02.15.03 · analysis / stochastic-analysis

Stochastic differential equations, diffusions, and the infinitesimal generator

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Anchor (Master): Karatzas-Shreve Brownian Motion and Stochastic Calculus Ch. 5 (strong/weak solutions, martingale problem); Stroock-Varadhan Multidimensional Diffusion Processes (1979); Friedman Stochastic Differential Equations and Applications Vol. 1

Intuition Beginner

Picture a particle pushed along by a steady wind and, at the same time, kicked around by random jitter. The wind is predictable: at each location it has a definite direction and strength. The jitter is not: it comes from countless tiny unseen collisions, the same buffeting that makes a speck of pollen tremble in still water. A stochastic differential equation is the rule that blends these two forces into a single recipe for how the particle moves moment to moment.

Over a short interval the particle drifts a little in the wind's direction and also takes a small random step whose size depends on where it is. Add up all these short moves and you get a jagged, never-smooth path. Run the same recipe twice from the same start and you get two different paths, because the random kicks differ each time. What does not differ is the statistical pattern: the cloud of possible paths spreads in a way fixed entirely by the wind field and the jitter field.

The remarkable fact this unit builds is that this cloud of random paths is governed by an ordinary, deterministic equation for its probabilities. The randomness lives in the individual path; the law of the cloud is as orderly as a heat-spreading rule.

Visual Beginner

A two-dimensional plane carries a faint field of arrows showing the wind that pushes the particle. Several jagged trajectories all start from one dot and wander outward, each taking a different jittery route, fanning into a spreading cloud. Beside the cloud a smooth grey shading shows where the particle is likely to be found at a fixed later time: dark where many paths pile up, pale at the fringes.

The picture holds the whole story in one frame. On the left, the individual jagged paths show the random motion of one particle: rough, unrepeatable, pushed by the arrows and shaken by noise. On the right, the smooth shading shows the law of the whole ensemble: a density that flows and spreads in a perfectly definite way. The bridge between the rough left and the smooth right is the object this unit is about.

Worked example Beginner

Take the simplest jitter rule with a pull toward the origin: a particle on a line that is tugged back to zero in proportion to how far it has strayed, while being kicked by noise of constant strength. The wind at position $x$ points toward zero with strength proportional to $x$ ; the jitter has the same size everywhere.

Start the particle at $x = 2$ . The pull tries to bring it home to zero, so on average the particle drifts back toward the origin and settles into hovering near it. But the kicks never stop, so it never sits exactly at zero — it rattles around in a small region whose width is set by the contest between how hard the wind pulls and how hard the noise shakes.

Now run a thousand copies of this particle from $x = 2$ . Early on they all lean back toward zero together. After a while the thousand positions form a bell-shaped cloud centred at zero, neither growing nor shrinking: a balance has been struck. That settled bell shape is the long-run law of the motion. We did not have to follow any single rattling path to find it — the balance of pull against shake fixed the shape directly, which is the first hint that an orderly equation controls the cloud.

Check your understanding Beginner

Exercise (easy, multiple choice).

You run the same noisy-motion recipe twice, starting both times from the very same point. What should you expect?

A. The two paths are identical
B. The two paths differ, but their statistical spread is the same
C. The two paths differ, and their statistical spread also differs each run
D. One path is smooth and the other is jagged

Hint

The random kicks differ from run to run, but the rule generating them — the wind field and the jitter size — does not change.

Answer

B. The two paths differ, but their statistical spread is the same. Each run draws fresh random kicks, so the individual jagged paths are different. The recipe that generates them — the wind field and the jitter strength — is fixed, so the cloud of possible outcomes, and its spread, is the same every time.

Formal definition Intermediate+

Fix a filtered probability space $(Ω, F, (F_{t})_{t \geq 0}, P)$ carrying an $m$ -dimensional $(F_{t})$ -Brownian motion $B = (B_{t})$ , with the Itô integral and Itô's formula as developed in 02.15.02. Let $$ b : \mathbb{R}^n \to \mathbb{R}^n, \qquad \sigma : \mathbb{R}^n \to \mathbb{R}^{n \times m} $$ be the drift and dispersion coefficients. A stochastic differential equation (SDE) is the integral relation written informally as $$ dX_t = b(X_t), dt + \sigma(X_t), dB_t, $$ whose rigorous meaning is the requirement that an $(F_{t})$ -adapted, continuous process $X = (X_{t})$ satisfy, $P$ -almost surely and for every $t \geq 0$ , $$ X_t = X_0 + \int_0^t b(X_s), ds + \int_0^t \sigma(X_s), dB_s, $$ the last term an Itô integral.

