The Itô Diffusion
The track so far has treated stochastic differential equations one at a
time. An equation was posed, a solution was hunted down or constructed, and
the page closed on that solution. This page changes the object of study. Fix
the coefficients once, solve from every starting point and every starting
time, and regard the resulting family of solutions as a single object. That
object has a name, and it is the protagonist of the remaining pages of the
track.
The setting is the one the
construction page left behind.
Throughout, \(w_t\) is the standard one-dimensional Brownian motion,
\(\{\mathcal{F}_t\}\) is the filtration generated by the motion and the
null sets, and the coefficients are Borel functions of the state alone. We
work on the real line, and the several-component version of each statement
is deferred to a remark in the final section, following the precedent of
the existence theory.
Definition: Itô Diffusion
Let \(b, \sigma : \mathbb{R} \to \mathbb{R}\) be
Lipschitz continuous,
so that there is a constant \(D\) with
\[
|b(x) - b(y)| + |\sigma(x) - \sigma(y)|
\leq D\, |x - y|
\quad \text{for all } x, y \in \mathbb{R} .
\]
The (time-homogeneous) Itô diffusion with coefficients
\(b\) and \(\sigma\), started at the point \(x \in \mathbb{R}\) at the
time \(s \geq 0\), is the unique solution
\(X^{s,x} = ( X^{s,x}_t )_{t \geq s}\) of
\[
d X_t = b( X_t )\, dt + \sigma( X_t )\, d w_t ,
\quad t \geq s ,
\quad X_s = x .
\]
When the start is at time zero we abbreviate \(X^x = X^{0,x}\).
Two features of the definition deserve immediate comment. The coefficients
do not depend on time. This is a genuine restriction relative to the
equations solved so far, whose coefficients were functions of the pair
\((t, x)\), and the final section of this page builds the bridge back to
that generality. And the definition speaks of the unique solution,
a claim that must be earned before the notation \(X^{s,x}\) is honest. The
rest of this section earns it.
From Lipschitz to Well-Posedness
The
existence and uniqueness theorem
asks for two bounds on the coefficients, the linear growth condition (H1)
and the Lipschitz condition (H2). The single displayed bound above
delivers both. With the time variable frozen, condition (H2) is the
displayed bound verbatim, with the same constant \(D\) for every \(t\).
For (H1), the triangle inequality gives
\[
\begin{align*}
|b(x)| + |\sigma(x)|
&\leq |b(0)| + |\sigma(0)| + |b(x) - b(0)| + |\sigma(x) - \sigma(0)| \\\\
&\leq |b(0)| + |\sigma(0)| + D\, |x| \\\\
&\leq C \bigl( 1 + |x| \bigr) ,
\end{align*}
\]
with \(C = |b(0)| + |\sigma(0)| + D\). A Lipschitz pair of coefficients
therefore sits inside the hypotheses of the theorem, and the whole
existence and uniqueness theory applies.
For the start \((0, x)\) the initial value is the constant \(Z \equiv x\).
A constant is measurable with respect to every \(\sigma\)-algebra, is
independent of the entire motion, and has finite second moment, all three
trivially, so the enlargement that the existence theory built around a
random start degenerates. The enlarged filtration collapses onto
\(\{\mathcal{F}_t\}\) itself, and on each horizon \([0, T]\) the theorem
hands over a solution satisfying the four clauses of the
solution definition,
unique up to indistinguishability.
Two horizons agree where they overlap, by the uniqueness half applied on
the shorter one, so the solutions over all horizons splice into a single
process \(X^{0,x}\) on \([0, \infty)\), and the definite article is
justified at \(s = 0\).
Starting the Clock Later
A start at time \(s \gt 0\) rides on shifted noise.
Lemma: The Shifted Motion Is a Standard Brownian Motion
Fix \(s \geq 0\) and define
\[
\tilde{w}_v = w_{s+v} - w_s ,
\quad v \geq 0 .
\]
Then \(\tilde{w}\) is a standard Brownian motion.
Proof.
The process \(\tilde{w}\) starts at zero and inherits continuous paths. Its
finite-dimensional distributions are Gaussian, being linear images of those
of \(w\), and its covariance is
\[
\begin{align*}
\operatorname{Cov}\bigl( \tilde{w}_u, \tilde{w}_v \bigr)
&= \min( s + u, s + v ) - \min( s + u, s ) - \min( s, s + v ) + \min( s, s ) \\\\
&= \min( s + u, s + v ) - s \\\\
&= \min( u, v ) ,
\end{align*}
\]
so \(\tilde{w}\) has mean zero and the covariance of the
Wiener process
started at the origin. Clause (W1) of that definition holds, (W2) follows
from it for a Gaussian process, as the Brownian motion page recorded, and
(W3) is the inherited continuity.
Pair \(\tilde{w}\) with the shifted filtration
\(\mathcal{G}_v = \mathcal{F}_{s+v}\). Then \(\tilde{w}\) is adapted to
\(\{\mathcal{G}_v\}\), and each increment
\(\tilde{w}_b - \tilde{w}_a = w_{s+b} - w_{s+a}\) is
independent of the running past
\(\mathcal{G}_a = \mathcal{F}_{s+a}\). These are exactly the two
properties of the ambient filtration that the existence proof and the
uniqueness half rely on, as the
uniqueness in law lemma
recorded when it made the construction travel. Both halves therefore run
verbatim over \((\tilde{w}, \{\mathcal{G}_v\})\) and produce, spliced
across horizons exactly as before, the unique solution \(Y\) of
\[
d Y_v = b( Y_v )\, dv + \sigma( Y_v )\, d \tilde{w}_v ,
\quad v \geq 0 ,
\quad Y_0 = x ,
\]
and we set \(X^{s,x}_t := Y_{t-s}\) for \(t \geq s\). This is the process
the definition names, and uniqueness transfers along the same
identification, so the notation \(X^{s,x}\) is earned for every admissible
start.
One accounting matter remains open. The display in the definition is
read on the original clock, while \(Y\) solves an equation read on the
shifted one. Reconciling the two readings is immediate for the time
integral and a genuine lemma for the stochastic integral. The next
section proves that lemma before first using it. Until then the shifted
reading is the definition.
One further piece of notation records what the diffusion generates on its
own. Write \(\mathcal{F}^X_t = \sigma( X^x_r : r \leq t )\) for the
\(\sigma\)-algebra generated by the path of \(X^x\) up to time \(t\),
for the fixed start \(x\), which the notation suppresses. The
solution is adapted, by the first clause of the solution definition, so
\[
\mathcal{F}^X_t \subseteq \mathcal{F}_t
\quad \text{for every } t \geq 0 .
