Generator of an Itô Diffusion

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The Generator Computing the Generator Two Diffusions and Their Operators

The Generator

Four pages have gone into turning the solution of a stochastic differential equation into an object worth naming. The Itô diffusion is not one process but a family indexed by the starting point; the family forgets the clock, and by the Markov property it restarts from its own present position carrying no memory of the route. The strong Markov property extended the restart to times read off the path itself, and the stopped expectation identity turned Itô's formula into a statement about averages at such times. The first three of these are bookkeeping about the law of the process, and none of them computes anything. Ask for the expected value of a function of the diffusion at a fixed time, or for the mean time the diffusion takes to leave a region, and the theory offers no procedure, only the assurance that such quantities are well posed. The fourth is the instrument that will change this, and it is already sharp; what is missing is the object it should be aimed at.

The procedure comes from differentiating at time zero. Fix a function \(f\) on the state space and watch the average \(E^x[ f( X_t ) ]\) as \(t\) decreases to zero. At \(t = 0\) the average is \(f(x)\), since the diffusion starts at \(x\); for small positive \(t\) it has moved off that value by an amount which, if the process is not too wild, should be proportional to \(t\). The proportionality constant is a number attached to the point \(x\), so it is again a function on the state space, and the assignment sending \(f\) to that function is an operator. It is the derivative of the process at the instant of its start, felt through the test function \(f\). Everything the diffusion does in its first instant is recorded there. That this should be a complete description of the process, and not merely its leading term, is what the two properties already in hand make plausible. Time homogeneity makes the first instant of the restarted diffusion the same object as the first instant of the original, and the Markov property makes the restarted diffusion the entire future. What the process does at any later instant is therefore what it does at the first, read from wherever it then happens to be.

That is the object this page defines and computes. The definition is immediate. The computation is the substance: the operator turns out to be a second-order partial differential operator whose coefficients are the drift and the noise coefficient of the equation. From that point on, a question about averages of a diffusion is a question about a differential operator, and the two subjects that the track has been developing in parallel become one subject.

The Working Setting Is Several-Component

From here the state carries several coordinates. The state space is \(\mathbb{R}^m\), the driving motion \(w\) is an \(n\)-dimensional Brownian motion with independent components, the drift is a vector field \(\mathbf{b} : \mathbb{R}^m \to \mathbb{R}^m\), and the noise coefficient is a matrix field \(\sigma : \mathbb{R}^m \to \mathbb{R}^{m \times n}\), both Lipschitz. The equation

\[ d X_t = \mathbf{b}( X_t )\, dt + \sigma( X_t )\, d w_t , \quad X_0 = x \in \mathbb{R}^m , \]

is read row by row, the \(i\)-th coordinate of the state being driven by the \(i\)-th row of \(\sigma\) against all \(n\) components of the noise. This is the several-component reading recorded earlier in the track. Existence and uniqueness, the clock shift, time homogeneity, and the Markov property were declared to hold in this generality with the same statements and the same proofs; they were held as declared rather than proved, the one-dimensional versions remaining the proved objects. Nothing on this page repairs that declaration. What this page does instead is keep the debt visible: each time a proof leans on the declared reading rather than on a theorem proved in several components, the step says so. The distinction is not pedantry, and the two ingredients of the computation below fall on opposite sides of it. The stopped expectation identity concerns an arbitrary Itô process and rests on theorems proved in several components, so it owes nothing. What is owed lies entirely on the other side: that the solution exists and is unique, that it forgets the clock, that it restarts from its present position. Everything said here about the diffusion as a family is said on the strength of the declaration.

The notation for averages is the start-indexed convention already in force. For a starting point \(x \in \mathbb{R}^m\) and a Borel function \(f : \mathbb{R}^m \to \mathbb{R}\) we write \(E^x[ f( X_t ) ] := \mathbb{E}[ f( X^x_t ) ]\), the superscript naming the start and the expectation being the ordinary one on the underlying probability space.