Definition (strong solution). Given the Brownian motion $B$ and an $F_{0}$ -measurable initial condition $X_{0}$ , a strong solution is a process $X$ as above, adapted to the augmented filtration generated by $X_{0}$ and $B$ , satisfying the integral equation with $\int_{0}^{t} E ∣ σ (X_{s}) ∣^{2} d s < \infty$ on every $[0, t]$ . Pathwise uniqueness holds when any two strong solutions $X, \tilde{X}$ on the same space with $X_{0} = \tilde{X}_{0}$ satisfy $P (X_{t} = \tilde{X}_{t} for all t) = 1$ .

Definition (weak solution and the martingale problem). A weak solution is a pair $(X, B)$ on some filtered space solving the integral equation; here the Brownian motion is part of the output, not the input. Equivalently (Stroock-Varadhan), a probability measure $P_{x}$ on path space $C ([0, \infty); R^{n})$ with $P_{x} (X_{0} = x) = 1$ solves the martingale problem for the operator $L$ below if, for every $f \in C_{c}^{\infty} (R^{n})$ , $$ M_t^f := f(X_t) - f(x) - \int_0^t (Lf)(X_s), ds $$ is an $(F_{t})$ -martingale under $P_{x}$ .

Definition (infinitesimal generator). With $a := σ σ^{T} : R^{n} \to R^{n \times n}$ the (symmetric, positive-semidefinite) diffusion matrix, the generator of the SDE is the second-order operator $$ L = \tfrac{1}{2} \sum_{i,j=1}^n a^{ij}(x), \frac{\partial^2}{\partial x_i \partial x_j} + \sum_{i=1}^n b^i(x), \frac{\partial}{\partial x_i}, $$ acting on $C^{2}$ functions. When $a (x)$ is positive-definite at every $x$ , $L$ is a (non-divergence-form) elliptic operator of the kind studied through the maximum principle in 02.13.01; this is the diffusion-to-elliptic-operator correspondence the unit develops.

Counterexamples to common slips

Strong existence is strictly stronger than weak existence. Tanaka's equation $d X_{t} = sgn (X_{t}) d B_{t}$ has a weak solution (a Brownian motion, since $σ^{2} = 1$ ) but no strong solution and pathwise uniqueness fails — Lipschitz continuity of $σ$ is what this example lacks.
The generator depends on $σ$ only through $a = σ σ^{T}$ . Two different dispersion matrices with the same $a$ produce diffusions with identical laws; the SDE carries more information than its generator, but the law of the process does not.
Without the linear-growth bound, solutions can explode in finite time: $d X_{t} = X_{t}^{2} d t$ (no noise) already blows up. Linear growth is what confines the solution to all of $[0, \infty)$ .

Key theorem with proof Intermediate+

Theorem (strong existence and pathwise uniqueness; Itô). Suppose $b$ and $σ$ are globally Lipschitz and of linear growth: there is $K > 0$ with $$ |b(x) - b(y)| + |\sigma(x) - \sigma(y)| \le K|x - y|, \qquad |b(x)| + |\sigma(x)| \le K(1 + |x|), $$ for all $x, y \in R^{n}$ . Let $X_{0}$ be $F_{0}$ -measurable with $E ∣ X_{0} ∣^{2} < \infty$ . Then the SDE has a strong solution $X$ , unique up to indistinguishability, with $E [sup_{s \leq t} ∣ X_{s} ∣^{2}] < \infty$ for every $t$ .

Proof. Fix a horizon $T$ and work in the Banach space $H_{T}$ of continuous adapted processes with norm $∥ Y ∥^{2} = E [sup_{s \leq T} ∣ Y_{s} ∣^{2}]$ . Define the Picard map $$ (\Phi Y)t = X_0 + \int_0^t b(Y_s), ds + \int_0^t \sigma(Y_s), dB_s, $$ the exact stochastic analogue of the deterministic iteration of 02.08.01. We estimate $∥Φ Y - Φ Z ∥$ . For the drift term, the Cauchy-Schwarz inequality and Lipschitz continuity give $$ \mathbb{E}\sup{s \le t}\Big| \int_0^s (b(Y_r) - b(Z_r)), dr \Big|^2 \le t, K^2 \int_0^t \mathbb{E}\sup_{r \le u}|Y_r - Z_r|^2, du . $$ For the dispersion term, Doob's maximal inequality followed by the Itô isometry 02.15.02 converts the supremum of the stochastic integral into a time integral of the integrand's square: $$ \mathbb{E}\sup_{s \le t}\Big| \int_0^s (\sigma(Y_r) - \sigma(Z_r)), dB_r \Big|^2 \le 4, K^2 \int_0^t \mathbb{E}\sup_{r \le u}|Y_r - Z_r|^2, du . $$ Writing $g (t) = E sup_{s \leq t} ∣ (Φ Y)_{s} - (Φ Z)_{s} ∣^{2}$ and $h (t) = E sup_{s \leq t} ∣ Y_{s} - Z_{s} ∣^{2}$ , the two bounds combine to $g (t) \leq C \int_{0}^{t} h (u) d u$ with $C = 2 (T + 4) K^{2}$ . Iterating this inequality $n$ times along $Φ^{n}$ produces the factorially small bound $C^{n} t^{n} / n!$ , so for $T$ fixed $Φ^{n}$ is a contraction once $n$ is large. The Banach fixed-point theorem then yields a unique fixed point $X$ in $H_{T}$ , the strong solution; the linear-growth bound, fed through the same isometry and Gronwall's lemma, gives the moment estimate $E sup_{s \leq T} ∣ X_{s} ∣^{2} \leq C_{T} (1 + E ∣ X_{0} ∣^{2}) < \infty$ , ruling out explosion.