\]
The motion can know more than the diffusion it drives, and the distinction
between the two filtrations does real work later in the page, when the
Markov property is stated with respect to each of them in turn.
Time Homogeneity
The previous section left a debt. The definition displays its integrals on
the original clock, running from \(s\) to \(t\) against \(w\), while the
object it names was built on the shifted clock, running from \(0\) against
\(\tilde{w}\). For the time integral the two readings agree by the
substitution \(u = s + v\) inside a pathwise Lebesgue integral. For the
stochastic integral the agreement is a statement about the Itô
construction itself, and we prove it first.
The payoff is larger than the debt. Once the definition reads the same on
either clock, the diffusion watched from a late start becomes comparable,
as a solution, with a fresh diffusion run from time zero, and the
comparison is this section's theorem.
Shifting the Clock Inside the Integral
Lemma: Clock Shift for the Itô Integral
Let \(s \geq 0\) and \(h \gt 0\), and let
\(f \in \mathcal{V}(s, s + h)\) over the filtration
\(\{\mathcal{F}_t\}\). Define, for \(v \geq 0\), the shifted data
\[
\begin{align*}
\tilde{f}_v &= f_{s+v} , \\\\
\tilde{w}_v &= w_{s+v} - w_s , \\\\
\mathcal{G}_v &= \mathcal{F}_{s+v} .
\end{align*}
\]
Then \(\tilde{f} \in \mathcal{V}(0, h)\) over \(\{\mathcal{G}_v\}\),
and
\[
\int_s^{s+h} f_u \, d w_u
= \int_0^h \tilde{f}_v \, d \tilde{w}_v
\quad \text{almost surely.}
\]
Proof.
Membership.
The map \((v, \omega) \mapsto (s + v, \omega)\) is measurable, so
\(\tilde{f}\) inherits joint measurability from \(f\). Adaptedness
transfers by relabeling, since \(f_{s+v}\) is
\(\mathcal{F}_{s+v}\)-measurable and \(\mathcal{F}_{s+v}\) is what
\(\mathcal{G}_v\) means. The integrability clause transfers by the
substitution \(u = s + v\) applied to the time variable,
\[
\mathbb{E} \int_0^h \tilde{f}_v^{\,2} \, dv
= \mathbb{E} \int_s^{s+h} f_u^{\,2} \, du
\lt \infty ,
\]
so \(\tilde{f}\) belongs to \(\mathcal{V}(0, h)\) over
\(\{\mathcal{G}_v\}\). The pair \((\tilde{w}, \{\mathcal{G}_v\})\)
carries the two structural properties that the integral theory
requires, adaptedness of the motion and independence of its increments
from the running past. Both were verified in the previous section, so
every quoted result below is available on the shifted clock in its
general filtration reading.
Elementary integrands.
Let \(f\) be a bounded
elementary process
on \([s, s + h]\), built from a partition
\(s = t_0 \lt t_1 \lt \cdots \lt t_k = s + h\) and
\(\mathcal{F}_{t_j}\)-measurable coefficients \(e_j\). Shifting the
partition by \(s\) exhibits \(\tilde{f}\) as an elementary process on
\([0, h]\) over \(\{\mathcal{G}_v\}\), with the same coefficients
attached to the times \(t_j - s\). The two integrals are the two sums,
each the definitional value of the integral on the bounded elementary
class,
\[
\begin{align*}
\int_s^{s+h} f_u \, d w_u
&= \sum_{j} e_j \bigl( w_{t_{j+1}} - w_{t_j} \bigr) , \\\\
\int_0^h \tilde{f}_v \, d \tilde{w}_v
&= \sum_{j} e_j \bigl( \tilde{w}_{t_{j+1} - s} - \tilde{w}_{t_j - s} \bigr) ,
\end{align*}
\]
and the increments match term by term, because
\(\tilde{w}_{t_{j+1} - s} - \tilde{w}_{t_j - s}
= w_{t_{j+1}} - w_{t_j}\) directly from the definition of
\(\tilde{w}\), the subtracted \(w_s\) canceling. For elementary
integrands the two sides are the same random variable, not merely
equal in law or almost surely.
General integrands.
For \(f \in \mathcal{V}(s, s + h)\), the
construction of the integral
supplies bounded elementary processes \(f_n\) with
\(\| f_n - f \|_{L^2(\lambda \otimes \mathbb{P})} \to 0\) over the
strip \([s, s+h] \times \Omega\). Their existence for a general
integrand is the content of the
three approximation steps
that precede that definition. The substitution \(u = s + v\) in the
time variable converts this into
\(\| \tilde{f}_n - \tilde{f} \|_{L^2(\lambda \otimes \mathbb{P})} \to 0\)
over \([0, h] \times \Omega\). By the
Itô isometry,
applied on each clock to the difference of integrands,
\[
\begin{align*}
\int_s^{s+h} f_{n,u} \, d w_u
&\longrightarrow \int_s^{s+h} f_u \, d w_u
\quad \text{in } L^2(\mathbb{P}) , \\\\
\int_0^h \tilde{f}_{n,v} \, d \tilde{w}_v
&\longrightarrow \int_0^h \tilde{f}_v \, d \tilde{w}_v
\quad \text{in } L^2(\mathbb{P}) .
\end{align*}
\]
By the elementary step the two sequences are the same sequence of
random variables, and two \(L^2\) limits of one sequence agree almost
surely.
The debt is discharged. We apply the lemma with
\(f_u = \sigma( X^{s,x}_u )\), whose shifted-clock membership is the third
clause of the solution definition, together with its mirror image under
the inverse relabeling \(v = u - s\). With the pathwise substitution for
the drift term added, this shows that the display in the definition and
the shifted equation that constructed \(X^{s,x}\) are the same statement.
The definition may now be read on either clock.
The Law Forgets the Clock
Time-independent coefficients ought to mean that the dynamics do not care
when the clock was started. The precise statement compares the diffusion
started at \(x\) at time \(s\), watched from time \(s\) onward, with the
diffusion started at \(x\) at time zero.
Theorem: Time Homogeneity
Let \(X\) be the Itô diffusion of the definition. For every
\(s \geq 0\), every \(x \in \mathbb{R}\), and all times
\(0 \leq v_1 \lt v_2 \lt \cdots \lt v_k\), the random vectors
\[
\bigl( X^{s,x}_{s+v_1}, \ldots, X^{s,x}_{s+v_k} \bigr)
\quad \text{and} \quad
\bigl( X^{x}_{v_1}, \ldots, X^{x}_{v_k} \bigr)
\]
have the same distribution. The processes
\(( X^{s,x}_{s+v} )_{v \geq 0}\) and \(( X^{x}_{v} )_{v \geq 0}\)
share their finite-dimensional distributions.
