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 at the end of the page, 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. Freezing the time variable, condition (H2) is the displayed bound
verbatim, with the same constant \(D\) for every \(t\). For (H1), the
triangle inequality gives
\[
\begin{align*}
|b(x)| + |\sigma(x)|
&\leq |b(0)| + |\sigma(0)| + |b(x) - b(0)| + |\sigma(x) - \sigma(0)| \\\\
&\leq |b(0)| + |\sigma(0)| + D\, |x| \\\\
&\leq C \bigl( 1 + |x| \bigr) ,
\quad C = |b(0)| + |\sigma(0)| + D .
\end{align*}
\]
A Lipschitz pair of coefficients therefore sits inside the hypotheses of
the theorem, and everything proved on the previous two pages 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. Define
\[
\tilde{w}_v = w_{s+v} - w_s ,
\quad v \geq 0 .
\]
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. So \(\tilde{w}\) is itself a standard
Brownian motion. Pair it 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
reads its integrals on the original clock, from \(s\) to \(t\) against
\(w\), while \(Y\) solves an equation read on the shifted clock, from
\(0\) to \(t - s\) against \(\tilde{w}\). For the time integral the two
readings agree path by path, by the substitution \(u = s + v\) inside an
ordinary Lebesgue integral. For the stochastic integral the agreement is a
genuine lemma about the Itô construction, and the next section proves it
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 the shifted data
\[
\tilde{f}_v = f_{s+v} ,
\quad
\tilde{w}_v = w_{s+v} - w_s ,
\quad
\mathcal{G}_v = \mathcal{F}_{s+v} ,
\quad v \geq 0 .
\]
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 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 \([s, s+h] \times \Omega\). The reduction of a general
integrand to this bounded class is the
truncation bridge.
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. Applied, together with its mirror image under
the inverse relabeling \(v = u - s\), with \(f_u = \sigma( X^{s,x}_u )\),
whose shifted-clock membership is the third clause of the solution
definition, and combined with the pathwise substitution for the drift
term, the lemma says 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 filtration has
the two properties of the enlargement lemma, verified for
\((\tilde{w}, \{\mathcal{G}_v\})\) in the previous section, whose
motion is standard. 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 \(t \in [0, T]\), 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 \([0, T]\). 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 \(t\), and the
conditional law of the future position \(X^x_{t+h}\) depends on that past
through the present position \(X^x_t\) alone, and 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
\[
E^y \bigl[ f( X_h ) \bigr]
:= \mathbb{E} \bigl[ f( X^{y}_h ) \bigr] ,
\quad
u(y) := E^y \bigl[ f( X_h ) \bigr] .
\]
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. This section erects the first two pillars, and the next
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. 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 every measurability assertion in the rest of the page is
manufactured from 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 the horizon
\(T = h\) with its constant \(A = 3 ( 1 + h )\, D^2\), 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 summable in \(m\) and 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, and the stability
bound it needs is available there because the uniqueness argument
travels, as recorded twice already. 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 last pillar of the proof 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, so
every event built from finitely many increment constraints over this
grid 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
\(\varphi\) dominating the whole sequence on the finite measure
\(\mathbb{P}\), so \(h_\varphi\) 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
monotone-class step this track has taken on faith since the
construction of the integral. We extend the same trust to this one
application.
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 next section 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 its sentence 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 independent of the entire
shifted motion, by the independence lemma, so the existence theorem
applies over the shifted setup in its literal reading to 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 \(y_i\),
\[
\Xi^Z_h
= \sum_i \mathbf{1}_{A_i}\, \tilde{Y}^{y_i}_h
= \sum_i \mathbf{1}_{A_i}\, \tilde{\Phi}_h( y_i, \cdot )
= \tilde{\Phi}_h( Z, \cdot )
\quad \text{almost surely.}
\]
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
\(\varphi_M( (u, v), \omega )
= \min\bigl( | \tilde{\Phi}_h( u, \omega )
- \tilde{\Phi}_h( v, \omega ) |^2, M \bigr)\)
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
\[
\mathbb{E} \bigl[ \varphi_M( ( U, V ), \cdot ) \bigr]
= \mathbb{E} \bigl[ h_M( U, V ) \bigr] ,
\quad
h_M( u, v ) = \mathbb{E} \bigl[ \varphi_M( ( u, v ), \cdot ) \bigr] .
\]
For fixed \(u\) and \(v\) the version property converts
\(h_M( u, v )\) 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 gives
\(h_M( u, v ) \leq 3\, | u - v |^2 e^{A h}\). 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.
Subtracting the integral equation of \(X^x\) at time \(t\) from the
one at time \(t + v\), using
additivity over intervals,
and converting both remainders to the shifted clock, the drift by the
pathwise substitution \(u = t + r\) and the noise by the
clock shift lemma, gives
\[
X^x_{t+v}
= Z
+ \int_0^v b( X^x_{t+r} )\, dr
+ \int_0^v \sigma( X^x_{t+r} )\, d \tilde{w}_r ,
\quad v \geq 0 .
\]
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
\[
\mathbb{E} \bigl[ f( X^x_{t+h} ) \bigm| \mathcal{F}_t \bigr]
= \mathbb{E} \bigl[ \varphi( Z, \cdot ) \bigm| \mathcal{F}_t \bigr]
= h_f( Z )
\quad \text{almost surely,}
\]
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,
\[
h_f( y )
= \mathbb{E} \bigl[ f( X^{t,y}_{t+h} ) \bigr]
= \mathbb{E} \bigl[ f( X^{y}_{h} ) \bigr]
= u( y )
\quad \text{for every } y \in \mathbb{R} .
\]
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
\[
\mathbb{E} \bigl[ f( X^x_{t+h} ) \bigm| \mathcal{F}^X_t \bigr]
= \mathbb{E} \bigl[ u( X^x_t ) \bigm| \mathcal{F}^X_t \bigr]
= u( X^x_t )
\quad \text{almost surely,}
\]
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, and the work of this section is
exactly what that upgrade costs. Second, a notational
convention. Downstream 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. A process that watches the clock is a
time-homogeneous process in one more dimension, 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 running componentwise and the constants
absorbing 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 the coefficients
ignore the clock coordinate, so they are 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
\(d T_t = \varepsilon \, dt\) with \(T_0 = \tau_0\) 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 downstream 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, extend to \(\mathbb{R}^2\) by freezing the clock below
zero, which preserves the bound, and 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 (H1) and (H2), which ask only measurability in the time
variable at fixed state. 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 split the sampling methods of modern
machine learning cleanly down the middle. Langevin dynamics targets a
fixed distribution, its drift is the gradient of the target's log
density, and nothing in the dynamics depends on when the process is
observed. It is a time-homogeneous Itô diffusion in exactly the sense
of this page, and the questions one asks about it, 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 is what lets one theory serve both. A smooth
schedule rides into the homogeneous framework as a clock coordinate,
and the machinery of this page, restart included, applies to the
enlarged state. When a later page derives the equation governing the
evolving marginal densities of such processes, the time variable in
that equation is this section's extra coordinate doing its work.
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. One question presses immediately. The restart time \(t\) was
deterministic throughout, yet the natural restart moments of a diffusion
are read off the path itself, the first exit from an interval above all.
Whether the Markov property survives such random restart times, and what
a random time must satisfy for the question to make sense, is settled by
the theory of
stopping times and the Brownian restart.