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¤ Ke»t β|x0 ’ y0 |.
f x(t) ’ f y(t)

Since this is true for each ¬xed w path, the estimate remains true when
we take expectations, so that

|P t f (x0 ) ’ P t f (y0 )| ¤ Ke»t β|x0 ’ y0 |.

Therefore, if f is a Lipschitz function in C(‚ ) then P t f is a bounded
continuous function. The Lipschitz functions are dense in C(‚ ) and P t
is a bounded linear operator. Consequently, if f is in C(‚ ) then P t f is
a bounded continuous function. We still need to show that it vanishes at
in¬nity. By uniqueness,
t
b x(r) dr + w(t) ’ w(s)
x(t) = x(s) +
s

for all 0 ¤ s ¤ t, so that
t
|x(t) ’ x(s)| ¤ [b x(r) ’ b x(t) ] dr + (t ’ s) b x(t)
s
+ |w(t) ’ w(s)|
t
¤κ |x(r) ’ x(t)| dr + t b x(t) + |w(t) ’ w(s)|
s
¤ κ sup |x(r) ’ x(t)| + t b x(t) + sup |w(t) ’ w(r)|.
0¤r¤t 0¤r¤t

Since this is true for each s, 0 ¤ s ¤ t,

sup |x(t) ’ x(s)| ¤ γ t b x(t) + sup |w(t) ’ w(s)| ,
0¤s¤t 0¤s¤t

where γ = 1/(1 ’ κt), provided that κt < 1. In particular, if κt < 1 then

|x(t) ’ x0 | ¤ γ t b x(t) + sup |w(t) ’ w(s)| . (8.7)
0¤s¤t
40 CHAPTER 8

Now let f be in Ccom (‚ ), let κt < 1, and let δ be the supremum of |b(z0 )|
for z0 in the support of f . By (8.7), f x(t) = 0 unless

|z0 ’ x0 | ¤ γ[ tδ + sup |w(t) ’ w(s)| ].
inf (8.8)
z0 ∈supp f 0¤s¤t


But as x0 tends to in¬nity, the probability that w will satisfy (8.8) tends
to 0. Since f is bounded, this means that Ef x(t) = P t f (x0 ) tends to 0
as x0 tends to in¬nity. We have already seen that P t f is continuous, so
P t f is in C(‚ ). Since Ccom (‚ ) is dense in C(‚ ) and P t is a bounded
linear operator, P t maps C(‚ ) into itself, provided κt < 1. This restric-
tion could have been avoided by introducing an exponential factor, but
this is not necessary, as we shall show that the P t form a semigroup.
Let 0 ¤ s ¤ t. The conditional distribution of x(t), with x(r) for all
0 ¤ r ¤ s given, is a function of x(s) alone, since the equation
t
b x(s ) ds + w(t) ’ w(s),
x(t) = x(s) +
s

has a unique solution. Thus the x process is a Markov process, and

E{f x(t) | x(r), 0 ¤ r ¤ s} = E{f x(t) | x(s)} = P t’s f x(s)

for f in C(‚ ), 0 ¤ s ¤ t. Therefore,

P t+s f (x0 ) = Ef (x(t + s)
= EE{f x(t + s) | x(r), 0 ¤ r ¤ s}
= EP t f x(s)
= P s P t f (x0 ),

so that P t+s = P t P s . It is clear that

sup P t f (x0 ) = 1
0¤f ¤1


for all x0 and t.
2 2
It remains only to prove (8.4) for f in Ccom (‚ ). (Since Ccom (‚ ) is
dense in C(‚ ) and the P t have norm one, this will imply that P t f ’ f
as t ’ 0 for all f in C(‚ ), so that P t is a Markovian semigroup.)
2
Let f be in Ccom (‚ ), and let K be a compact set containing the sup-
port of f in its interior. An argument entirely analogous to the derivation
A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS 41

of (8.7), with the subtraction and addition of b(x0 ) instead of b x(t) ,
gives
|x(t) ’ x0 | ¤ γ[ t |b(x0 )| + sup |w(0) ’ w(s)| ], (8.9)
0¤s¤t

provided κt < 1 (which we shall assume to be the case). Let x0 be
in the complement of K. Then f (x0 ) = 0 and f x(t) is also 0 unless
µ ¤ |x(t) ’ x0 |, where µ is the distance from the support of f to the
complement of K. But the probability that the right hand side of (8.9)
will be bigger than µ is o(t) (in fact, o(tn ) for all n) by familiar properties
of the Wiener process. Since f is bounded, this means that P t f (x0 ) is
uniformly o(t) for x0 in the complement of K, so that
P t f (x0 ) ’ f (x0 )
’ b(x0 ) · f (x0 ) + Cf (x0 ) = 0
t
uniformly for x0 in the complement of K. Now let x0 be in K. We have
t
t
b x(s) ds + w(t) ’ w(0) .
P f (x0 ) = Ef x(t) = Ef x0 +
0