Definition: The Generator of an Itô Diffusion

Let \(X\) be a time-homogeneous Itô diffusion in \(\mathbb{R}^m\). The generator, or infinitesimal generator, \(A\) of \(X\) is defined at a point \(x \in \mathbb{R}^m\) by

\[ A f( x ) = \lim_{t \downarrow 0} \frac{ E^x[ f( X_t ) ] - f( x ) }{ t } , \]

the function \(f : \mathbb{R}^m \to \mathbb{R}\) being Borel and such that \(E^x[ | f( X_t ) | ] \lt \infty\) for all sufficiently small \(t \gt 0\), so that the difference quotient is defined. We write \(\mathcal{D}_A( x )\) for the set of such \(f\) for which the limit exists at \(x\), and \(\mathcal{D}_A\) for the set of those for which it exists at every \(x \in \mathbb{R}^m\). The name is shared with the infinitesimal generator of a flow; the relation between the two constructions is taken up below.

Two features of the definition are worth pausing over before anything is computed. The limit is one-sided, and it has to be: the diffusion is started at \(x\) at time zero and has no past, so the quotient exists only for \(t \gt 0\) and the limit can only be taken from above. And the integrability clause is our reading of the definition rather than a separate hypothesis. Without it the quotient whose limit is being taken need not exist at all. The condition is automatically satisfied for every function actually used below, since a bounded Borel \(f\) satisfies it at every starting point; it is stated because the definition is otherwise silent about which functions may be fed to \(A\), and a silent domain is where unstated assumptions accumulate.

Three observations fix the shape of the object. First, \(A\) is not an operator on a space of functions but a partially defined one, and the domain is part of the data. The two domains recorded above need not coincide, a limit being able to exist at one point and fail at another, and it is the global domain \(\mathcal{D}_A\) that will serve as the natural class. Second, on its domain \(A\) is linear: if \(f\) and \(g\) lie in \(\mathcal{D}_A( x )\) and \(\alpha, \beta\) are real, then linearity of the expectation makes the difference quotient of \(\alpha f + \beta g\) the corresponding combination of the two quotients, so \(\alpha f + \beta g \in \mathcal{D}_A( x )\) and \(A( \alpha f + \beta g )( x ) = \alpha A f( x ) + \beta A g( x )\). In particular a constant function lies in \(\mathcal{D}_A\) and is annihilated by \(A\), the diffusion having nothing to report about a function that does not vary. Third, and least comfortably, the definition as it stands exhibits no function in the domain beyond the constants. That the domain contains a rich supply, and that on that supply the operator has a closed form, is exactly what has to be earned.

The manoeuvre of differentiating a family at time zero has appeared on this curriculum before. A smooth flow on a manifold has an infinitesimal generator, obtained by taking the velocity at time zero of the trajectory through each point, and the result is a vector field. Here the same manoeuvre is applied to a family of random trajectories, and it is applied to the average of a test function rather than to the trajectory itself, since a Brownian path has no velocity to take. The two constructions agree in spirit and differ in output: a flow returns a tangent vector at each point, which acts on functions as a first-order derivative along the flow, whereas the diffusion returns an operator on functions directly. Whether the noise leaves a trace on that operator beyond the first-order term the flow would contribute, and what that trace is, is settled by the computation below.

The route to it is short, because the hard work was done before the generator was named. The identity already in hand expresses the average of a function of an Itô process at a stopping time as its initial value plus the average of a second-order expression accumulated along the path. Specializing the stopping time to a fixed deterministic time and dividing by it produces the difference quotient of the definition, and the limit can then be taken.

Computing the Generator

The diffusion is an Itô process with a particular pair of coefficients: it satisfies

\[ X_t = x + \int_0^t \mathbf{b}( X_s )\, ds + \int_0^t \sigma( X_s )\, d w_s , \]

so the general coefficients \(\mathbf{u}\) and \(\mathbf{v}\) of the identity become \(\mathbf{b}( X_s )\) and \(\sigma( X_s )\), evaluated along the path rather than prescribed independently of it. Substituting them into the identity and taking the stopping time to be a fixed time is the entire computation. What has to be checked is that the hypotheses survive the substitution, and what has to be supplied is the limit that the definition of the generator asks for and the identity does not perform.