Proof.
Write \(Y_v = X^{s,x}_{s+v}\). By the construction of the previous
section, \(Y\) is the solution of
\[
d Y_v = b( Y_v )\, dv + \sigma( Y_v )\, d \tilde{w}_v ,
\quad Y_0 = x ,
\]
over the filtration \(\{\mathcal{G}_v\}\), satisfying the four
clauses of the solution definition on every horizon \([0, T]\). We
check that \(Y\) qualifies as a
weak solution
with initial distribution \(\delta_x\). The carrying pair
\((\tilde{w}, \{\mathcal{G}_v\})\) has a standard motion and the two
properties of the enlargement lemma, all established in the previous
section. The process is continuous and adapted with \(Y_0 = x\)
distributed according to \(\delta_x\). Clauses (S2) and (S3) hold
because the stronger four-clause package does. The integral equation
holds simultaneously for every \(v \in [0, T]\), being clause (S4).
Every requirement of the weak-solution definition is met. The
process \(X^{x}\), read over \((w, \{\mathcal{F}_t\})\), qualifies
by the identical checklist.
The initial law \(\delta_x\) is a Borel probability measure with
\(\int_{\mathbb{R}} y^2 \, d\delta_x = x^2 \lt \infty\), and the
coefficients satisfy (H1) and (H2), as the first section derived from
the Lipschitz bound. The
uniqueness in law lemma
therefore applies on each horizon \([0, T]\), and the theorem inherits
the one step that the lemma's sketch takes on trust. Two solutions of
the same equation with the same initial law, living on possibly
different filtrations, share their finite-dimensional distributions.
Choosing \(T \gt v_k\) covers any prescribed collection of times, and
the theorem follows.
From now on the starting time is dead weight. Every distributional
question about \(X^{s,x}\) reduces to the same question about \(X^{x}\),
and the two-index family collapses onto the one-index family
\(( X^x )_{x \in \mathbb{R}}\) of diffusions started at time zero. What
survives of the index \(s\) is the conditioning structure it suggests,
and that is the subject of the next section. The Markov property says
that the diffusion restarts from its own present position, and time
homogeneity is what lets the restarted future be compared with a fresh
copy run from time zero.
The Markov Property
The Markov property is the statement that the diffusion restarts.
Condition on everything the driving noise has revealed up to time \(t\).
The conditional law of the future position \(X^x_{t+h}\) then depends on
that past through the present position \(X^x_t\) alone, and it agrees
with the law of a fresh diffusion run from that position for a duration
\(h\). Time homogeneity is what makes the second half of the sentence
meaningful, since the restarted future runs on a late clock and the fresh
copy runs from zero.
Two pieces of notation serve the whole remaining track. For a starting
point \(y \in \mathbb{R}\) and a bounded Borel function
\(f : \mathbb{R} \to \mathbb{R}\), write
\[
\begin{align*}
E^y \bigl[ f( X_h ) \bigr] &:= \mathbb{E} \bigl[ f( X^{y}_h ) \bigr] , \\\\
u(y) &:= E^y \bigl[ f( X_h ) \bigr] .
\end{align*}
\]
The superscript carries the starting point, the expectation is the
ordinary one, and \(u\) packages the answer as a function of the start.
The function \(u\) depends on \(f\) and on \(h\), both suppressed in the
notation. In this notation the theorem to be proved reads
\[
\mathbb{E} \bigl[ f( X^x_{t+h} ) \bigm| \mathcal{F}_t \bigr]
= u( X^x_t )
\quad \text{almost surely,}
\]
and the proof has three pillars. The solution must depend continuously on
its starting point, in mean square. The assignment of starting points to
outcomes must be packaged as a single jointly measurable object, so that a
random starting point can be substituted into it. And the substituted
object must be identified with the diffusion itself, by pathwise
uniqueness. The first two subsections below erect the first two pillars,
and the closing subsection completes the proof.
Continuity in the Starting Point
The closing of the
uniqueness argument
proved more than uniqueness. For two solutions of the same equation over
the same setup, with square-integrable initial values \(Z\) and
\(\hat{Z}\), it established the mean-square stability bound
\(\mathbb{E}[ | X_t - \hat{X}_t |^2 ]
\leq 3\, \mathbb{E}[ | Z - \hat{Z} |^2 ]\, e^{A t}\)
on each horizon \([0, T]\), with \(A = 3 ( 1 + T )\, D^2\) built from the
Lipschitz constant and the horizon. Two constant starts qualify as initial
values, so the bound specializes to the diffusion family.
Lemma: Mean-Square Stability in the Starting Point
Fix a horizon \(T \gt 0\) and let \(A = 3 ( 1 + T )\, D^2\) be the
constant above. For all \(y, z \in \mathbb{R}\) and
\(0 \leq t \leq T\),
\[
\mathbb{E} \bigl[ | X^y_t - X^z_t |^2 \bigr]
\leq 3\, | y - z |^2 \, e^{A t} .
\]
The start-to-state map \(y \mapsto X^y_t\) is Lipschitz from the real line
into \(L^2(\mathbb{P})\), with a constant uniform over \(t \leq T\). This
is the quantitative form of the statement that nearby starts stay
comparable, and both the measurable version constructed next and the
substitution of a random start rest on it.
A Measurable Version of the Start-to-State Map
Fix a time \(h \geq 0\). The family \(\{ X^y_h \}_{y \in \mathbb{R}}\) is,
as it stands, one random variable per starting point, each defined up to
its own null set. To substitute a random start into the family we need a
single function of the pair \((y, \omega)\), measurable in both variables
jointly. Mean-square continuity is exactly the currency that buys such a
version.
Lemma: A Jointly Measurable Version of the Start-to-State Map
Fix \(h \geq 0\). There is a function
\(\Phi_h : \mathbb{R} \times \Omega \to \mathbb{R}\), measurable with
respect to \(\mathcal{B}(\mathbb{R}) \otimes \mathcal{F}\), such that
for every \(y \in \mathbb{R}\),
\[
\Phi_h( y, \cdot ) = X^{y}_h
\quad \text{almost surely.}
\]
Proof.
Dyadic snapping.
For \(m \in \mathbb{N}\) let \(\lfloor y \rfloor_m = 2^{-m} \lfloor 2^m y \rfloor\)
denote \(y\) snapped down to the dyadic grid of mesh \(2^{-m}\), and
define
\[
\Phi^{(m)}( y, \omega ) = X^{\lfloor y \rfloor_m}_h( \omega ) .
