinear way, it is simultaneously very restrictive and underdetermined. It tells you transition probabilities must compose, but it does not tell you how fast — the rate of composition, encoded in a generator (jump rates, or drift and diffusion coefficients), is extra physical input. That is why CK alone is a consistency law and the master and Fokker–Planck equations are the dynamical laws. The punchline figure below shows the cleanest realisation: two Gaussian diffusion propagators convolving into one wider Gaussian, the variances simply adding.

Figure

Punchline — two diffusion propagators convolve into one wider Gaussian (variances add)

The core equations

The central object is the two-time conditional transition density $P(x_2,t_2\mid x_1,t_1)$ : the density of finding the system at $x_2$ at the later time $t_2$ given it sat at $x_1$ at the earlier time $t_1$ . The CK equation constrains this family of kernels. It is a single equation for a two-argument function, so it restricts without fully determining them.

P(x_3,t_3\mid x_1,t_1)=\int \mathrm{d}x_2\,P(x_3,t_3\mid x_2,t_2)\,P(x_2,t_2\mid x_1,t_1),\qquad t_1<t_2<t_3 \qquad (1)

where: $P(x_b,t_b\mid x_a,t_a)$ — conditional transition probability density (density of state $x_b$ at $t_b$ given state $x_a$ at earlier $t_a$ ); $x_2,t_2$ — intermediate state and time, integrated over; $\int\mathrm{d}x_2$ — sum over all possible intermediate states, i.e. the law of total probability. In a discrete state space the transition probabilities form a matrix, and equation (1) becomes ordinary matrix multiplication. Reading the rows as "from state $i$ " and columns as "to state $k$ ", inserting an intermediate index $j$ and summing is exactly the matrix product.

P_{ik}(t_1,t_3)=\sum_j P_{ij}(t_1,t_2)\,P_{jk}(t_2,t_3),\qquad \mathbf P(t_1,t_3)=\mathbf P(t_1,t_2)\,\mathbf P(t_2,t_3) \qquad (2)

where: $P_{ij}$ — probability of transition from state $i$ to state $j$ over the indicated interval; $\sum_j$ — sum over the intermediate discrete state $j$ ; $\mathbf P$ — stochastic matrix (non-negative entries, rows summing to one). For the kernel to be a genuine probability it must remain non-negative and normalised at all times, and it must collapse to a spike at coincident times. These constraints are preserved by CK composition — the semigroup maps stochastic kernels to stochastic kernels.

P(x,t\mid x_0,t_0)\ge 0,\qquad \int\mathrm{d}x\,P(x,t\mid x_0,t_0)=1,\qquad \lim_{t\to t_0}P(x,t\mid x_0,t_0)=\delta(x-x_0) \qquad (3)

where: $\ge 0$ — positivity preservation; $\int\mathrm{d}x\,P=1$ — conservation of total probability; $\delta(x-x_0)$ — Dirac delta enforcing certainty at coincident times. For a time-homogeneous process the kernel depends only on the elapsed time difference, and placing the intermediate time a small step $\Delta t$ after the start turns CK into an iteration whose $\Delta t\to0$ expansion defines the generator. This short-time relation is the seed of the continuum equations.

P(x,\tau\mid x_0)=\int \mathrm{d}x'\,P(x,\tau-\Delta t\mid x')\,P(x',\Delta t\mid x_0) \qquad (4)

where: $\tau$ — elapsed time in a homogeneous process; $\Delta t$ — infinitesimal step whose expansion produces the master and Fokker–Planck equations; $x'$ — state at the intermediate infinitesimal time. The character of that generator is captured by the Kramers–Moyal jump moments — the expected powers of the increment per unit time. Only the first two survive for continuous processes, and they are the drift and diffusion.

\alpha_n(x)\equiv\lim_{\Delta t\to0}\frac{1}{\Delta t}\int\mathrm{d}x'\,(x'-x)^n\,P(x',\Delta t\mid x) \qquad (5)

where: $\alpha_n$ — $n$ -th jump moment per unit time; $\alpha_1$ = drift, $\alpha_2$ = diffusion; $(x'-x)^n$ — $n$ -th power of the increment over the step $\Delta t$ .

The physical picture in one line

CK says the propagators compose; the generator (jump rates for the master equation, drift and diffusion for Fokker–Planck) says how fast. The infinitesimal-time limit of CK yields the master equation — a gain–loss balance of transition rates, derived and owned in `aq-sm-master-equation-sm11` — when the elementary steps are finite jumps, and the Fokker–Planck equation — a drift–diffusion partial differential equation obtained via the Kramers–Moyal expansion and Pawula truncation, derived and owned in `aq-sm-fokker-planck-equation-sm11`/`fokker-planck-equation` — when the steps are small and Gaussian. We do not re-derive either continuum equation here; CK's own content is the composition law and its semigroup realisation. The Markov property that equation (1) rests on is treated in `aq-sm-markov-processes-and-markov-property`.