Theorem: The Generator Is a Second-Order Operator

Let \(X\) be the Itô diffusion with drift \(\mathbf{b}\) and noise coefficient \(\sigma\), and let \(f \in C_0^2( \mathbb{R}^m )\). Then \(f \in \mathcal{D}_A\), and for every \(x \in \mathbb{R}^m\),

\[ A f( x ) = \sum_{i = 1}^{m} b_i( x )\, \partial_i f( x ) + \tfrac{1}{2} \sum_{i, l = 1}^{m} \bigl( \sigma \sigma^\top \bigr)_{il}( x )\, \partial_{il} f( x ) . \]

Proof.

The identity applies.
Fix \(x\) and write \(K\) for the support of \(f\). The process \(X = X^x\) is an Itô process with \(\mathbf{u}( s, \omega ) = \mathbf{b}( X_s )\) and \(\mathbf{v}( s, \omega ) = \sigma( X_s )\): the coefficients grow at most linearly, being Lipschitz, so the second-moment bound places \(\sigma( X_\cdot )\) in the admissible class on every horizon. That bound is one of the results held as declared in several components. The drift side of the membership costs nothing: each \(b_i( X_\cdot )\) is continuous in time and adapted, hence progressively measurable by the criteria for progressive measurability, and its pathwise integrals are finite by continuity. A fixed time \(t\) is a stopping time with \(E^x[ t ] = t \lt \infty\). The coefficients are Lipschitz, hence continuous, hence bounded on the compact set \(K\); since \(\mathbf{u}( s, \omega )\) and \(\mathbf{v}( s, \omega )\) are the values of \(\mathbf{b}\) and \(\sigma\) at \(X_s( \omega )\), they are bounded on the set of pairs for which \(X_s( \omega )\) lies in \(K\). Every hypothesis of the stopped expectation identity is met, and it gives

\[ E^x \bigl[ f( X_t ) \bigr] = f( x ) + E^x \Bigl[ \int_0^t g( X_s )\, ds \Bigr] , \quad g( y ) = \sum_{i} b_i( y )\, \partial_i f( y ) + \tfrac{1}{2} \sum_{i, l} \bigl( \sigma \sigma^\top \bigr)_{il}( y )\, \partial_{il} f( y ) . \]

The function \(g\) is continuous with compact support.
Its ingredients are the entries of \(\mathbf{b}\) and of \(\sigma \sigma^\top\), continuous because \(\mathbf{b}\) and \(\sigma\) are, and the first and second derivatives of \(f\), continuous by assumption. All derivatives of \(f\) vanish off \(K\), so \(g\) does too. A continuous function vanishing off a compact set is bounded; write \(C = \sup | g |\).

The difference quotient.
Divide the identity by \(t \gt 0\) and subtract \(g(x)\), which integrates to \(t\, g(x)\) over \([0, t]\). This gives

\[ \frac{ E^x[ f( X_t ) ] - f( x ) }{ t } - g( x ) = \frac{1}{t} E^x \Bigl[ \int_0^t \bigl( g( X_s ) - g( x ) \bigr)\, ds \Bigr] . \]

The integrand is jointly measurable and bounded by \(2C\), so Tonelli's theorem applied to its absolute value exchanges the two integrations and bounds the right side in absolute value by

\[ \frac{1}{t} \int_0^t E^x \bigl[\, \bigl| g( X_s ) - g( x ) \bigr| \,\bigr]\, ds =: \frac{1}{t} \int_0^t \varphi( s )\, ds . \]

The average of \(\varphi\) vanishes with \(t\).
The paths of \(X\) are continuous and start at \(x\), so \(X_s \to x\) almost surely as \(s \downarrow 0\), and continuity of \(g\) gives \(g( X_s ) \to g( x )\) almost surely. The differences are bounded by the constant \(2C\), so dominated convergence gives \(\varphi( s ) \to 0\) as \(s \downarrow 0\). Given \(\varepsilon \gt 0\), choose \(\delta\) with \(\varphi( s ) \lt \varepsilon\) for \(0 \lt s \lt \delta\); for \(0 \lt t \lt \delta\) the average of \(\varphi\) over \([0, t]\) is then smaller than \(\varepsilon\) as well. Hence the difference quotient converges to \(g(x)\) as \(t \downarrow 0\). The limit exists at every \(x\), so \(f \in \mathcal{D}_A\) and \(A f = g\), which is the display of the theorem.