Pathwise uniqueness is the $Y = X$ , $Z = \tilde{X}$ case of the same estimate: $h = g$ forces $g (t) \leq C \int_{0}^{t} g$ , and Gronwall gives $g \equiv 0$ . Patching the unique solutions across $[0, T]$ , $[T, 2 T], \dots$ extends $X$ to all of $[0, \infty)$ . $□$

Bridge. This existence theorem builds toward the entire diffusion theory of the Master tier and appears again in 02.15.04, where the same solution flow is run against a potential to produce the Feynman-Kac representation. The foundational reason the proof works is that the Itô isometry turns the unfamiliar stochastic integral into a familiar $L^{2}$ time-integral, so the whole argument becomes the deterministic Picard-Lindelöf contraction of 02.08.01 with one extra term controlled by Doob's inequality — this is exactly the deterministic existence proof with a noise correction. Pathwise uniqueness generalises ODE uniqueness, and the Markov property we extract next is dual to the elliptic operator $L$ of 02.13.01: the central insight, putting these together, is that the random flow $X$ and the deterministic operator $L$ are two faces of one object, the bridge being Dynkin's formula, which equates an expectation over paths with a time-integral of $L f$ .

Exercises Intermediate+

Exercise 2 (medium, symbolic).

Use Itô's formula to verify that for a $C^{2}$ function $f$ , the process $M_{t}^{f} = f (X_{t}) - f (X_{0}) - \int_{0}^{t} (L f) (X_{s}) d s$ is a local martingale, where $X$ solves the SDE.

Hint

Apply Itô's formula to $f (X_{t})$ . The $d t$ terms it produces are exactly $L f$ ; the $d B_{t}$ terms are the martingale part.

Answer

Itô's formula gives $df (X_{t}) = \sum_{i} \partial_{i} f d X_{t}^{i} + \frac{1}{2} \sum_{i, j} \partial_{i} \partial_{j} f d ⟨ X^{i}, X^{j} ⟩_{t}$ . Substituting $d X^{i} = b^{i} d t + \sum_{k} σ^{ik} d B^{k}$ and $d ⟨ X^{i}, X^{j} ⟩ = a^{ij} d t$ with $a = σ σ^{T}$ , the $d t$ part is $(\sum_{i} b^{i} \partial_{i} f + \frac{1}{2} \sum_{i, j} a^{ij} \partial_{i} \partial_{j} f) d t = (L f) d t$ , and the remainder is $\sum_{i, k} \partial_{i} f σ^{ik} d B^{k}$ , a stochastic integral hence a local martingale. Subtracting $\int_{0}^{t} L f d s$ leaves exactly that martingale part, so $M^{f}$ is a local martingale. Rubric: full credit for the identification of $a^{ij}$ as the quadratic covariation and the cancellation of the $d t$ terms.

Exercise 3 (medium, short-answer).

Explain why two dispersion coefficients $σ_{1} \neq = σ_{2}$ with $σ_{1} σ_{1}^{T} = σ_{2} σ_{2}^{T}$ give diffusions with the same generator, and what this says about strong versus weak solutions.

Hint

The generator sees $σ$ only through $a = σ σ^{T}$ . The martingale problem is posed in terms of $L$ , hence in terms of $a$ .

Answer

Because $L$ depends on $σ$ only via $a = σ σ^{T}$ , equal $a$ yields the same $L$ and hence the same martingale problem, so $X^{(1)}$ and $X^{(2)}$ have the same law on path space. The strong solutions may genuinely differ as functions of a fixed Brownian motion — they are driven differently — but their distributions coincide. This is the precise sense in which weak solutions (laws) carry less information than strong solutions (path-by-path constructions): the generator determines the law but not the strong solution. Rubric: full credit for the $a = σ σ^{T}$ dependence and the law-versus-pathwise distinction.