The core derivations

Derivation AChapman–Kolmogorov from the Markov property

The goal is to show that equation (1) is a theorem, not an assumption — it follows from the chain rule of probability applied to a three-time joint density, the single use of the Markov property to drop the earliest time from a conditional, and marginalising over the intermediate state. Nothing about equilibrium, stationarity, or detailed balance enters; the derivation is pure probability plus the one physical input that defines a Markov process. Establishing this cleanly is what makes CK a diagnostic for Markovianity rather than a postulate.

Step 1. Write the three-time joint density using the chain rule of probability, exact for any process:

P_3(x_3,t_3;x_2,t_2;x_1,t_1)=P(x_3,t_3\mid x_2,t_2;x_1,t_1)\,P_2(x_2,t_2;x_1,t_1).

Why: conditioning factorises any joint into a conditional times a lower joint; no approximation yet.

Step 2. Apply the Markov property to the leading conditional:

P(x_3,t_3\mid x_2,t_2;x_1,t_1)=P(x_3,t_3\mid x_2,t_2).

Why: the future given the present $x_2,t_2$ does not additionally depend on the past $x_1,t_1$ — this is the only physical assumption, and it is exactly the definition of a Markov process.

Step 3. Integrate out the intermediate state $x_2$ :

P_2(x_3,t_3;x_1,t_1)=\int\mathrm{d}x_2\,P(x_3,t_3\mid x_2,t_2)\,P_2(x_2,t_2;x_1,t_1).

Why: the law of total probability — the system passes through some $x_2$ , so summing over it recovers the marginal joint of $x_1$ and $x_3$ .

Step 4. Divide both sides by the single-time density $P_1(x_1,t_1)$ to convert joints to conditionals:

P(x_3,t_3\mid x_1,t_1)=\int\mathrm{d}x_2\,P(x_3,t_3\mid x_2,t_2)\,P(x_2,t_2\mid x_1,t_1),

which is equation (1). Why: $P_2(a;b)/P_1(b)=P(a\mid b)$ applied to both sides simultaneously; the common factor cancels. Verified: the two-assumption structure (Markov property + total probability) and the resulting semigroup composition are checked symbolically in verify_aq-sm-chapman-kolmogorov-equation.py (matrix associativity P·P·P = P³ and normalization preservation).

Go deeper · Derivation B · 1 minThe Gaussian diffusion kernel is a convolution semigroup

Derivation BThe Gaussian diffusion kernel is a convolution semigroup

The goal here is to exhibit the simplest non-trivial solution of CK explicitly and to make the semigroup abstraction concrete: free diffusion. The diffusion propagator is a Gaussian whose variance grows linearly in time, and composing two intervals is a convolution of Gaussians. Convolving Gaussians adds their variances, so the composed kernel is again the diffusion propagator over the total elapsed time — CK satisfied on the nose. This is the calculation behind the punchline figure and the worked example, and it is the template for every continuous Markov process near its diffusive limit.

Step 1. Take the free-diffusion kernel, the delta-initialised solution of $\partial_t P=D\,\partial_x^2 P$ :

G_\tau(x)=\frac{1}{\sqrt{4\pi D\tau}}\exp\!\Big(-\frac{x^2}{4D\tau}\Big).

Why: this is the Green's function of the heat equation; its variance is $\langle x^2\rangle=2D\tau$ , linear in time.

Step 2. Compose two intervals $\tau_1$ then $\tau_2$ by the CK integral, which for a translation-invariant kernel is a convolution:

(G_{\tau_2}*G_{\tau_1})(x)=\int\mathrm{d}x_2\,G_{\tau_2}(x-x_2)\,G_{\tau_1}(x_2).

Why: homogeneity makes the kernel depend only on displacement, turning equation (1) into a convolution in the spatial variable.

Step 3. Evaluate the Gaussian convolution; variances add:

(G_{\tau_2}*G_{\tau_1})(x)=\frac{1}{\sqrt{4\pi D(\tau_1+\tau_2)}}\exp\!\Big(-\frac{x^2}{4D(\tau_1+\tau_2)}\Big)=G_{\tau_1+\tau_2}(x).

Why: the convolution of two zero-mean Gaussians with variances $2D\tau_1$ and $2D\tau_2$ is a Gaussian with variance $2D\tau_1+2D\tau_2=2D(\tau_1+\tau_2)$ — exactly the CK requirement.