Two Operators and What the Theorem Does Not Say

The right-hand side of the theorem is a differential operator in its own right, defined by the same formula on any function that is twice differentiable, with no probability in sight. It deserves a name of its own, because from here on the two operators must be kept apart.

Definition: The Differential Operator of a Diffusion

Let \(\mathbf{b}\) and \(\sigma\) be the coefficients of an Itô diffusion in \(\mathbb{R}^m\). The associated differential operator \(L\) acts on \(f \in C^2( \mathbb{R}^m )\) by

\[ L f = \sum_{i = 1}^{m} b_i\, \partial_i f + \tfrac{1}{2} \sum_{i, l = 1}^{m} \bigl( \sigma \sigma^\top \bigr)_{il}\, \partial_{il} f . \]

In this notation the theorem reads: \(C_0^2( \mathbb{R}^m ) \subseteq \mathcal{D}_A\), and \(A\) and \(L\) agree there. Both halves are informative and neither is more than it says. The theorem does not assert that \(\mathcal{D}_A\) equals \(C_0^2\); the domain is larger, the nonzero constants already lying in it, and how much larger is a question the definition of \(A\) does not settle. Nor does it assert that \(A\) and \(L\) agree wherever both make sense. What it provides is a supply of functions on which the probabilistic operator is computable, and that supply is rich enough to determine everything the track will ask of \(A\).

The second-order coefficients deserve a look. They form the matrix \(a = \sigma \sigma^\top\), which is symmetric, and for any vector \(\xi\) satisfies \(\xi^\top a\, \xi = | \sigma^\top \xi |^2 \geq 0\), so \(a\) is positive semidefinite at every point. The operator \(L\) is therefore elliptic in the weak sense, and non-degenerate exactly where \(\sigma\) has full row rank. Degeneracy is not pathology here but the normal state of affairs: whenever the noise has fewer components than the state, \(a\) is singular everywhere. Along the degenerate directions the noise injects no randomness directly, which is not to say the state stays deterministic along them: the drift can couple them to coordinates the noise does reach, and carry randomness in.

One feature of the formula should be stated plainly, because it is the shape of everything that follows. The noise enters only through \(a\), never through \(\sigma\) itself. Two diffusions built with different noise coefficients but the same \(a\) have the same generator: the direction each component of the driving motion pushed in is information the operator has thrown away. Whether two such diffusions must in fact have the same law is a sharper question, and nothing on this page decides it; what is decided is that the operator, and everything built from the operator alone, carries no trace of how the noise was wired.

Two Diffusions and Their Operators

Two instances show what the formula delivers. Both have coefficients simple enough that the computation is a matter of reading entries off a matrix, and both produce operators that were on this curriculum long before any stochastic process was.

Example: Brownian motion.

Take the state and the noise to have the same number of components and let the equation be \(d X_t = d w_t\), so that \(\mathbf{b} = 0\) and \(\sigma = I_n\). The coefficients are Lipschitz, and the solution started at \(x\) is \(X_t = x + w_t\): Brownian motion begun at \(x\). Here \(\sigma \sigma^\top = I_n\), so the first-order sum is absent and the second-order sum keeps only its diagonal,

\[ A f = \tfrac{1}{2} \sum_{i = 1}^{n} \partial_{ii} f = \tfrac{1}{2} \Delta f , \quad f \in C_0^2( \mathbb{R}^n ) , \]

with \(\Delta\) the Laplace operator. The generator of Brownian motion is half the Laplacian.

The factor of one half is not a blemish but the record of a convention made much earlier. Brownian motion was normalized so that \(w_t\) has variance \(t\) in each component, and the second-order coefficient of the generator is half that variance rate. Had the motion been scaled differently the constant would move with it. The heat kernel on the line makes the agreement visible: at diffusivity \(k = \tfrac{1}{2}\) its formula collapses to \(( 2 \pi t )^{-1/2} \exp( - x^2 / 2 t )\), which is precisely the density of \(w_t\). The operator computed here and the equation that kernel solves carry the same constant because they describe the same process.