Exercise 4 (hard, short-answer).

Derive Dynkin's formula $E_{x} [f (X_{τ})] - f (x) = E_{x} [\int_{0}^{τ} (L f) (X_{s}) d s]$ for $f \in C_{c}^{2}$ and a stopping time $τ$ with $E_{x} [τ] < \infty$ .

Hint

Take expectations in the martingale identity of Exercise 2 and apply optional stopping; the compact support and finite-expectation hypotheses give the integrability needed.

Answer

By Exercise 2, $M_{t}^{f} = f (X_{t}) - f (x) - \int_{0}^{t} L f (X_{s}) d s$ is a local martingale started at $0$ under $P_{x}$ . For $f \in C_{c}^{2}$ , both $f$ and $L f$ are bounded, so $sup_{t \leq τ} ∣ M_{t}^{f} ∣ \leq 2∥ f ∥_{\infty} + ∥ L f ∥_{\infty} τ$ , which is integrable because $E_{x} [τ] < \infty$ ; hence $M^{f}$ is a genuine (uniformly integrable on $[0, τ]$ ) martingale and optional stopping applies: $E_{x} [M_{τ}^{f}] = M_{0}^{f} = 0$ . Rearranging, $E_{x} [f (X_{τ})] - f (x) = E_{x} [\int_{0}^{τ} L f (X_{s}) d s]$ . Rubric: full credit for the boundedness bound legitimising optional stopping and the final rearrangement.

Exercise 5 (hard, short-answer).

Let $D \subset R^{n}$ be bounded with smooth boundary and let $u$ solve the elliptic Dirichlet problem $Lu = 0$ in $D$ , $u = g$ on $\partial D$ . Using Dynkin's formula with the exit time $τ_{D}$ , show that $u (x) = E_{x} [g (X_{τ_{D}})]$ .

Hint

Apply Dynkin's formula to $u$ and the stopping time $τ_{D}$ ; the hypothesis $Lu = 0$ kills the integral term.

Answer

Dynkin's formula applied to $u$ and $τ_{D}$ gives $E_{x} [u (X_{τ_{D}})] - u (x) = E_{x} [\int_{0}^{τ_{D}} Lu (X_{s}) d s] = 0$ , since $Lu \equiv 0$ on $D$ and $X_{s} \in D$ for $s < τ_{D}$ . (Ellipticity and the bounded smooth domain ensure $E_{x} [τ_{D}] < \infty$ , so the formula applies after a localisation argument.) At the exit time $X_{τ_{D}} \in \partial D$ , where $u = g$ , so $E_{x} [u (X_{τ_{D}})] = E_{x} [g (X_{τ_{D}})]$ . Therefore $u (x) = E_{x} [g (X_{τ_{D}})]$ : the elliptic boundary-value problem is solved by averaging the boundary data over the exit distribution of the diffusion. This generalises the mean-value property of harmonic functions in 02.13.01, to which it reduces when $L = \frac{1}{2} Δ$ . Rubric: full credit for killing the integral via $Lu = 0$ and reading off the boundary values at the exit time.

Exercise 6 (medium, short-answer).

State the Kolmogorov backward and forward equations for the transition density $p (t, x, y)$ of the diffusion, and identify the operator appearing in each.

Hint

The backward equation differentiates in the starting point $x$ and uses $L$ ; the forward equation differentiates in the end point $y$ and uses the formal adjoint $L^{*}$ .

Answer

Backward: $\partial_{t} p (t, x, y) = L_{x} p (t, x, y)$ , with $L$ acting on the starting variable $x$ — equivalently $\partial_{t} u = Lu$ for $u (t, x) = E_{x} [f (X_{t})]$ . Forward (Fokker-Planck): $\partial_{t} p (t, x, y) = L_{y}^{*} p (t, x, y)$ , where $L^{*} ρ = \frac{1}{2} \sum_{i, j} \partial_{i} \partial_{j} (a^{ij} ρ) - \sum_{i} \partial_{i} (b^{i} ρ)$ is the formal adjoint, acting on the end variable $y$ and governing the evolution of the density. Rubric: full credit for both equations with the correct operator ( $L$ backward, $L^{*}$ forward) and variable.

