Now choose β so large that

4κ 1

¤; (10.20)

β 2

i.e., let β ≥ 8κ. Then (10.19) implies that

|v(t)| ¤ 2|v(tn )| + 4|b x(tn ) |

sup

tn ¤t¤tn+1

|w(t) ’ w(tn )|

+ 4κ sup

(10.21)

tn ¤t¤tn+1

t

e’β(t’s) dw(s)|.

|β

+2 sup

tn ¤t¤tn+1 tn

Let

v(t)

·n = sup , (10.22)

β

tn ¤t¤tn+1

b x(tn )

ζn = , (10.23)

β

t

e’β(t’s) dw(s)

µn =2 sup

tn ¤t¤tn+1 tn

(10.24)

4κ

sup |w(t) ’ w(tn )|.

+

β tn ¤t¤tn+1

Recall that our task is to show that ·n ’ 0 with probability one for all n

as β ’ ∞. Suppose we can show that

µn ’ 0 (10.25)

BROWNIAN MOTION IN A FORCE FIELD 63

with probability one for all n as β ’ ∞. By (10.21),

·n ¤ 2·n’1 + 4ζn + µn (10.26)

where ·’1 = |v0 |/β, and by (10.18) for n ’ 1 and (10.20),

1 1

ζn ¤ 2ζn’1 + ·n’1 + µn . (10.27)

2 2

Now ζ0 = |b(x0 )|/β ’ 0 and ·’1 = |v0 |/β ’ 0, ζ1 ’ 0 by (10.27) and

(10.25), and consequently ·1 ’ 0. By induction, ζn ’ 0 and ·n ’ 0 for

all n. Therefore, we need only prove (10.25).

It is clear that the second term on the right hand side of (10.24)

converges to 0 with probability one as β ’ ∞, since w is continuous with

probability one. Let

w(t) ’ w(tn ), t ≥ tn ,

z(t) =

0, t < tn .

Then

t t

’β(t’s)

e’β(t’s) dz(s)

e dw(s) =

’∞

tn

t

e’β(t’s) z(s) ds + z(t).

= ’β

∞

This converges to 0 uniformly for tn ¤ t ¤ tn+1 with probability one, since

z is continuous with probability one. Therefore (10.25) holds. QED.

A possible physical objection to the theorem is that the initial velocity

v0 should not be held ¬xed as β varies but should have a Maxwellian

distribution (Gaussian with mean 0 and variance Dβ). Let v00 have a

1

Maxwellian distribution for a ¬xed value β = β0 . Then v0 = (β/β0 ) 2 v00

has a Maxwellian distribution for all β. Since it is still true that v0 /β ’ 0

as β ’ ∞, the theorem remains true with a Maxwellian initial velocity.

Theorem 10.1 has a corollary that can be expressed purely in the lan-

guage of partial equations:

PSEUDOTHEOREM 10.2 Let b : ‚ ’ ‚ satisfy a global Lipschitz con-

dition, and let D and β be strictly positive constants. Let f0 be a bounded

64 CHAPTER 10

‚. Let f on [0, ∞) — ‚ be the bounded solution

continuous function on

of

‚

f (t, x) = D∆x + b(x) · f (t, x); f (0, x) = f0 (x). (10.28)

x

‚t

Let gβ on [0, ∞) — ‚ — ‚ be the bounded solution of

‚

gβ (t, x, v) = β 2 D∆v + v · + β(b(x) ’ v) · gβ (t, x, v);

x v

‚t

gβ (0, x, v) = f0 (x). (10.29)

Then for all t, x, and v,

lim gβ (t, x, v) = f (t, x). (10.30)

β’∞

To prove this, notice that f (t, x0 ) = Ef0 y(t) and gβ (t, x0 , v0 ) =

Ef0 x(t) , since (10.28) and (10.29) are the backward Kolmogorov equa-

tions of the two processes. The result follows from Theorem 10.1 and the

Lebesgue dominated convergence theorem.

There is nothing wrong with this proof”only the formulation of the

result is at fault. Equation (10.28) is a parabolic equation with smooth

coe¬cients, and it is a classical result that it has a unique bounded so-

lution. However, (10.29) is not parabolic (it is of ¬rst order in x), so we

do not know that it has a unique bounded solution. One way around this

problem would be to let gβ,µ be the unique bounded solution of (10.29)

with the additional operator µ∆x on the right hand side and to prove that

gβ,µ (t, x0 , v0 ) ’ gβ (t, x0 , v0 ) = Ef0 x(t) as µ ’ 0. This would give us

a characterization of gβ purely in terms of partial di¬erential equations.

We shall not do this.

Reference

[26]. Eugen Kappler, Versuche zur Messung der Avogadro-Loschmidt-

schen Zahl aus der Brownschen Bewegung einer Drehwaage, Annalen der

Physik, 11 (1931), 233“256.

Chapter 11

Kinematics of stochastic

motion

We shall investigate the kinematics of motion in which chance plays a

rˆle (stochastic motion).

o

Let x(t) be the position of a particle at time t. What does it mean

to say that the particle has a velocity x(t)? It means that if ∆t is a very

™

short time interval then

x(t + ∆t) ’ x(t) = x(t)∆t + µ,

™

where µ is a very small percentage error. This is an assumption about

actual motion of particles that may not be true. Let us be conservative

and suppose that it is not necessarily true. (“Conservative” is a useful

word for mathematicians. It is used when introducing a hypothesis that

a physicist would regard as highly implausible.)