\]
On each dyadic cell the map \(y \mapsto \lfloor y \rfloor_m\) is
constant, so \(\Phi^{(m)}\) is a countable patchwork of sets of the
form (cell) \(\times\) (event), measurable in the pair. Each
\(\Phi^{(m)}\) is jointly measurable.
A rate that does not depend on the start.
Snapping moves the start by at most \(2^{-m}\), so the mean-square
stability bound of the previous subsection, read on a horizon
\(T \geq h\) with \(T \gt 0\) and its constant
\(A = 3 ( 1 + T )\, D^2\), gives, for every \(y \in \mathbb{R}\),
\[
\mathbb{E} \Bigl[ \bigl| \Phi^{(m)}( y, \cdot ) - X^{y}_h \bigr|^2 \Bigr]
\leq 3 \cdot 2^{-2m} \, e^{A h} .
\]
The right side is free of \(y\). By the
Chebyshev inequality
with threshold \(2^{-m/2}\),
\[
\mathbb{P} \Bigl[ \bigl| \Phi^{(m)}( y, \cdot ) - X^{y}_h \bigr| \geq 2^{-m/2} \Bigr]
\leq 3 \cdot 2^{-m} \, e^{A h} ,
\]
and the right side sums over \(m\). By the
Borel-Cantelli lemma,
for each fixed \(y\) the event that
\(| \Phi^{(m)}( y, \cdot ) - X^{y}_h | \geq 2^{-m/2}\) for infinitely
many \(m\) is null, so
\[
\Phi^{(m)}( y, \cdot ) \longrightarrow X^{y}_h
\quad \text{almost surely, for every fixed } y .
\]
The version.
Define
\[
\Phi_h( y, \omega )
= \limsup_{m \to \infty} \Phi^{(m)}( y, \omega ) ,
\]
with the convention that the value is \(0\) on the set where the
\(\limsup\) is not finite. A pointwise \(\limsup\) of jointly
measurable functions is jointly measurable, and the exceptional set is
measurable, so \(\Phi_h\) is
\(\mathcal{B}(\mathbb{R}) \otimes \mathcal{F}\)-measurable. For each
fixed \(y\) the sequence converges to \(X^{y}_h\) off a null set, and
on that full-measure set the \(\limsup\) is the limit, so
\(\Phi_h( y, \cdot ) = X^{y}_h\) almost surely.
Nothing about the argument is specific to the original clock. Run over
the shifted pair \((\tilde{w}, \{\mathcal{G}_v\})\) of the previous
sections, with the shifted diffusions in place of \(X^{y}\), it produces
a jointly measurable version of the shifted start-to-state map. The
stability bound it needs is available there because the uniqueness
argument travels to the shifted pair, as recorded when that pair was
introduced. The rest of the section substitutes the present position of
the diffusion into this measurable object, identifies the result with the
true future by pathwise uniqueness, and then integrates out the
independent noise. That last step deserves its own lemma, and conditional
expectation is where the work happens.
Independent Noise and the Freezing Principle
The proof needs one more piece beyond the three pillars, and it is
probabilistic rather than analytic. The restarted diffusion is
manufactured from two ingredients, the present position, which the past
knows completely, and the shifted noise, which the past should not know at
all. The next lemma makes the second half precise, and the one after it
converts the split into a computation rule for conditional expectations.
Lemma: The Shifted Noise Is Independent of the Past
Fix \(t \geq 0\) and let \(\tilde{w}_v = w_{t+v} - w_t\) for
\(v \geq 0\). Then the \(\sigma\)-algebra
\(\sigma( \tilde{w}_v : v \geq 0 )\) is independent of
\(\mathcal{F}_t\).
Proof.
One grid.
Fix times \(0 = v_0 \lt v_1 \lt \cdots \lt v_k\) and write
\(\Delta_i = w_{t+v_i} - w_{t+v_{i-1}}\) for the increments of the
motion along the shifted grid. Let \(\mathcal{F}^0_t\) denote the raw
past \(\sigma( w_s : s \leq t )\), fix \(B \in \mathcal{F}^0_t\) and
Borel sets \(B_1, \ldots, B_k\), and peel increments from the right.
The indicator of \(B \cap \{ \Delta_1 \in B_1 \} \cap \cdots \cap
\{ \Delta_{k-1} \in B_{k-1} \}\) is measurable with respect to
\(\mathcal{F}^0_{t+v_{k-1}}\), and the last increment is
independent of that past,
so the probability of the full intersection factors as the probability
of the first \(k - 1\) constraints times
\(\mathbb{P}[ \Delta_k \in B_k ]\). Repeating the step \(k\) times,
\[
\mathbb{P} \Bigl[ B \cap \bigcap_{i=1}^{k} \{ \Delta_i \in B_i \} \Bigr]
= \mathbb{P}[ B ] \, \prod_{i=1}^{k} \mathbb{P}[ \Delta_i \in B_i ] .
\]
Taking \(B = \Omega\) shows that the product on the right is the
probability of the intersection of the increment constraints alone.
Every event built from finitely many increment constraints over
this grid therefore factors against every event of
\(\mathcal{F}^0_t\), once the factorization is extended from the
products displayed to the \(\sigma\)-algebra
\(\sigma( \Delta_1, \ldots, \Delta_k )\) that they generate.
All grids.
The vector \(( \tilde{w}_{v_1}, \ldots, \tilde{w}_{v_k} )\) is the
image of \(( \Delta_1, \ldots, \Delta_k )\) under cumulative
summation, a linear bijection, so
\(\sigma( \tilde{w}_{v_1}, \ldots, \tilde{w}_{v_k} )
= \sigma( \Delta_1, \ldots, \Delta_k )\), and the factorization of
the previous step covers every event determined by the shifted motion
at finitely many times. Events of this kind, taken over all finite
grids, are closed under intersection, since two grids merge into one,
and they generate \(\sigma( \tilde{w}_v : v \geq 0 )\). The two
extensions this argument owes, from displayed products to one grid's
\(\sigma\)-algebra and from the merged family to the full
\(\sigma\)-algebra of the shifted motion, are the monotone-class step
this track has taken on faith since the construction of the integral.
We extend the same trust to these two applications.
Null sets.
Every event of \(\mathcal{F}_t\) differs from an event of
\(\mathcal{F}^0_t\) by a null set, by the same symmetric-difference
argument that the representation page ran for its augmented
\(\sigma\)-algebra, and altering an event by a null set changes no
probability. Independence therefore holds against \(\mathcal{F}_t\)
itself.