De¬ne R(t) by
t
b x(s) ds + w(t) ’ w(0)
f x0 +
0
= f (x0 ) + tb(x0 ) · f (x0 ) + [w(t) ’ w(0)] · f (x0 )
‚2
1 i i j j
[w (t) ’ w (0)][w (t) ’ w (0)] i j f (x0 ) + R(t).
+
2 i,j ‚x ‚x

Then
P t f (x0 ) ’ f (x0 ) 1
= b(x0 ) · f (x0 ) + Cf (x0 ) + ER(t).
t t
By Taylor™s formula,
t
2
R(t) = o(|w(t) ’ w(0)| ) + o b x(s) ’ b(x0 ) ds .
0

Since E(|w(t) ’ w(0)|2 ) ¤ const. t, we need only show that
t
1
|b x(s) ’ b(x0 )|ds
E sup (8.10)
x0 ∈K t 0
42 CHAPTER 8

tends to 0. But (8.10) is less than
t
1
|x(s) ’ x0 |ds,
E sup κ
x0 ∈K t 0

which by (8.9) is less than
E sup κγ[ t |b(x0 )| + sup |w(0) ’ w(s)|]. (8.11)
x0 ∈K 0¤s¤t

The integrand in (8.11) is integrable and decreases to 0 as t ’ 0. QED.

Theorem 8.1 can be generalized in various ways. The ¬rst paragraph
of the theorem remains true if b is a continuous function of x and t that
satis¬es a global Lipschitz condition in x with a uniform Lipschitz con-
stant for each compact t-interval. The second paragraph needs to be
slightly modi¬ed as we no longer have a semigroup, but the proofs are
the same. Doob [15, §6, pp. 273“291], using K. Itˆ™s stochastic integrals
o
(see Chapter 11), has a much deeper generalization in which the matrix
cij depends on x and t. The restriction that b satisfy a global Lipschitz
condition is necessary in general. For example, if the matrix cij is 0 then
we have a system of ordinary di¬erential equations. However, if C is el-
liptic (that is, if the matrix cij is of positive type and non-singular) the
smoothness conditions on b can be greatly relaxed (cf. [20]).
We make the convention that
dx(t) = b x(t) dt + dw(t)
means that
t
x(t) ’ x(s) = b x(r) dr + w(t) ’ w(s)
s

for all t and s.

THEOREM 8.2 Let A : ‚ ’ ‚ be linear, let w be a Wiener process
on ‚ with in¬nitesimal generator (8.1), and let f : [0, ∞) ’ ‚ be
continuous. Then the solution of
dx(t) = Ax(t)dt + f (t)dt + dw(t), x(0) = x0 , (8.12)
for t ≥ 0 is
t t
At A(t’s)
eA(t’s) dw(s).
x(t) = e x0 + e f (s) ds + (8.13)
0 0
A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS 43

The x(t) are Gaussian with mean
t
At
eA(t’s) f (s) ds
Ex(t) = e x0 + (8.14)
0

and covariance r(t, s) = E x(t) ’ Ex(t) x(s) ’ Ex(s) given by
s T
eA(t’s) 0 eAr 2ceA r dr, t ≥ s
r(t, s) = (8.15)
t Ar T
e 2ceA r dreA(s’t) , t ¤ s.
0


The latter integral in (8.13) is a Wiener integral (as in Chapter 7). In
(8.15), AT denotes the transpose of A and c is the matrix with entries cij
occurring in (8.1).

Proof. De¬ne x(t) by (8.13). Integrate the last term in (8.13) by parts,
obtaining
t t
s=t
A(t’s)
AeA(t’s) w(s) ds + eA(t’s) w(s)
e dw(s) = s=0
0 0
t
AeA(t’s) w(s) ds + w(t) ’ eAt w(0).
=
0

It follows that x(t)’w(t) is di¬erentiable, and has derivative Ax(t)+f (t).
This proves that (8.12) holds.
The x(t) are clearly Gaussian with the mean (8.14). Suppose that
t ≥ s. Then the covariance is given by

Exi (t)xj (s) ’ Exi (t)Exj (s)
t s
A(t’t1 )
eA(s’s1 )
=E e dwk (t1 ) dwh (s1 )
ik jh
0 0
k h
s
eA(t’r) 2ckh eA(s’r)
= dr
ik jh
0 k,h
s
T (s’r)
eA(t’r) 2ceA
= dr
ij
0
s
T
A(t’s)
eAr 2ceA r dr
= e .
0 ij

The case t ¤ s is analogous. QED.
44 CHAPTER 8

Reference

[20]. Edward Nelson, Les ´coulements incompressibles d™´nergie ¬nie,
e e
Colloques internationaux du Centre national de la recherche scienti¬que
´
No 117, “Les ´quations aux d´riv´es partielles”, Editions du C.N.R.S.,
e ee
Paris, 1962. (The last statement in section II is incorrect.)
Chapter 9

The Ornstein-Uhlenbeck
theory of Brownian motion

The theory of Brownian motion developed by Einstein and Smolu-

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. 6
( 18 .)



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