Pacific Journal of
Mathematics
ON BROWNIAN MOTION IN A HOMOGENEOUS
RIEMANNIAN SPACE
KOSAKU YOSIDA
Vol. 2, No. 2
February 1952
ON BROWNIAN MOTION IN A HOMOGENEOUS
RIEMANNIAN SPACE
KOSAKU YOSIDA
1. Introduction. Let R be an /z-dimensional, orientable, infinitely differentiable Riemannian space such that the group G of isometric transformations S of
R onto R constitutes a Lie group transitive on R Consider a temporally homogeneous Markoff process in A , and let P (t, x, E) be the transition probability that
the point x £ R is, by this process, transferred into a Borel set E C R after the
lapse of t units of time, t > 0. We assume that P (t, x, E) is, for fixed (ί, x),
countably additive for Borel sets E and, for fixed {t, E), Borel measurable in x.
Then we must have the probability condition
(1.1)
P(t,x,E)
> 0 , P(t,x,R)
= 1,
and Smoluchowski's equation
(1.2)
P(t + s,x,E)
= fRP(t,x,dy)
(t, s > 0 ) .
P(s,y,E)
We further assume that the process is spatially homogeneous:
(1.3)
P (ί, x, E) = P (ί, S*x, S*£)
for every S* C G.
The purpose of the present note is to prove the following:
THEOREM 1, Let x0
be any point of R and assume that the Lie subgroup
{S* G G; S*x0 - χQ\ of G is compact1. Let us denote by d(xf y) the distance of
two points x, y ζl R. Then the continuity
(1.4)
lim
1
—
f
J
£-0+
t
d (x, y) > β
condition:
P (ί, x, dy) = 0
for any
6 > 0,
implies the condition of Lindeberg's type:
At first, this condition was overlooked. Mr. Seizo Ito kindly remarked that this
condition is necessary for the convergence of the integral (2.11) below.
Received August 30, 1951.
Pacific J. Math. 2 (1952), 263-270
263
264
KOSAKU YOSIDA
d(x,γ)2
7
ΓJ
1 r
" JR "
h m
(1.5)
P(t,x,dy)
From t h i s theorem we may deduce:
THEOREM 2. The finite
(1.6)
α* (*) = Hm
-
ί -* 0+
61'' (*) = lim
(1.7)
{x = ( x 1 , x 2 , . . . , xn))
limits
d (%, y) < €
—
—
ί-»0+
ί
(yι - %') (γi - *>> P (ί, x, dy)
//
rf(x,y)<6
exists independently of the sufficiently
function fo(x)
(y* - %') P (ί, %, rfy),
/
£
be such that fo(x),
small € > 0. Moreover, if a real-valued
dfo/dxι,
d2fQ/dxιdx]
are bounded and uni-
formly continuous on R, then
lim
(1.8)
t
->0+
1 ( ί fQ (y) P (t, x, dy) - fQ (x)) = a1 (*) ψr
R
t
ox
+ W (x)
- ^ j
dxι
dxJ
REMARK. In the literature [4; 2; 6], (1.8) is derived by assuming the condition of Lindeberg's type:
(1.5)'
lim
(ί'd(x,y)
0+
ί » 0+
and
3
P ( t , x , dy)/ f
R
some differentiability
d(x, y)2 P ( t , x , d γ ) )
=0
K
hypothesis
concerning P (t, x, E). Considering the
Brownian motion on the real line, Seizδ Itδ raised the question whether, under
the condition of the spatial homogeneity (1.3), "the (almost sure) continuity of
the sample motions of the temporally homogeneous Markoff process" which is
equivalent to the continuity condition (1.4), would be sufficient to derive Theorem 2. And he proved Theorem 2 in the special case where R = G and G is a
maximally almost periodic Lie group. The present note gives an extension of his
result to general homogeneous space, without the hypothesis of the maximal
almost periodicity of the group G of motions of the space R. Thus we may define
the Brownian motions in a homogeneous Riemannian space R as temporally and
spatially homogeneous Markoff processes satisfying the condition (1.4) of continuity.
2. Preliminaries. Let us denote by C (R) the totality of real-valued bounded
functions f(x)
on R which are uniformly continuous on /?. The space C(R) is a
Banach space by the norm
ON BROWNIAN MOTION IN A HOMOGENEOUS RIEMANNIAN SPACE
(2.D
11/11 = sup
265
\f(χ)\.