The computation rule now follows in full generality. It says that when a
quantity is assembled from an ingredient the conditioning
\(\sigma\)-algebra knows and an ingredient independent of it, the
conditional expectation is computed by freezing the known ingredient at
its realized value and averaging over the independent one. The statement
is measure-theoretic and owes nothing to diffusions, so we prove it for
an arbitrary pair of \(\sigma\)-algebras.
Lemma: Freezing the Measurable Ingredient
Let \(\mathcal{G}\) and \(\mathcal{H}\) be independent
sub-\(\sigma\)-algebras of \(\mathcal{F}\). Let \(X\) be a
\(\mathcal{G}\)-measurable random variable with values in a
measurable space \((E, \mathcal{E})\), and let
\(\varphi : E \times \Omega \to \mathbb{R}\) be bounded and
\(\mathcal{E} \otimes \mathcal{H}\)-measurable. Then the function
\[
h( y ) = \mathbb{E} \bigl[ \varphi( y, \cdot ) \bigr] ,
\quad y \in E ,
\]
is bounded and \(\mathcal{E}\)-measurable, and
\[
\mathbb{E} \bigl[ \varphi( X, \cdot ) \bigm| \mathcal{G} \bigr]
= h( X )
\quad \text{almost surely.}
\]
Proof.
The composed map \(\omega \mapsto \varphi( X(\omega), \omega )\) is
measurable and bounded, being the composition of
\(\omega \mapsto ( X(\omega), \omega )\) with \(\varphi\), so the
conditional expectation on the left exists. Let \(\mathcal{D}\) be
the class of bounded
\(\mathcal{E} \otimes \mathcal{H}\)-measurable \(\varphi\) for which
the conclusion holds in both halves, meaning that \(h\) is
\(\mathcal{E}\)-measurable and that
\[
\mathbb{E} \bigl[ \mathbf{1}_G \, \varphi( X, \cdot ) \bigr]
= \mathbb{E} \bigl[ \mathbf{1}_G \, h( X ) \bigr]
\quad \text{for every } G \in \mathcal{G} .
\]
Rectangles.
Let \(\varphi = \mathbf{1}_{A \times B}\) with \(A \in \mathcal{E}\)
and \(B \in \mathcal{H}\). Then \(h( y ) = \mathbf{1}_A( y )\,
\mathbb{P}[ B ]\), which is \(\mathcal{E}\)-measurable. For
\(G \in \mathcal{G}\), the event \(G \cap \{ X \in A \}\) belongs to
\(\mathcal{G}\), because \(\sigma( X ) \subseteq \mathcal{G}\), and
\(B\) is independent of it, so
\[
\begin{align*}
\mathbb{E} \bigl[ \mathbf{1}_G \, \mathbf{1}_A( X ) \, \mathbf{1}_B \bigr]
&= \mathbb{P} \bigl[ G \cap \{ X \in A \} \bigr] \, \mathbb{P}[ B ] \\\\
&= \mathbb{E} \bigl[ \mathbf{1}_G \, \mathbf{1}_A( X ) \, \mathbb{P}[ B ] \bigr] \\\\
&= \mathbb{E} \bigl[ \mathbf{1}_G \, h( X ) \bigr] .
\end{align*}
\]
Rectangles belong to \(\mathcal{D}\), and they are closed under
intersection and generate \(\mathcal{E} \otimes \mathcal{H}\).
Closure.
The class \(\mathcal{D}\) is a linear space containing the
constants. If \(\varphi_n \in \mathcal{D}\) increase pointwise to
a bounded \(\varphi\), then \(h_{\varphi_n} \to h_\varphi\)
pointwise by the
dominated convergence theorem
applied at each \(y\). The larger of the bounds of \(\varphi_1\)
and \(\varphi\) dominates the whole sequence on the finite measure
\(\mathbb{P}\). Hence \(h_\varphi\) is a pointwise limit of
\(\mathcal{E}\)-measurable functions and is measurable. The
integral identity passes to the limit on both sides by the same
theorem and the same dominating constant. The passage from a class
with these closure properties, containing the rectangles, to all
bounded \(\mathcal{E} \otimes \mathcal{H}\)-measurable functions
is the same monotone-class step, taken on the same trust, as in
the independence lemma.
Conclusion.
For general \(\varphi\), the function \(h( X )\) is bounded and
\(\sigma( X )\)-measurable, hence \(\mathcal{G}\)-measurable, and the
displayed identity is exactly the defining property of the
conditional expectation
of \(\varphi( X, \cdot )\) given \(\mathcal{G}\). By almost-sure
uniqueness of that object,
\(\mathbb{E}[ \varphi( X, \cdot ) \mid \mathcal{G} ] = h( X )\)
almost surely.
Taking expectations on both sides recovers an unconditional companion
that the substitution lemma below uses on its own. For \(X\) and
\(\varphi\) as in the lemma,
\(\mathbb{E}[ \varphi( X, \cdot ) ] = \mathbb{E}[ h( X ) ]\), by the
tower property
with the trivial \(\sigma\)-algebra downstairs. Averaging a mixed
quantity may be done in two stages, first over the independent ingredient
with the measurable one frozen, then over the frozen one. All the
machinery is now in place, and the closing subsection assembles it into
the Markov property.
Assembling the Markov Property
Fix the restart time \(t \geq 0\) and carry over the shifted trio
\(\tilde{w}_v = w_{t+v} - w_t\), \(\mathcal{G}_v = \mathcal{F}_{t+v}\),
and, for each \(y \in \mathbb{R}\), the shifted diffusion \(\tilde{Y}^y\)
solving the equation from \(y\) over \(( \tilde{w}, \{\mathcal{G}_v\} )\).
One refinement of the measurable version is worth recording before the
substitution. The construction of \(\tilde{Y}^y\) may be run over the
smaller filtration \(\mathcal{F}^{\tilde{w}}_v\), generated by the shifted
motion up to \(v\) and the null sets. The motion is adapted to it by
construction, and its increments are independent of it because
\(\mathcal{F}^{\tilde{w}}_a \subseteq \mathcal{G}_a\) and independence
against the larger \(\sigma\)-algebra restricts to the smaller. A solution
over the smaller filtration is a solution over the larger one, its
integrands being adapted to both and its integrals agreeing, since
elementary approximations over the smaller filtration are elementary over
the larger and their limits coincide. Pathwise uniqueness then identifies
the two, so \(\tilde{Y}^y_h\) has a version measurable with respect to
\(\mathcal{H} := \sigma( \tilde{w}_v : v \geq 0 ) \vee \mathcal{N}\), the
shifted noise augmented by the null sets.
