Chaudesaigues [6, §3]. Rather surprisingly, considering the number of

assumptions that went into the argument, the result obtained for Avo-

gadro™s number agreed to within 19% of the modern value obtained by

other means. Notice how the points 3“6 of Chapter 3 are re¬‚ected in the

formula (4.7).

Einstein™s argument does not give a dynamical theory of Brownian

motion; it only determines the nature of the motion and the value of

the di¬usion coe¬cient on the basis of some assumptions. Smoluchowski,

independently of Einstein, attempted a dynamical theory, and arrived

at (4.5) with a factor of 32/27 of the right hand side. Langevin gave

ALBERT EINSTEIN 17

another derivation of (4.5) which was the starting point for the work of

Ornstein and Uhlenbeck, which we shall discuss later (Chapters 9“10).

Langevin is the founder of the theory of stochastic di¬erential equations

(which is the subject matter of these lectures).

Einstein™s work was of great importance in physics, for it showed in

a visible and concrete way that atoms are real. Quoting from Einstein™s

Autobiographical Notes again [12, p. 49]:

“The agreement of these considerations with experience together with

Planck™s determination of the true molecular size from the law of radiation

(for high temperatures) convinced the sceptics, who were quite numerous

at that time (Ostwald, Mach) of the reality of atoms. The antipathy of

these scholars towards atomic theory can indubitably be traced back to

their positivistic philosophical attitude. This is an interesting example

of the fact that even scholars of audacious spirit and ¬ne instinct can be

obstructed in the interpretation of facts by philosophical prejudices.”

Let us not be too hasty in adducing any other interesting example

that may spring to mind.

Reference

[12]. Paul Arthur Schilpp, editor, “Albert Einstein: Philosopher-Scien-

tist”, The Library of Living Philosophers, Vol. VII, The Library of Living

Philosophers, Inc., Evanston, Illinois, 1949.

Chapter 5

Derivation of the Wiener

process

Einstein™s basic assumption is that the following is possible [10, §3,

p. 13]: “We will introduce a time-interval „ in our discussion, which is to

be very small compared with the observed interval of time [i.e., the inter-

val of time between observations], but, nevertheless, of such a magnitude

that the movements executed by a particle in two consecutive intervals

of time „ are to be considered as mutually independent phenomena.”

He then implicitly considers the limiting case „ ’ 0. This assumption

has been criticized by many people, including Einstein himself, and later

on (Chapter 9“10) we shall discuss a theory in which this assumption is

modi¬ed. Einstein™s derivation of the transition probabilities proceeds by

formal manipulations of power series. His neglect of higher order terms is

tantamount to the assumption (5.2) below. In the theorem below, pt may

be thought of as the probability distribution at time t of the x-coordinate

of a Brownian particle starting at x = 0 at t = 0. The proof is taken from

a paper of Hunt [13], who showed that Fourier analysis is not the natural

tool for problems of this type.

THEOREM 5.1 Let pt , 0 ¤ t < ∞, be a family of probability measures

on the real line ‚ such that

pt — ps = pt+s ; 0 ¤ t, s < ∞, (5.1)

where — denotes convolution; for each µ > 0,

pt ({x : |x| ≥ µ}) = o(t), t ’ 0; (5.2)

19

20 CHAPTER 5

and for each t > 0, pt is invariant under the transformation x ’ ’x.

Then either pt = δ for all t ≥ 0 or there is a D > 0 such that, for all

t > 0, pt has the density

1 x2

’ 4Dt

p(t, x) = √ e ,

4πDt

so that p satis¬es the di¬usion equation

‚2p

‚p

= D 2, t > 0.

‚t ‚x

First we need a lemma:

THEOREM 5.2 Let X be a real Banach space, f ∈ X , D a dense linear

subspace of X , u1 , . . . , un continuous linear functionals on X , δ > 0.

Then there exists a g ∈ D with

f ’g ¤δ

(u1 , f ) = (u1 , g), . . . , (un , f ) = (un , g).

Proof. Let us instead prove that if X is a real Banach space, D a

dense convex subset, M a closed a¬ne hyperplane, then D © M is dense

in M . Then the general case of ¬nite co-dimension follows by induction.

Without loss of generality, we can assume that M is linear (0 ∈ M ),

so that, if we let e be an element of X not in M ,

X = M • ‚e.

Let f ∈ M , µ > 0. Choose g+ in D so that

(f + e) ’ g+ ¤ µ

and choose g’ in D so that

(f ’ e) ’ g’ ¤ µ.

Set

m+ ∈ M

g+ = m+ + r+ e,

m’ ∈ M .

g’ = m’ + r’ e,

DERIVATION OF THE WIENER PROCESS 21

Since M is closed, the linear functional that assigns to each element of X

the corresponding coe¬cient of e is continuous. Therefore r+ and r’ tend

to 1 as µ ’ 0 and so are strictly positive for µ su¬ciently small. By the

convexity of D,

r’ g+ + r+ g’

g=

r’ + r +

is then in D. But

r’ m + + r + m ’

g=

r’ + r +

is also in M , and it converges to f as µ ’ 0. QED.