X
We define, for any / C C{R),
(2.2)
(7ίf)(*)= fR
P(t,x,dγ)f(y);
then we have, by (1.1),
(2.3)
s u p\{Ttf)
(x)\ < s u p
X
\f(x)\.
X
We have, by (1.3),
(2.3)
(Ttf) (S**) = ΓP(t, S*x, dy) f(y) = ί P[t, S* xf d(S*y)]
R
f{S* y)
R
= JΓ P(t,x,dy)f(S*γ),
and hence the commutativity
(2.4)
TtS = STt9
where S is defined by
(2.5)
{Sf) {%) = f(S*x),
S*CG.
Thus, if S* C G be such that S*% = x\ we have
(2.6)
(Ttf)(x) - (Ttf)(xΊ = σ,/)(*) ^ (STtf)(x) = Tt(f-Sf){x).
By the uniform continuity of /(%), and by (2.3) and (2.6), we see that (Ttf)(x)
is
bounded and uniformly continuous in x. Hence Tt defines a bounded linear transformation on C (R) into C {R) such that
(2-7)
| | Γ t | | = sup | | Γ t / | | = 1 .
||/ lί — A
We have, from (1.2),
(2.8)
Tt+S = Tt Ts
(t,s > 0 ) .
We have also, from (1.1),
(Ttf)
(x) -fix)'
LP (ί, * , dy) [/(y) - / ( * ) ]
R
S
d(x,y)<€
P(t,x,dy)[f(γ)
-f{x)]+
S
d(x,y)>e
P(t,x,dy)[f(y)-f{x)}.
266
KOSAKU YOSΪDA
Thus, in view of conditions (1.4) and (1.1), and the uniform continuity of f(x)9
we have
(2.9)
lim
t-> o +
(Ttf) (x) = f(χ)
boundedly in*.
Hence Tt is weakly continuous in ί, and therefore, by (2.8) and N. Dunford's
theorem [l], Tt is strongly continuous in t and
(2.9)'
strong lim
ί~»o+
Ttf « /; that is, lim
t-+ o +
|| Ttf-f\\ = 0.
Therefore we may apply the theory [3; 5] of one-parameter semigroups of
bounded linear operators to the semigroup {Tt\. In particular, we have the result:
(2.10) strong lim (Ttf-f)/t
= Af exists, for those / which constitute a
linear subset D (A) of C {R) which is dense in C{R). Moreover, A is a
closed linear operator defined on D (A) C C (R) with values in C (/?).
LEMMA. Let g(x) £ C(R) vanish outside a compact set. Then the convolution
(2.1D
(/ ® g) (*) = JΓ f{S*x) g(S*xo)dy
belongs to D{A) if f belongs to D{A). Here S^ is a general element of G, dy is
a right invariant Haar measure of G, and x0 is any fixed point of R.
Proof. The integral may be approximated by the Riemann sum
(2.12)
T /(S* x)ci
i = 1
uniformly in x. This we see by the uniform continuity of f(x) and the fact that
g{x) vanishes outside a compact set. We know, from (2.4), that A is commutative
with every Sy\
(2.13)
f€D(A) implies Syf£D(A) and SyAf=ASyf.
Hence (2.12) belongs to D (A), and we have
ΪTV
771
(2.14)
A ( Y f(St x ) c i ) = A { Y
1 =1
1= 1
Πl
(Sγ f) { x ) a ) = V ( S y h) ( x ) c i t
1 = 1
where h = Af. Therefore, since h £ C(R), we see that (2.14) converges, when
ON BROWNIAN MOTION IN A HOMOGENEOUS RIEMANNIAN SPACE
267
m—>oo, to a function £ C (R) uniformly in x. Since A is a closed operator, we
must have (f ® g) (x) £ D (A).
COROLLARY 1. The convolution (f ® g) (x) is infinitely differentiable if
g{x) is infinitely differentiable.
Proof. It is possible, for sufficiently small d{x, x0), to choose S* (x) £G
such that
(2.15)
S*(x)x = x0 and S* (x)x0 depends analytically on a 1 , . . . , xn .