Example: the graph of Brownian motion.

Let \(w\) be one-dimensional and consider the pair consisting of a running clock and the motion itself, \(\widehat{S}_t = ( \tau_0 + \varepsilon t, x_0 + w_t )\) with \(\varepsilon \in \{ +1, -1 \}\). The space-time extension identifies this as an Itô diffusion of state dimension two and driving dimension one, with

\[ \widehat{\mathbf{b}}( \tau, x ) = \begin{pmatrix} \varepsilon \\\\ 0 \end{pmatrix} , \quad \widehat{\sigma}( \tau, x ) = \begin{pmatrix} 0 \\\\ 1 \end{pmatrix} , \quad \widehat{\sigma} \widehat{\sigma}^\top = \begin{pmatrix} 0 & 0 \\\\ 0 & 1 \end{pmatrix} , \]

the coefficients of the base diffusion being \(b = 0\) and \(\sigma = 1\). Only one entry of the second-order matrix survives, and only the first entry of the drift is nonzero, so for \(f = f( \tau, x ) \in C_0^2( \mathbb{R}^2 )\),

\[ A f = \varepsilon\, \partial_\tau f + \tfrac{1}{2}\, \partial_{xx} f . \]

With \(\varepsilon = +1\) this is the backward heat operator; passing to \(L\), which extends the same formula to all of \(C^2\), the functions it annihilates are those satisfying \(\partial_\tau f = - \tfrac{1}{2} \partial_{xx} f\). With \(\varepsilon = -1\) the operator is the negative of the heat operator, and the functions annihilated are the solutions of the heat equation itself, \(\partial_\tau f = \tfrac{1}{2} \partial_{xx} f\).

This example is where the declared several-component reading is most heavily spent, the first having needed it only for the definition of the object, its solution being explicit. The space-time extension was proved to be a diffusion only in that reading, and the theorem of this page was applied to it as such. The computation itself is a matrix multiplication and owes nothing.

The example also repays the trouble of having introduced the enlarged state at all. A derivative in the clock direction has appeared in the generator, yet nothing about the construction has become time-dependent. The first coordinate is a coordinate of the state like any other, the process moving along it at unit speed, and \(\partial_\tau\) is a spatial derivative of the enlarged problem. That is the mechanism by which a question with time in it becomes a question about a time-homogeneous object. The sign \(\varepsilon\) decides whether the clock of the enlarged state runs with the equation or against it, so the two operators above are two operators and not one written twice.

The Operator Behind a Noise Schedule

The study of diffusions as a family divided the sampling methods of machine learning by whether their dynamics watch the clock, and promised that a prescribed noise schedule could ride into the homogeneous framework as a clock coordinate. The second example above is that promise being kept in the smallest case, and it shows what the enlarged state buys. Provided the enlarged coefficient fields are Lipschitz in the enlarged variables jointly, a scheduled noising process is a diffusion in the sense of this page once time is made a coordinate, and the theorem above applies to it. That proviso is a genuine requirement and is stronger than what the solution theory alone asks: a drift that couples the schedule to the state, as the variance-preserving family does, fails it however smooth the schedule, while a schedule entering the noise coefficient alone can meet it. Where it holds, the generator carries a clock derivative alongside the second-order term, exactly as the example computed.

The operator is not yet the thing such a model needs. What is wanted is the evolution of the density of the noised data, and the density is governed by the adjoint of the operator rather than by the operator itself. Nothing on this page identifies that adjoint or the equation it produces. What this page supplies is the object to take the adjoint of, which until now did not exist.

With the generator attached and computed, the identity behind it can be used at full strength. Its stopping time was specialized to a constant here, and that single choice was enough to produce the operator. Restoring the generality changes what the identity is for. Asked what it says when the clock is stopped at a moment read off the path, it converts the operator into a tool for computing quantities the theory could previously only declare well posed: the average of a function at an exit time, and the expected duration of the exit itself.