We recall that if X is a Banach space, then a contraction semigroup

on X (in our terminology) is a family of bounded linear transforma-

tions P t of X into itself, de¬ned for 0 ¤ t < ∞, such that P 0 = 1,

P t P s = P t+s , P t f ’ f ’ 0, and P t ¤ 1, for all 0 ¤ t, s < ∞ and

all f in X . The in¬nitesimal generator A is de¬ned by

P tf ’ f

Af = lim

t

t’0+

on the domain D(A) of all f for which the limit exists.

If X is a locally compact Hausdor¬ space, C(X) denotes the Banach

space of all continuous functions vanishing at in¬nity in the norm

f = sup |f (x)|,

x∈X

™

and X denotes the one-point compacti¬cation of X. We denote by

Ccom (‚ ) the set of all functions of class C 2 with compact support on ‚ ,

2

by C 2 (‚ ) its completion in the norm

‚2f

‚f

†

f =f+ + ,

‚xi ‚xi ‚xj

i=1 i,j=1

and by C 2 (‚ ) the completion of Ccom (‚ ) together with the constants,

™ ™

2

in the same norm.

22 CHAPTER 5

A Markovian semigroup on C(X) is a contraction semigroup on C(X)

such that f ≥ 0 implies P t f ≥ 0 for 0 ¤ t < ∞, and such that for all x

in X and 0 ¤ t < ∞,

sup P t f (x) = 1.

0¤f ¤1

f ∈C(X)

If X is compact, the last condition is equivalent to P t 1 = 1, 0 ¤ t < ∞.

By the Riesz theorem, there is a unique regular Borel probability measure

pt (x, ·) such that

P t f (x) = f (y) pt (x, dy),

and pt is called the kernel of P t .

THEOREM 5.3 Let P t be a Markovian semigroup on C(‚ ) commuting

™

with translations, and let A be the in¬nitesimal generator of P t . Then

C 2 (‚ ) ⊆ D(A).

™

Proof. Since P t commutes with translations, P t leaves C 2 (‚ ) invari-

™

ant and is a contraction semigroup on it. Let A† be the in¬nitesimal

generator of P t on C 2 (‚ ). Clearly D(A† ) ⊆ D(A), and since the domain

™

of the in¬nitesimal generator is always dense, D(A) © C 2 (‚ ) is dense in

™

C 2 (‚ ).

™

Let ψ be in C 2 (‚ ) and such that ψ(x) = |x|2 in a neighborhood

™

of 0, ψ(x) = 1 in a neighborhood of in¬nity, and ψ is strictly positive

on ‚ ’ {0}. Apply Theorem 5.2 to X = C 2 (‚ ), D = D(A) © C 2 (‚ ),

™ ™ ™

f = ψ, and to the continuous linear functionals mapping • in X to

‚2•

‚•

•(0), (0), (0).

‚xi ‚xi ‚xj

Then, for all µ > 0, there is a • in D(A) © C 2 (‚ ) with

™

‚2•

‚•

•(0) = (0) = 0, (0) = 2δij ,

‚xi ‚xi ‚xj

DERIVATION OF THE WIENER PROCESS 23

and • ’ ψ ¤ µ. If µ is small enough, • must be strictly positive on

‚ ’ {0}. Fix such a •, and let δ > 0, f ∈ C 2(‚ ). By Theorem 5.2 again

™ ™

there is a g in D(A) © C 2 (‚ ) with

™

|f (y) ’ g(y)| ¤ δ•(y)

for all y in ‚ . Now

™

1 δ

|f (y) ’ g(y)| pt (0, dy) ¤ •(y) pt (0, dy)

t t

and since • ∈ D(A) with •(0) = 0, the right hand side is O(δ). Therefore

1

[f (y) ’ f (0)] pt (0, dy) (5.3)

t

and

1

[g(y) ’ g(0)] pt (0, dy) (5.4)

t

di¬er by O(δ). Since g ∈ D(A), (5.4) has a limit as t ’ 0. Since δ is

arbitrary, (5.3) has a limit as t ’ 0. Therefore (5.3) is bounded as t ’ 0.

Since this is true for each f in the Banach space C 2 (‚ ), by the principle

™

of uniform boundedness there is a constant K such that for all f in C 2 (‚ )

™

and t > 0,

1t

(P f ’ f )(0) ¤ K f † .

t

By translation invariance,

1t

(P f ’ f ) ¤ K f † .

t

Now 1 (P t g ’ g) ’ Ag for all g in the dense set D(A† ) © C 2 (‚ ), so by

™

t

the Banach-Steinhaus theorem, 1 (P t f ’ f ) converges in C(‚ ) for all f