Advanced results Master

The strong-existence theorem makes the solution flow into a Markov family: for Lipschitz coefficients the solution $X_{t}^{x}$ started at $x$ satisfies $E [f (X_{t + s}^{x}) ∣ F_{t}] = (P_{s} f) (X_{t}^{x})$ , where $P_{s} f (y) = E_{y} [f (X_{s})]$ is the transition semigroup. Pathwise uniqueness is what makes the flow consistent — restarting at $X_{t}$ reproduces the same law — and the moment bound makes $P_{s}$ a Feller semigroup on $C_{0} (R^{n})$ . The infinitesimal generator $L$ is then literally the derivative of this semigroup, $L f = lim_{s ↓ 0} (P_{s} f - f) / s$ on a suitable domain, and Dynkin's formula $E_{x} [f (X_{t})] - f (x) = E_{x} [\int_{0}^{t} L f (X_{s}) d s]$ is the integrated form of that derivative. This is Oksendal Ch. 7 and Dynkin's Markov Processes (1965); it is the precise sense in which a diffusion is a second-order elliptic operator wearing probabilistic clothes.

The martingale problem of Stroock and Varadhan reorganises the entire theory around the generator alone. Rather than build $X$ from a Brownian motion, one seeks a measure $P_{x}$ on path space under which $f (X_{t}) - f (x) - \int_{0}^{t} L f (X_{s}) d s$ is a martingale for all test $f$ . Existence of a solution to the martingale problem is equivalent to weak existence for the SDE; well-posedness (existence and uniqueness) of the martingale problem is equivalent to the diffusion being a strong Markov Feller process. Stroock and Varadhan proved well-posedness under merely continuous, non-degenerate $a$ and bounded measurable $b$ — far weaker than Lipschitz — by an elliptic-regularity argument on $L$ itself. This is the deepest payoff of the operator viewpoint: uniqueness of the process is reduced to uniqueness for a partial differential equation, namely the resolvent equation $(λ - L) u = g$ , whose solvability is governed by the maximum principle of 02.13.01.

The two Kolmogorov equations are the analytic shadow of the probabilistic semigroup. Writing $u (t, x) = P_{t} f (x) = E_{x} [f (X_{t})]$ , differentiating the semigroup gives the backward equation $\partial_{t} u = Lu$ , $u (0, \cdot) = f$ — an evolution equation in the starting point, with the generator $L$ . Dually, the forward (Fokker-Planck) equation $\partial_{t} p = L^{*} p$ governs the density $p (t, \cdot)$ of $X_{t}$ , with $L^{*}$ the formal adjoint in divergence-correcting form. Kolmogorov derived both in 1931 by purely analytic means, before Itô built the paths; the SDE is the path-space realisation of Kolmogorov's analytic diffusion. When the diffusion is Brownian motion, $L = \frac{1}{2} Δ$ and both equations collapse to the heat equation $\partial_{t} u = \frac{1}{2} Δ u$ of 02.13.03 — self-adjointness erases the distinction between forward and backward, and the transition density is the Gaussian heat kernel. Every diffusion is thus a variable-coefficient, possibly non-self-adjoint, generalisation of heat flow.

A final structural fact: positivity of $a$ (non-degeneracy) is exactly uniform ellipticity of $L$ , and it buys smoothing. Hörmander's theorem extends this to degenerate $a$ provided the drift and dispersion vector fields, together with their iterated Lie brackets, span the tangent space — a bracket-generating condition tying diffusion smoothing to the geometry of 02.12.01's vector fields. Where ellipticity holds, the transition density exists, is smooth, and the Dirichlet problem $Lu = 0$ is solved probabilistically by the exit-distribution average of Exercise 5.

Synthesis. The infinitesimal generator is the foundational reason a jagged, unrepeatable random path is governed by smooth deterministic equations: $L$ packages the drift and diffusion into a single second-order elliptic operator, and the entire analytic theory of the diffusion is read off from it. This is exactly the duality that converts the probabilistic object — the Markov family of solution flows — into the analytic object — the semigroup $P_{t} = e^{t L}$ and its two Kolmogorov equations; the central insight is that strong existence (Picard plus the Itô isometry) and elliptic regularity (the maximum principle on $L$ ) are the same uniqueness theorem viewed from the path side and the operator side. Putting these together, Dynkin's formula is the bridge: it equates a path-average $E_{x} [f (X_{t})]$ with a time-integral of $L f$ , and from it flow the Feynman-Kac formula of 02.15.04, the probabilistic solution of the Dirichlet problem, and the collapse to the heat equation of 02.13.03 when $L = \frac{1}{2} Δ$ . The martingale problem generalises strong existence to merely continuous coefficients, and the bracket-generating condition generalises ellipticity to the degenerate case; what is invariant across all of it is that a diffusion is an elliptic operator made dynamical, the probabilistic face of the analysis built in 02.13.01 and 02.13.03.