The particle should have some tendency to persist in uniform rectilin-

ear motion for very small intervals of time. Let us use Dx(t) to denote

the best prediction we can make, given any relevant information available

at time t, of

x(t + ∆t) ’ x(t)

∆t

for in¬nitely small positive ∆t.

Let us make this notion precise.

Let I be an interval that is open on the right, let x be an ‚ -valued

stochastic process indexed by I, and let Pt for t in I be an increasing

family of σ- algebras such that each x(t) is Pt -measurable. (This implies

65

66 CHAPTER 11

that Pt contains the σ-algebra generated by the x(s) with s ¤ t, s ∈ I.

Conversely, this family of σ-algebras satis¬es the hypotheses.) We shall

have occasion to introduce various regularity assumptions, denoted by

(R0), (R1), etc.

(R0). Each x(t) is in L 1 and t ’ x(t) is continuous from I into L 1 .

This is a very weak assumption and by no means implies that the

sample functions (trajectories) of the x process are continuous.

(R1). The condition (R0) holds and for each t in I,

x(t + ∆t) ’ x(t)

Pt

Dx(t) = lim E

∆t

∆t’0+

exists as a limit in L 1 , and t ’ Dx(t) is continuous from I into L 1 .

Here E{ |Pt } denotes the conditional expectation; cf. Doob [15, §6].

The notation ∆t ’ 0+ means that ∆t tends to 0 through positive values.

The random variable Dx(t) is automatically Pt -measurable. It is called

the mean forward derivative (or mean forward velocity if x(t) represents

position).

As an example of an (R1) process, let I = (’∞, ∞), let x(t) be the

position in the Ornstein-Uhlenbeck process, and let Pt be the σ-algebra

generated by the x(s) with s ¤ t. Then Dx(t) = dx(t)/dt = v(t). In

fact, if t ’ x(t) has a continuous strong derivative dx(t)/dt in L 1 , then

Dx(t) = dx(t)/dt. A second example of an (R1) process is a process x(t)

of the form discussed in Theorem 8.1, with I = [0, ∞), x(0) = x0 , and

Pt the σ-algebra generated by the x(s) with 0 ¤ s ¤ t. In this case

Dx(t) = b x(t) . The derivative dx(t)/dt does not exist in this exam-

ple unless w is identically 0. For a third example, let P t be a Markovian

semigroup on a locally compact Hausdor¬ space X with in¬nitesimal gen-

erator A, let I = [0, ∞), let ξ(t) be the X-valued random variables of the

Markov process for some initial measure, and let Pt be the σ-algebra

generated by the ξ(s) with 0 ¤ s ¤ t. If f is in the domain of the

in¬nitesimal generator A then x(t) = f ξ(t) is an (R1) process, and

Df ξ(t) = Af ξ(t) .

KINEMATICS OF STOCHASTIC MOTION 67

THEOREM 11.1 Let x be an (R1) process, and let a ¤ b, a ∈ I, b ∈ I.

Then

b

E{x(b) ’ x(a) Pa } = E Dx(s) ds Pa (11.1)

a

Notice that since s ’ Dx(s) is continuous in L 1 , the integral exists

as a Riemann integral in L 1 .

Proof. Let µ > 0 and let J be the set of all t in [a, b] such that

s

E{x(s) ’ x(a) | Pa } ’ E Dx(r) dr Pa ¤ µ(s ’ a) (11.2)

a 1

1 denotes the L norm. Clearly, a is in J,

1

for all a ¤ s ¤ t, where

and J is a closed subinterval of [a, b]. Let t be the right end-point of J,

and suppose that t < b. By the de¬nition of Dx(t), there is a δ > 0 such

that t + δ ¤ b and

µ

E{x(t + ∆t) ’ x(t) | Pt } ’ Dx(t)∆t ¤ ∆t

1

2

for 0 ¤ ∆t ¤ δ. Since conditional expectations reduce the L 1 norm and

since Pt © Pa = Pa ,

µ

E{x(t + ∆t) ’ x(t) | Pa } ’ E{Dx(t)∆t | Pa } ¤ ∆t (11.3)

1

2

for 0 ¤ ∆t ¤ δ. By reducing δ if necessary, we ¬nd

t+∆t

µ

Dx(t)∆t ’ ¤ ∆t

Dx(s) ds

2

t 1

for 0 ¤ ∆t ¤ δ, since s ’ Dx(s) is L 1 continuous. Therefore,

t+∆t

µ

E{Dx(t)∆t Pa } ’ E Dx(s) ds Pa ¤ ∆t (11.4)

2

t 1

for 0 ¤ ∆t ¤ δ. From (11.2) for s = t, (11.3), and (11.4), it follows

that (11.2) folds for all t + ∆t with 0 ¤ ∆t ¤ δ. This contradicts the

assumption that t is the end-point of J, so we must have t = b. Since µ