Step 4. Read off the semigroup statement: $G_{\tau_2}\!*\,G_{\tau_1}=G_{\tau_1+\tau_2}$ . Why: free diffusion forms a one-parameter convolution semigroup in time, the cleanest explicit realisation of Chapman–Kolmogorov and the origin of Brownian variance additivity. Verified: the symbolic convolution equals the direct kernel for several $(D,\tau_1,\tau_2)$ , variance additivity $2D(\tau_1+\tau_2)$ , and a numerical convolution on a grid all pass in verify_aq-sm-chapman-kolmogorov-equation.py.

Limiting cases

Chapman–Kolmogorov specialises into the working equations of stochastic physics according to the nature of the elementary step; the recall paragraph above names the owners of the two continuum limits.

Case	Result	Meaning
Discrete states	$P_{ik}(t_1,t_3)=\sum_j P_{ij}(t_1,t_2)\,P_{jk}(t_2,t_3)$	CK is matrix multiplication of stochastic matrices — the Markov semigroup underlying Monte-Carlo and Metropolis–Hastings dynamics.
Jump (rate) limit	$\partial_t P=\int\mathrm{d}x'\,[\,W(x\mid x')P(x')-W(x'\mid x)P(x)\,]$	Finite jump rates $W$ give the master equation (owned by `aq-sm-master-equation-sm11`).
Diffusive (Gaussian) limit	$\partial_t P=-\partial_x[\alpha_1 P]+\tfrac12\partial_x^2[\alpha_2 P]$	Small Gaussian increments (first two Kramers–Moyal moments) give the Fokker–Planck equation (owned by `fokker-planck-equation`).
Homogeneous / stationary	$P(x_2,t_2\mid x_1,t_1)=P(x_2,\tau\mid x_1),\ \tau=t_2-t_1$	Time-translation invariance reduces the two-time kernel to a one-time-difference kernel — CK becomes a convolution semigroup.

Worked examples

Example 1 — Composing the free-diffusion propagator (a real number). A colloidal tracer diffuses freely with $D=0.5\ \mu\mathrm{m}^2/\mathrm{s}$ . Verify Chapman–Kolmogorov by composing a $\tau_1=1\ \mathrm{s}$ step with a $\tau_2=2\ \mathrm{s}$ step and comparing to the direct $3\ \mathrm{s}$ propagator.

Step 1. The variance after the first interval is $2D\tau_1=2(0.5)(1)=1.0\ \mu\mathrm{m}^2$ ; after the second, $2D\tau_2=2(0.5)(2)=2.0\ \mu\mathrm{m}^2$ .

Step 2. Convolving the two Gaussian propagators adds the variances: $1.0+2.0=\mathbf{3.0\ \mu\mathrm{m}^2}$ .

Step 3. The direct propagator over the total $\tau_1+\tau_2=3\ \mathrm{s}$ has variance $2D(\tau_1+\tau_2)=2(0.5)(3)=\mathbf{3.0\ \mu\mathrm{m}^2}$ . The two agree exactly — Chapman–Kolmogorov holds, and the width grows as $\sqrt{2Dt}$ : after $3\ \mathrm{s}$ the tracer's positional standard deviation is $\sqrt{3}\approx\mathbf{1.73\ \mu\mathrm{m}}$ .

Example 2 — Discrete two-state chain, semigroup composition. A telegraph process hops between states $1$ and $2$ with one-step matrix over $[t_1,t_2]$ equal to $\mathbf P_{12}=\begin{pmatrix}0.7&0.3\\0.4&0.6\end{pmatrix}$ and over $[t_2,t_3]$ equal to $\mathbf P_{23}=\begin{pmatrix}0.9&0.1\\0.5&0.5\end{pmatrix}$ . Find the direct kernel over $[t_1,t_3]$ .

Step 1. By CK, $\mathbf P_{13}=\mathbf P_{12}\,\mathbf P_{23}$ . Multiply: the $1\!\to\!1$ entry is $0.7(0.9)+0.3(0.5)=0.63+0.15=0.78$ .

Step 2. The full product is $\mathbf P_{13}=\begin{pmatrix}0.78&0.22\\0.66&0.34\end{pmatrix}$ ; each row sums to $1$ , so probability is conserved by the composition.

Step 3. The probability of being in state $1$ at $t_3$ having started in state $1$ is $\mathbf{0.78}$ — computed without ever visiting the intermediate time explicitly, exactly what the semigroup law guarantees.

Going further

Texts. Pathria & Beale, Statistical Mechanics, 3rd ed., ch. 15 (Stochastic processes) · Kardar, Statistical Physics of Particles, ch. on stochastic processes · Reif, Fundamentals of Statistical and Thermal Physics, ch. 15 · van Kampen, Stochastic Processes in Physics and Chemistry, ch. IV–V · Gardiner, Handbook of Stochastic Methods, ch. 3 · Risken, The Fokker–Planck Equation, ch. 1 · Sethna, Statistical Mechanics: Entropy, Order Parameters and Complexity.