Running the dyadic construction of the measurable version on these
representatives makes \(\tilde{\Phi}_h\) measurable with respect to
\(\mathcal{B}(\mathbb{R}) \otimes \mathcal{H}\), and \(\mathcal{H}\) is
independent of \(\mathcal{F}_t\), by the independence lemma together with
one more application of the null-set argument that closed its proof.
Lemma: Substituting a Random Start
Fix \(t \geq 0\) and \(h \geq 0\), and let \(Z\) be an
\(\mathcal{F}_t\)-measurable random variable with
\(\mathbb{E}[ Z^2 ] \lt \infty\). Let \(\Xi^Z\) be the solution of
\[
d \Xi_v = b( \Xi_v )\, dv + \sigma( \Xi_v )\, d \tilde{w}_v ,
\quad \Xi_0 = Z ,
\]
over \(( \tilde{w}, \{\mathcal{G}_v\} )\). Then
\(\Xi^Z_h = \tilde{\Phi}_h( Z, \cdot )\) almost surely.
Proof.
Before the steps, one hypothesis check serves the whole proof. Every
\(\mathcal{F}_t\)-measurable start is \(\mathcal{G}_0\)-measurable,
and the existence proof and the uniqueness half, which rest only on
the two properties of the carrying pair and on a square-integrable
initial value known at time zero, run over the shifted setup for every
start appearing below.
Simple starts.
Let \(Z = \sum_{i=1}^{N} y_i \mathbf{1}_{A_i}\) with
\(\{ A_i \}\) a finite \(\mathcal{F}_t\)-measurable partition of
\(\Omega\). Set \(S_v = \sum_i \mathbf{1}_{A_i} \tilde{Y}^{y_i}_v\).
On each \(A_i\) the process \(S\) coincides with
\(\tilde{Y}^{y_i}\), so \(b( S_v ) = \sum_i \mathbf{1}_{A_i}\,
b( \tilde{Y}^{y_i}_v )\) pointwise, and likewise for \(\sigma\). The
time integral of the drift splits along the partition path by path.
For the noise term, linearity of the integral and the lemma on
taking out an initially known factor,
applied with the \(\mathcal{G}_0\)-measurable factor
\(\mathbf{1}_{A_i}\), give
\[
\int_0^v \sigma( S_r )\, d \tilde{w}_r
= \sum_i \mathbf{1}_{A_i} \int_0^v \sigma( \tilde{Y}^{y_i}_r )\, d \tilde{w}_r
\quad \text{almost surely,}
\]
the products \(\mathbf{1}_{A_i}\, \sigma( \tilde{Y}^{y_i} )\)
remaining in the integrand class because the factors are bounded and
initially known. Multiplying the integral equation of
\(\tilde{Y}^{y_i}\) by \(\mathbf{1}_{A_i}\) and summing over the
partition therefore exhibits \(S\) as a solution with initial value
\(\sum_i y_i \mathbf{1}_{A_i} = Z\), satisfying the four clauses as a
finite sum of processes that satisfy them. Pathwise uniqueness over
the shifted setup identifies \(\Xi^Z\) with \(S\). Evaluating at
\(h\) and using the version property of \(\tilde{\Phi}_h\) at each of
the finitely many points \(y_i\),
\[
\begin{align*}
\Xi^Z_h
&= \sum_i \mathbf{1}_{A_i}\, \tilde{Y}^{y_i}_h \\\\
&= \sum_i \mathbf{1}_{A_i}\, \tilde{\Phi}_h( y_i, \cdot ) \\\\
&= \tilde{\Phi}_h( Z, \cdot )
\quad \text{almost surely.}
\end{align*}
\]
A stability bound with random arguments.
Let \(U\) and \(V\) be \(\mathcal{F}_t\)-measurable and
square-integrable, and fix a truncation level \(M \gt 0\). The
function
\(\varphi_M( (y, z), \omega )
= \min\bigl( | \tilde{\Phi}_h( y, \omega )
- \tilde{\Phi}_h( z, \omega ) |^2, M \bigr)\)
is bounded and
\(\mathcal{B}(\mathbb{R}^2) \otimes \mathcal{H}\)-measurable,
and the pair \(( U, V )\) is \(\mathcal{F}_t\)-measurable with
values in \(\mathbb{R}^2\). The unconditional companion of the
freezing lemma gives
\[
\begin{align*}
\mathbb{E} \bigl[ \varphi_M( ( U, V ), \cdot ) \bigr] &= \mathbb{E} \bigl[ h_M( U, V ) \bigr] , \\\\
h_M( y, z ) &= \mathbb{E} \bigl[ \varphi_M( ( y, z ), \cdot ) \bigr] .
\end{align*}
\]
For fixed \(y\) and \(z\) the version property converts
\(h_M( y, z )\) into an expectation over the two shifted diffusions,
and dropping the truncation only increases it, so the mean-square
stability bound on the shifted clock, read on the horizon chosen for
the measurable version, gives
\(h_M( y, z ) \leq 3\, | y - z |^2 e^{A h}\). Therefore
\(\mathbb{E}[ \varphi_M( ( U, V ), \cdot ) ]
\leq 3\, \mathbb{E}[ | U - V |^2 ]\, e^{A h}\) for every \(M\), and
letting \(M \uparrow \infty\) by the
monotone convergence theorem,
\[
\mathbb{E} \Bigl[ \bigl| \tilde{\Phi}_h( U, \cdot )
- \tilde{\Phi}_h( V, \cdot ) \bigr|^2 \Bigr]
\leq 3\, \mathbb{E} \bigl[ | U - V |^2 \bigr] \, e^{A h} .
\]
General starts.
Truncate \(Z\) at level \(m\) and snap the result to the dyadic grid
of mesh \(2^{-m}\). The outcome \(Z_m\) is a simple
\(\mathcal{F}_t\)-measurable variable with \(Z_m \to Z\) pointwise
and \(| Z_m | \leq | Z | + 1\), so \(Z_m \to Z\) in \(L^2\) by
dominated convergence. By the first step,
\(\Xi^{Z_m}_h = \tilde{\Phi}_h( Z_m, \cdot )\) for every \(m\). The
left side converges to \(\Xi^Z_h\) in \(L^2\), by the mean-square
stability bound of the uniqueness argument run on the shifted setup
with the initial pair \(( Z_m, Z )\). The right side converges to
\(\tilde{\Phi}_h( Z, \cdot )\) in \(L^2\), by the bound just proved
with \(( U, V ) = ( Z_m, Z )\). Two \(L^2\) limits of one sequence
agree almost surely, which is the claim.