This we see from the fact that the set {Sί £ G; Stx = xo\ forms an analytic
submanifold of G; it is one of the cosets of G with respect to the Lie subgroup
{Sy £ G; SyX0 ~ x0}. Hence, by the right invariance of dy, we have
(2.16)
(f® g) U) = j£ f(S*S*{x)x)
(g(S*yS*(x)x0)dy
= fGf(S*yχ0)g(S*yS*{χ)χ0)dy
The right side is infinitely differentiable in the vicinity of x0, and
(2.17)
a +
dHl
+ Hnn
(f ® g) (x)
1
q
^
= JG f(Syx0)
(
ι
belongs to C(/?)
COROLLARY 2. (i) There exist infinitely differentiable
n
F (%),
, F (x) £ D (A) such that the Jacobian
functions
ι
F (x),
2
(9^o^
\ΛΛo)
d(Fι(x), ...
,Fn(x))
_ _ ^ _ _ _ ^ _ _ _
does not vanish at x — x0 .
(ii) There exists an infinitely differentiable function Fo (x) £ D (A) such that
()
u £ 444
)(>4)
2
Σ
(
'
4
)
Σ
4
d
ι
dx 0
J
dx 0
i -l
Proof. In (2.16),/belongs to D (A), which is dense in C(R); andg(*) £ C (R)
268
KOSAKU YOSIDA
is arbitrary except that g(x)
must vanish outside a compact set. Thus, by taking
F {%) - (f ® g) (x) suitably, we may prove (i) and (ii).
3. Proof of Theorem 1. Because of their functional independence, we may
take
Fι (x), ••• ,Fn{x)
d(x, x0)
(3.1)
as
local coordinates of the points x which
satisfy
ι
< € for sufficiently small £ > 0. Since F (x) C D (A),
l
a finite limit —
t -> o+ t
r
J (Fι (x) - Fι (x0))
P (ί, x0, dx) e x i s t s
R
(i = 1, .
, τι).
Because of (1.4), this limit is equal to
(3.1)'
lim
1
' - 0 +
(Fί(x)~Fi(x0))P(t,x,dx),
/
ί
d(x,xΌ)<€
independently of the positive constant 6. We shall denote t h e s e new local coordinates Fι(x),
F2(x),
(3.1)r/
—
lim
••• , Fn(x)
by the letters xι,
x2, . . . , * \ Then
(xι - xι0) P {t, x, dx) = aι(xQ)
/
exists
(£ = 1, ••• , n),
independently of 6 > 0. The function F0(x)
(3.2)
U F 0 ) ( * 0 ) = lim
*-o+ t
belongs to D(A); hence, by (1.4),
/
d(x,
{F0(x)-F0{xQ))P{t,x09dx)f
xo)<e
independently of 6 > 0. This limit is equal to
lim
—
ι
J
{x
d {x, xQ) < €
/
ι
- x 0)
,
dxι0
Uι4)(^4) (
td(x,xo)<6
P(t, x0, dx)
Θ F
.° )
l
\dx
]
dx jx
=XQ + θ(χ -χ0)
P(t,xo,dx),
J
0 < θ < 1.
The first term in [] has the limit
ON BROWNIAN MOTION IN A HOMOGENEOUS RIEMANNIAN SPACE
269
and hence the second term h a s a limit. Thus, by virtue of (1.1) and (2.19),
1
/.
lim —
J
- ° + ' d(x,χo)<e
(3.3)
"
£
|
i
l
2
(x -x o) P{t,xo,dx)<n.
f =ι
Hence, by (1.1) and Schwarz's inequality,
r
1
.
.
.
.
—
J
(xι - % Q ) (χJ ~*o) PU* *o> dx) is bounded in t > 0 .
t
d(χ, xQ) < e
(3.4)
Therefore, by (1.4), we obtain (1.5).