Full proof set Master

The strong existence and pathwise uniqueness theorem is proved in full in the Key theorem section. The remaining Master claims are recorded here.

Proposition (Dynkin's formula). Let $X$ solve the SDE with Lipschitz, linear-growth coefficients and generator $L$ . For $f \in C_{c}^{2} (R^{n})$ and a stopping time $τ$ with $E_{x} [τ] < \infty$ , $$ \mathbb{E}x[f(X\tau)] - f(x) = \mathbb{E}_x\Big[ \int_0^\tau (Lf)(X_s), ds \Big]. $$

Proof. Apply Itô's formula 02.15.02 to $f (X_{t})$ . Using $d X^{i} = b^{i} d t + \sum_{k} σ^{ik} d B^{k}$ and the quadratic covariation $d ⟨ X^{i}, X^{j} ⟩ = a^{ij} d t$ with $a = σ σ^{T}$ , $$ f(X_t) = f(x) + \int_0^t (Lf)(X_s), ds + \int_0^t \sum_{i,k} \partial_i f(X_s), \sigma^{ik}(X_s), dB_s^k , $$ the $d t$ terms assembling precisely into $L f$ by the definition of $L$ . The stochastic integral $N_{t} = \int_{0}^{t} \sum_{i, k} \partial_{i} f σ^{ik} d B^{k}$ is a martingale with $N_{0} = 0$ , because $\partial_{i} f$ has compact support and $σ$ has linear growth, so the integrand is bounded and the Itô isometry gives $E [N_{t}^{2}] < \infty$ . Since $f \in C_{c}^{2}$ makes $f$ and $L f$ bounded, $f (X_{t \land τ}) - f (x) - \int_{0}^{t \land τ} L f d s = N_{t \land τ}$ is a uniformly integrable martingale on $[0, τ]$ once $E_{x} [τ] < \infty$ (the dominating bound is $2∥ f ∥_{\infty} + ∥ L f ∥_{\infty} τ$ ). Optional stopping yields $E_{x} [N_{τ}] = 0$ , and taking expectations in the Itô identity at time $τ$ gives the claim. $□$

Proposition (semigroup property and the backward equation). Let $P_{t} f (x) = E_{x} [f (X_{t})]$ for $f \in C_{0} (R^{n})$ . Then ${P_{t}}_{t \geq 0}$ is a contraction semigroup, $P_{t + s} = P_{t} P_{s}$ , and for $f \in C_{c}^{2}$ the function $u (t, x) = P_{t} f (x)$ solves the Kolmogorov backward equation $\partial_{t} u = Lu$ with $u (0, \cdot) = f$ .

Proof. The Markov property of the unique solution flow gives, for $s, t \geq 0$ , $$ P_{t+s}f(x) = \mathbb{E}x[f(X{t+s})] = \mathbb{E}_x\big[\mathbb{E}x[f(X{t+s}) \mid \mathcal{F}_t]\big] = \mathbb{E}_x[(P_s f)(X_t)] = P_t(P_s f)(x), $$ the inner equality by the Markov property and pathwise uniqueness (restarting at $X_{t}$ gives a solution with the law of $X^{X_{t}}$ ). Contraction is $∣ P_{t} f ∣ \leq E_{x} ∣ f (X_{t}) ∣ \leq ∥ f ∥_{\infty}$ . For the generator, apply Dynkin's formula on the deterministic time $τ = t$ : $P_{t} f (x) - f (x) = E_{x} [\int_{0}^{t} L f (X_{s}) d s] = \int_{0}^{t} P_{s} (L f) (x) d s$ , using Fubini and the semigroup. Differentiating at $t$ , $\partial_{t} P_{t} f = P_{t} (L f)$ ; and by the semigroup commuting with $L$ on $C_{c}^{2}$ , $\partial_{t} P_{t} f = L (P_{t} f)$ . Hence $u = P_{t} f$ satisfies $\partial_{t} u = Lu$ , $u (0, \cdot) = f$ . $□$

Proposition (forward equation as the adjoint evolution). If $X_{t}$ has a density $p (t, \cdot)$ for each $t > 0$ and $L^{} $d e n o t es t h e f or ma l$ L^2 $- a d j o in t o f$ L $, t h e n$ p $so l v es$ \partial_t p = L^{}p$ weakly.