Everything is bought and paid for, and the theorem is now an assembly.
Theorem: The Markov Property of Itô Diffusions
Let \(f : \mathbb{R} \to \mathbb{R}\) be bounded and Borel, and let
\(x \in \mathbb{R}\) and \(t, h \geq 0\). Then the function
\(u( y ) = E^y[ f( X_h ) ]\) is bounded and Borel, and
\[
\mathbb{E} \bigl[ f( X^x_{t+h} ) \bigm| \mathcal{F}_t \bigr]
= u( X^x_t )
\quad \text{almost surely.}
\]
Proof.
The diffusion restarts from its present position.
Set \(Z = X^x_t\), which is \(\mathcal{F}_t\)-measurable with
\(\mathbb{E}[ Z^2 ] \lt \infty\) by the
second-moment bound.
We subtract the integral equation of \(X^x\) at time \(t\) from the
one at time \(t + v\), using
additivity over intervals,
and convert both remainders to the shifted clock, the drift by the pathwise
substitution of \(t + r\) for the running time and the noise by the
clock shift lemma.
The result is
\[
X^x_{t+v}
= Z
+ \int_0^v b( X^x_{t+r} )\, dr
+ \int_0^v \sigma( X^x_{t+r} )\, d \tilde{w}_r ,
\quad v \geq 0 .
\]
The process \(( X^x_{t+v} )_{v \geq 0}\) is continuous, adapted to
\(\{ \mathcal{G}_v \}\), and inherits the membership clauses from
those of \(X^x\) over \([t, t + v]\) by the same substitution. It is
therefore a solution of the shifted equation with initial value
\(Z\), and pathwise uniqueness over the shifted setup identifies it
with \(\Xi^Z\). In particular \(X^x_{t+h} = \Xi^Z_h\) almost surely,
and the substitution lemma upgrades this to
\[
X^x_{t+h} = \tilde{\Phi}_h( Z, \cdot )
\quad \text{almost surely.}
\]
Freeze and average.
The map \(\varphi( y, \omega ) = f\bigl( \tilde{\Phi}_h( y, \omega ) \bigr)\)
is bounded and
\(\mathcal{B}(\mathbb{R}) \otimes \mathcal{H}\)-measurable, the pair
\(( \mathcal{F}_t, \mathcal{H} )\) is independent, and \(Z\) is
\(\mathcal{F}_t\)-measurable. The freezing lemma yields
\[
\begin{align*}
\mathbb{E} \bigl[ f( X^x_{t+h} ) \bigm| \mathcal{F}_t \bigr]
&= \mathbb{E} \bigl[ \varphi( Z, \cdot ) \bigm| \mathcal{F}_t \bigr] \\\\
&= h_f( Z )
\quad \text{almost surely,}
\end{align*}
\]
where \(h_f( y ) = \mathbb{E}[ f( \tilde{\Phi}_h( y, \cdot ) ) ]\) is
bounded and Borel by the lemma's measurability half.
The frozen average is a fresh start.
For each fixed \(y\), the version property gives
\(h_f( y ) = \mathbb{E}[ f( \tilde{Y}^y_h ) ]\), and
\(\tilde{Y}^y_v = X^{t, y}_{t+v}\) is how the first section defined
the late-started diffusion. By
time homogeneity,
\[
\begin{align*}
h_f( y )
&= \mathbb{E} \bigl[ f( X^{t,y}_{t+h} ) \bigr] \\\\
&= \mathbb{E} \bigl[ f( X^{y}_{h} ) \bigr] \\\\
&= u( y )
\quad \text{for every } y \in \mathbb{R} .
\end{align*}
\]
So \(u = h_f\) everywhere, \(u\) is bounded and Borel, and the
display of the previous step reads
\(\mathbb{E}[ f( X^x_{t+h} ) \mid \mathcal{F}_t ] = u( X^x_t )\)
almost surely.
Two remarks close the section. First, the property descends to the
diffusion's own filtration. The variable \(u( X^x_t )\) is
\(\mathcal{F}^X_t\)-measurable, and
\(\mathcal{F}^X_t \subseteq \mathcal{F}_t\), so the
tower property
gives
\[
\begin{align*}
\mathbb{E} \bigl[ f( X^x_{t+h} ) \bigm| \mathcal{F}^X_t \bigr]
&= \mathbb{E} \bigl[ u( X^x_t ) \bigm| \mathcal{F}^X_t \bigr] \\\\
&= u( X^x_t )
\quad \text{almost surely,}
\end{align*}
\]
the last equality because \(u( X^x_t )\) is measurable for the
conditioning \(\sigma\)-algebra. Conditioning on the whole path of the
diffusion so far, or on everything the richer noise filtration has seen,
produces the same answer, and that answer reads only the present position.
Readers who met the
first-order Markov property
of graphical models will recognize the shape. The conditioning rule that
page stated for a finite row of variables, each depending on its past
through its immediate predecessor alone, is here running in continuous
time, over an uncountable family of positions. The work of this section is
exactly what that upgrade costs.
The second remark is a notational convention. Later pages write the
conclusion in the compact form
\[
E^x \bigl[ f( X_{t+h} ) \bigm| \mathcal{F}_t \bigr]
= E^{X_t} \bigl[ f( X_h ) \bigr] ,
\]
in which the left side abbreviates the conditional expectation just
proved, for the diffusion started at \(x\), and the right side is, by
definition, the function \(y \mapsto E^y[ f( X_h ) ]\) evaluated at the
random position \(X^x_t\). The superscript position does the bookkeeping
that the function \(u\) did throughout this section, and nothing beyond
this section's theorem is being claimed.
Space-Time Extension
One restriction has run through the whole page. The coefficients of an
Itô diffusion depend on the state and not on the clock, while the
equations that the existence pages solved were allowed coefficients
\(b( t, x )\) and \(\sigma( t, x )\). This closing section builds the
bridge between the two regimes, and the device is a single change of
viewpoint. Time becomes a coordinate. In one more dimension, a process
that watches the clock can be read as a time-homogeneous process, because
the enlarged state already contains the clock reading.
Several Components, and the Clock as One of Them
The bridge lives one dimension up, so this is the moment to record the
several-component reading of the page, as promised at the outset. Every
statement so far has a vector version. The state is
\(\mathbb{R}^m\)-valued, the driving motion is an \(n\)-dimensional
Brownian motion with independent components, the drift is a vector field
\(\mathbf{b} : \mathbb{R}^m \to \mathbb{R}^m\), the noise coefficient is
a matrix field \(\sigma : \mathbb{R}^m \to \mathbb{R}^{m \times n}\), and
the Lipschitz bound is read with Euclidean lengths on vectors and the
Frobenius length on matrices.