4. Proof of Theorem 2. Since cij(e) is of order 6, we have
(4.1)
=t
+ 1
UVΛ, *
0
,
+
(%ι'-χ£) (χi-χi0) P(t,xo,dx)
/
-
;x c
—
()x0
^
ΘXQ
f
"—
(*ι-xι0)PU,x0,dx)
/
d(χ, χo)<e
ι
ΘXJ0
(γ*'• __γi \ ( J __ J )r . . ( f) P (f γ Jγ)
J
\X
XQ ) \X
X
0 I Cιj
\^f ί
\ ί , XQ , LLX)
t d{x, xQ) < £
+ —
/
1
(fo(x) - fo(xQ))
P(t, x0, dx)
d(x,xo)>β
Now
(4.2)
lim
/ t ( ί , 6) = a ' ( x 0 )
^- by (3.1)";
lim
I4(t, e) = 0 by (1.4).
lim
/ 3 (ί, e) = 0 by (3.4);
On the other hand, by (1.4) and (3.4), the finite limits
ς
J_
1
/
-*o+
.
i
.
ί d(x, x0) < e
.
i\ j
j
.
.
)
ij
270
KOSAKU YOSIDA
exist and are independent of 6 > 0. Let u s , in place of fo(x),
the form (f Θ g) (x).
We may choose F0(x)
ι
such that dF0/dx 0
take F0(x)
dx^0
of
assumes
v a l u e s arbitrarily near to given c o n s t a n t s CCjy (i, / = 1, ••• , n). T h u s , by (4.1)
and (4.2) and the fact F0(x)
l
χ
b j ( o)»
G D (Λ)9
ι
we see that b ^{x0)
must be equal to
Hence (1.7) is proved.
Therefore, by (1.4), ( 3 . 1 ) " , and (4.2), we obtain (1.8).
REFERENCES
1. Nelson Dunford, On one-parameter groups of linear transformations, Ann. of Math.
39(1938), 569-573.
2. W. Feller,
113-160.
Zur Theorie der stochastischen
3. E. flille, Functional
New York, 1948.
analysis
Prozesse,
Math. Ann. 113
(1936),
and semi-groups, American Mathematical Society,
4. A. Kolmogoroff, Zur Theorie der stetigen
(1933), 149-160.
zufalligen
Prozesse,
Math. Ann. 108
5. K. Yosida, On the differentiability and the representation of one-parameter semigroups of linear operators, J. Math. Soc. Japan 1 (1948), 15-21.
6.
No. 9, 1-3.
, An extension of Fokker-Planck9s
MATHEMATICAL INSTITUTE,
NAGOYA
UNIVERSITY.
equation, Proc. Japan Acad. 25(1949),
PACIFIC JOURNAL OF MATHEMATICS
EDITORS
R. M. ROBINSON
*R. P. DILWORTH
University of California
Berkeley 4, California
California Institute of Technology
Pasadena 4, California
E. F. BECKENBACH, Managing Editor
University of California
Los Angeles 24, California
*During the absence of Herbert Busemann in 1952.
ASSOCIATE EDITORS
R. P . DILWORTH
P . R. HALMOS
B(?RGE JESSEN
J. J. STOKER
HERBERT FEDERER
HEINZ HOPF
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E . G. STRAUS
MARSHALL HALL
R. D. JAMES
GEORGE POLYA
KOSAKU YOSIDA
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COPYRIGHT 1952 BY PACIFIC JOURNAL OF MATHEMATICS
Pacific Journal of Mathematics
Vol. 2, No. 2
February, 1952
L. Carlitz, Some theorems on Bernoulli numbers of higher order . . . . . . . . . . .
Watson Bryan Fulks, On the boundary values of solutions of the heat
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
John W. Green, On the level surfaces of potentials of masses with fixed
center of gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Isidore Heller, Contributions to the theory of divergent series . . . . . . . . . . . . .
Melvin Henriksen, On the ideal structure of the ring of entire functions . . . .
James Richard Jackson, Some theorems concerning absolute neighborhood
retracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Everett H. Larguier, Homology bases with applications to local
connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Janet McDonald, Davis’s canonical pencils of lines . . . . . . . . . . . . . . . . . . . . . .
J. D. Niblett, Some hypergeometric identities . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Elmer Edwin Osborne, On matrices having the same characteristic
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robert Steinberg and Raymond Moos Redheffer, Analytic proof of the
Lindemann theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Edward Silverman, Set functions associated with Lebesgue area . . . . . . . . . . .
James G. Wendel, Left centralizers and isomorphisms of group algebras . . .
Kosaku Yosida, On Brownian motion in a homogeneous Riemannian
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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