Proof. For $f \in C_{c}^{\infty}$ , the backward equation gives $\frac{d}{d t} E_{x} [f (X_{t})] = E_{x} [L f (X_{t})]$ , i.e. $\frac{d}{d t} \int f p (t, y) d y = \int (L f) (y) p (t, y) d y$ . Integrating by parts to move the derivatives off $f$ onto $p$ — the boundary terms vanish since $f$ has compact support — the right side equals $\int f (y) (L^{*} p) (t, y) d y$ , where $L^{*} ρ = \frac{1}{2} \sum_{i, j} \partial_{i} \partial_{j} (a^{ij} ρ) - \sum_{i} \partial_{i} (b^{i} ρ)$ . Thus $\int f \partial_{t} p d y = \int f L^{*} p d y$ for all test $f$ , which is the weak form of $\partial_{t} p = L^{*} p$ . $□$

Connections Master

First-order ODE existence and uniqueness via Picard iteration 02.08.01 is the deterministic skeleton this unit dresses in noise. The Picard map $Φ$ , the contraction estimate, and the Banach fixed-point argument are imported verbatim; the only new ingredient is the stochastic integral term, controlled by the Itô isometry and Doob's inequality. Setting $σ \equiv 0$ collapses the strong-existence theorem back to Picard-Lindelöf, so the SDE theory is a strict extension of the ODE theory, and the linear-growth no-explosion bound is the stochastic counterpart of the ODE global-existence criterion.

Phase space, vector fields, and integral curves 02.12.01 supply the geometric meaning of the drift. The drift $b$ is a vector field on $R^{n}$ and the zero-noise solution is exactly its integral curve; the diffusion is the integral curve perturbed by Brownian forcing. The bracket-generating (Hörmander) condition for hypoellipticity is a statement about the Lie brackets of the drift and dispersion vector fields, tying the smoothing of the transition density directly to the vector-field geometry developed there.

The Laplace equation, harmonic functions, and the maximum principle 02.13.01 are the elliptic half of the diffusion-operator correspondence. The generator $L$ with positive-definite $a$ is a non-divergence-form elliptic operator, the Dirichlet problem $Lu = 0$ is solved probabilistically by the exit-distribution average of Exercise 5, and the maximum principle that governs harmonic functions is what powers Stroock-Varadhan uniqueness for the martingale problem. When $a = I$ and $b = 0$ , $L = \frac{1}{2} Δ$ and harmonic functions are exactly the $L$ -harmonic functions, with the mean-value property becoming the martingale property of $f (X_{t})$ .

The heat equation, heat kernel, and Duhamel's principle 02.13.03 are the special case $L = \frac{1}{2} Δ$ of the backward and forward equations. There both Kolmogorov equations collapse to $\partial_{t} u = \frac{1}{2} Δ u$ , the transition density is the Gaussian heat kernel, and the diffusion is Brownian motion. Every diffusion in this unit is therefore a variable-coefficient generalisation of the heat flow studied there, with the generator $L$ playing the role $\frac{1}{2} Δ$ plays for the heat kernel.

The Itô integral and Itô's formula 02.15.02, co-produced in this wave, are the machinery the whole unit runs on. The Itô isometry is the single estimate that makes the Picard contraction close; Itô's formula is what turns $f (X_{t})$ into $f (x) + \int L f d s + (martingale)$ , the identity behind both Dynkin's formula and the martingale-problem reformulation. This unit is the natural sequel: it takes the calculus of 02.15.02 and uses it to solve equations rather than merely differentiate along paths.

The Feynman-Kac formula 02.15.04 is the next step. It runs the very solution flow constructed here against a potential $V$ , so that $u (t, x) = E_{x} [f (X_{t}) exp (- \int_{0}^{t} V (X_{s}) d s)]$ solves $\partial_{t} u = Lu - V u$ — Dynkin's formula with a killing term. The existence and Markov property established in this unit are exactly the hypotheses that make that representation well-defined.

Historical & philosophical context Master

The diffusion equations came before the diffusion paths. Andrei Kolmogorov, in his 1931 Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung (Mathematische Annalen 104, 415–458) ^{[Kolmogorov 1931]}, derived the forward and backward parabolic equations for a Markov transition probability by purely analytic means, characterising a diffusion through its infinitesimal mean (drift) and infinitesimal variance (the matrix $a$ ) without ever constructing a single trajectory. Kiyosi Itô's 1951 memoir On stochastic differential equations (Memoirs of the AMS 4, 1–51) ^{[Ito 1951]} supplied the missing object: a pathwise integral against Brownian motion and a fixed-point proof building the trajectories Kolmogorov's equations only described statistically. The generator $L$ and the expectation formula equating $E_{x} [f (X_{t})] - f (x)$ with a time-integral of $L f$ were systematised by Eugene Dynkin in Markov Processes (1965) ^{[Dynkin 1965]}, completing the dictionary between the analytic and probabilistic descriptions.