Under that bound the existence and uniqueness theory, the clock shift,
time homogeneity, and the Markov property all hold with the same
statements and the same proofs. The single-variable estimates run
componentwise, and the constants absorb the dimension. As with the systems
remark that closed the construction page, we record this reading without
proof and hold the one-dimensional versions as the proved objects of the
track. The one theorem of this section is stated against this reading, and
its proof isolates exactly where the reading is spent.
Theorem: The Space-Time Extension Is an Itô Diffusion
Let \(X\) be the Itô diffusion of the definition, let
\(x \in \mathbb{R}\), let \(\tau_0 \in \mathbb{R}\), and let
\(\varepsilon \in \{ +1, -1 \}\). Then the
\(\mathbb{R}^2\)-valued process
\[
\widehat{S}_t = \bigl( \tau_0 + \varepsilon t , X^{x}_t \bigr) ,
\quad t \geq 0 ,
\]
is an Itô diffusion in the several-component reading, with state
dimension \(2\), driving dimension \(1\), coefficients
\[
\widehat{\mathbf{b}}( \tau, x )
= \begin{pmatrix} \varepsilon \\ b( x ) \end{pmatrix} ,
\quad
\widehat{\sigma}( \tau, x )
= \begin{pmatrix} 0 \\ \sigma( x ) \end{pmatrix} ,
\]
and starting point \(( \tau_0, x )\).
Proof.
The coefficients are Lipschitz.
The first components of \(\widehat{\mathbf{b}}\) and
\(\widehat{\sigma}\) are constant, so for any two enlarged states,
\[
\begin{align*}
\bigl| \widehat{\mathbf{b}}( \tau, x ) - \widehat{\mathbf{b}}( \tau', x' ) \bigr|
+ \bigl| \widehat{\sigma}( \tau, x ) - \widehat{\sigma}( \tau', x' ) \bigr|
&= | b( x ) - b( x' ) | + | \sigma( x ) - \sigma( x' ) | \\\\
&\leq D\, | x - x' | \\\\
&\leq D\, \bigl| ( \tau, x ) - ( \tau', x' ) \bigr| ,
\end{align*}
\]
the last step because a Euclidean length dominates the length of any
single component. The enlarged pair satisfies the several-component
Lipschitz bound with the same constant \(D\), and it consists of
functions of the enlarged state alone, so it is time-independent in
the enlarged sense.
The process solves the enlarged system.
Componentwise, the first coordinate of \(\widehat{S}\) is the
deterministic function \(\tau_0 + \varepsilon t\), which is
continuous, adapted to any filtration, and satisfies
\(\tau_0 + \varepsilon t = \tau_0 + \int_0^t \varepsilon \, dr\) as an
ordinary integral identity, its noise row being zero. The second
coordinate is \(X^x\) itself, which satisfies its equation with the
four clauses by the first section of this page. The pair therefore
satisfies the enlarged integral equation from the start
\(( \tau_0, x )\), clause by clause and component by component.
Where the reading is spent.
What has been verified outright is that the enlarged coefficients
meet the several-component hypotheses and that \(\widehat{S}\) solves
the enlarged system. That the solution is unique, and that the
resulting object carries the Markov structure of this page, is the
several-component reading recorded above, taken here as stated and
not reproved. Nothing else is used.
The direction that matters most for what follows runs the clock backwards.
With \(\varepsilon = -1\) the first coordinate counts down from
\(\tau_0\), and a function of the pair \(( \tau_0 - t, X_t )\) is how a
terminal-value problem watches a diffusion. That configuration is put to
work, on a later page of this track, in the uniqueness half of the
equation governing \(u( t, x ) = E^x[ f( X_t ) ]\), and the theorem above
is the license it runs on.
The same device runs in reverse, and this is the bridge back to the
time-dependent world. Suppose the coefficients \(b( t, x )\) and
\(\sigma( t, x )\) are jointly Lipschitz in the pair \(( t, x )\). Then
the enlarged fields \(( \tau, x ) \mapsto ( 1, b( \tau, x ) )\) and
\(( \tau, x ) \mapsto ( 0, \sigma( \tau, x ) )^\top\) are Lipschitz on the
half-plane, and freezing the clock below zero extends them to
\(\mathbb{R}^2\) with the same bound. They are time-independent, so they
define an Itô diffusion in the enlarged sense, and reading its
second component from the start \(( s, x )\) recovers the time-dependent
equation on the clock shifted to \(s\). A time-dependent equation with
jointly Lipschitz coefficients is a time-homogeneous diffusion that
carries its own clock.
We caution that joint Lipschitz continuity is a genuine strengthening of
the standing hypotheses, which ask only joint measurability together with
the bounds (H1) and (H2), and so place no regularity on the time variable.
The fully general reduction, with merely measurable time dependence, needs
machinery beyond this page and is not claimed here.
Schedules and Clocks
The two regimes of this section separate two families of sampling
methods in machine learning by whether their dynamics watch the
clock. Langevin dynamics targets a fixed distribution, its drift is
the gradient of the target's log density when the noise coefficient
is \(\sqrt{2}\), and nothing in the dynamics depends on when the
process is observed. When that gradient is Lipschitz, the dynamics
are a time-homogeneous Itô diffusion, in exactly the sense of
this page when the state is one-dimensional and in the
several-component reading otherwise, and the questions one asks about
them, stationarity above all, are questions about the one-index
family \(( X^x )\). The forward corruption process of a
diffusion model
is the opposite case by design. Its coefficients follow a prescribed
time-dependent noise schedule, chosen so that the data distribution
degrades to noise on a fixed budget, and the schedule is the point.
The space-time extension lets one theory serve both, within limits.
A schedule whose coefficients are jointly Lipschitz in time and
state rides into the homogeneous framework as a clock coordinate,
subject to the caution recorded above, and the machinery of this
page, restart included, applies to the enlarged state. The common
variance-preserving schedules fall outside this reach, because their
drift multiplies a time-varying rate by the unbounded state, so no
single constant bounds the change of the drift in time uniformly
over the state.
The page is complete, and it changes what kind of object the track
studies. Before it, a stochastic differential equation was a problem and a
solution was an answer. After it, the solutions form a single family
indexed by the starting point, the family forgets the clock, and it
restarts from its own present position with no memory of how it got there.