Provenance. Derivations of CK from the Markov property and total probability follow van Kampen ch. IV and Gardiner ch. 3. The master-equation and Fokker–Planck limits (Kramers–Moyal expansion, Pawula theorem) are derived and owned in `aq-sm-master-equation-sm11` and `aq-sm-fokker-planck-equation-sm11`/`fokker-planck-equation`, following Risken ch. 1, Gardiner ch. 3, and Pathria & Beale ch. 15. A verifier should: (i) reproduce the three-step joint→conditional→marginalise derivation; (ii) confirm the Gaussian convolution-semigroup identity and variance additivity $2D(\tau_1+\tau_2)$ ; (iii) check discrete stochastic-matrix composition preserves row normalization. All results are standard; no numerical constants are asserted beyond the diffusion variance relation.

Builds on: Markov Processes and the Markov Property · Conditional probability and the law of total probability. Leads to: The Master Equation · Fokker–Planck Equation · Detailed Balance and Microscopic Reversibility · Metropolis–Hastings Algorithm.

Is Chapman–Kolmogorov the same thing as the master equation?

No. Chapman–Kolmogorov is an integral consistency law relating transition kernels at three times; it holds for any Markov process and assumes nothing about rates. The master equation is the differential law you get by taking the infinitesimal-time limit of CK with finite jump rates. CK is the parent identity; the master equation (owned by `aq-sm-master-equation-sm11`) is one of its two children, the Fokker–Planck equation being the other. CK constrains the kernels; the master equation actually evolves the probabilities in time.

Why does CK not determine the dynamics by itself?

Equation (1) is a single functional relation for a two-argument object, so it is heavily underdetermined. It says the kernels must compose, but not at what rate. The rate of composition is the generator — the jump rates $W$ for a jump process, or the drift $\alpha_1$ and diffusion $\alpha_2$ for a diffusion. Only after you specify that extra physical input do you get a closed evolution equation. This is exactly why CK is called a consistency law and the master/Fokker–Planck equations are called the dynamical laws.

Does satisfying CK prove a process is Markovian?

No — CK is necessary but not sufficient. There are constructed non-Markovian processes whose marginal two-time transition densities happen to satisfy equation (1) for particular distributions. So a positive CK test cannot prove memorylessness; it can only fail to refute it. Because of this, in data analysis CK is used as a screening diagnostic: a clear violation rules out a Markovian description, while agreement is treated as consistent-with-Markov rather than proof.

What makes the Gaussian diffusion kernel such a clean example?

Free diffusion is translation-invariant in space and homogeneous in time, so the CK integral becomes an ordinary convolution. Convolving two Gaussians adds their variances, and the diffusion kernel's variance is exactly $2D\tau$ , linear in time. Therefore $2D\tau_1+2D\tau_2=2D(\tau_1+\tau_2)$ reproduces the single Gaussian over the total time — CK holds identically. It is the simplest continuous realisation of a convolution semigroup and the reason Brownian mean-square displacement is additive in time.

How do the master and Fokker–Planck equations both come out of one equation?

Both arise from the short-time limit of CK, differing only in the nature of the elementary step. If the process makes finite jumps at finite rates, the $\Delta t\to0$ expansion keeps a gain–loss integral over transition rates and you get the master equation. If instead the increments are small and Gaussian, you Taylor-expand in the increment, keep only the first (drift) and second (diffusion) Kramers–Moyal moments — Pawula's theorem forbids stopping at any other finite order while keeping $P\ge0$ — and you get the Fokker–Planck equation. The full derivations live in those articles; CK supplies the common starting point.

What is the semigroup, and why "semigroup" and not "group"?

The transition operators satisfy $U(t_3,t_1)=U(t_3,t_2)U(t_2,t_1)$ , the composition rule of a semigroup: closed under composition and associative, but with no guaranteed inverse. There is no inverse because the dynamics loses information — many starting distributions can flow to the same later distribution, so you cannot uniquely run time backwards. This irreversibility of the coarse-grained probabilistic description coexists with time-reversible microscopic mechanics, and it is the mathematical fingerprint of the arrow of time in stochastic dynamics.

How is CK used inside Markov-chain Monte Carlo?

In MCMC you invent a transition matrix $\mathbf P$ engineered (via detailed balance) to have your target distribution as its unique stationary state. Chapman–Kolmogorov guarantees that applying $\mathbf P$ repeatedly is consistent: $n$ steps equal the single kernel $\mathbf P^n$ . Convergence to the target is then governed by the spectral gap of this CK operator — the second-largest eigenvalue controls the autocorrelation time. Critical slowing down near phase transitions, which cluster algorithms are built to defeat, is precisely a shrinking of that gap. So CK underpins both the correctness and the efficiency analysis of sampling.