The philosophical content is a duality between two ways of knowing the same process. Kolmogorov's analytic picture treats a diffusion as an evolution of densities, governed by a deterministic PDE; Itô's pathwise picture treats it as a random trajectory, built by integrating noise. Neither is prior: Dynkin's formula is the precise hinge on which one rotates into the other, and the Stroock-Varadhan martingale problem (1979) ^{[Stroock-Varadhan 1979]} shows that the generator alone — the analytic datum — often determines the process uniquely, with no reference to any particular Brownian driver. This is why a diffusion can be characterised three equivalent ways: as a strong solution of an SDE, as a weak solution, or as a solution of a martingale problem. The lesson recurring through the lineage is that the right invariant of a diffusion is not its sample paths, nor its driving noise, but the second-order elliptic operator $L$ that generates it; everything else is a representation. That operator-centred view is what later carried diffusion theory into the geometry of manifolds, where $\frac{1}{2} Δ$ becomes the Laplace-Beltrami operator and Brownian motion becomes intrinsic to the metric.

Bibliography Master

@book{oksendal2003sde,
  author    = {{\O}ksendal, Bernt},
  title     = {Stochastic Differential Equations: An Introduction with Applications},
  edition   = {6th},
  publisher = {Springer-Verlag, Berlin},
  year      = {2003}
}

@book{karatzasshreve1991,
  author    = {Karatzas, Ioannis and Shreve, Steven E.},
  title     = {Brownian Motion and Stochastic Calculus},
  edition   = {2nd},
  series    = {Graduate Texts in Mathematics},
  volume    = {113},
  publisher = {Springer-Verlag, New York},
  year      = {1991}
}

@book{stroockvaradhan1979,
  author    = {Stroock, Daniel W. and Varadhan, S. R. Srinivasa},
  title     = {Multidimensional Diffusion Processes},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {233},
  publisher = {Springer-Verlag, Berlin},
  year      = {1979}
}

@article{ito1951sde,
  author  = {It\^o, Kiyosi},
  title   = {On stochastic differential equations},
  journal = {Memoirs of the American Mathematical Society},
  volume  = {4},
  pages   = {1--51},
  year    = {1951}
}

@article{kolmogorov1931,
  author  = {Kolmogorov, Andrei N.},
  title   = {\"Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung},
  journal = {Mathematische Annalen},
  volume  = {104},
  pages   = {415--458},
  year    = {1931}
}

@book{dynkin1965,
  author    = {Dynkin, Eugene B.},
  title     = {Markov Processes (Volume I)},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {121},
  publisher = {Springer-Verlag, Berlin},
  year      = {1965}
}

Prerequisites

02.08.01
02.12.01
02.13.01
02.13.03

Tier anchors

beginner: 3Blue1Brown / Strogatz informal noisy-trajectory picture; Higham 'An Algorithmic Introduction to Numerical Simulation of SDEs' (SIAM Review 2001) for the discrete Euler-Maruyama intuition
intermediate: Oksendal Stochastic Differential Equations Ch. 5 (existence/uniqueness), Ch. 7 (diffusions, generator, Dynkin); Evans An Introduction to SDEs Ch. 5-6
master: Karatzas-Shreve Brownian Motion and Stochastic Calculus Ch. 5 (strong/weak solutions, martingale problem); Stroock-Varadhan Multidimensional Diffusion Processes (1979); Friedman Stochastic Differential Equations and Applications Vol. 1

References

Oksendal — Stochastic Differential Equations: An Introduction with Applications (6th ed., 2003) · Ch. 5 (Thm 5.2.1 existence and uniqueness), Ch. 7 (diffusions, the generator, Dynkin's formula Thm 7.4.1), Ch. 8 (Kolmogorov backward equation)
Karatzas and Shreve — Brownian Motion and Stochastic Calculus (2nd ed., 1991) · Ch. 5 §2 (strong solutions), §3 (weak solutions and the martingale problem), §4 (the Markov property and the generator)
Stroock and Varadhan — Multidimensional Diffusion Processes (1979) · Ch. 6 (the martingale problem), Ch. 10 (uniqueness via the elliptic operator)
Ito 1951 — On stochastic differential equations · Memoirs of the American Mathematical Society 4, pp. 1--51; originator of the SDE and the Picard-iteration existence proof
Kolmogorov 1931 — Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung · Mathematische Annalen 104, pp. 415--458; originator of the forward and backward diffusion equations
Dynkin 1965 — Markov Processes (Vol. I) · Ch. V; the infinitesimal generator and the formula bearing his name

Estimated time

beginner: 18m
intermediate: 45m
master: 80m