On non-euclidean harmonic measure

A NNALES DE L ’I. H. P., SECTION B

M ^ARK A. P ^INSKY

On non-euclidean harmonic measure

Annales de l’I. H. P., section B, tome 21, n

^o

1 (1985), p. 39-46

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On Non-Euclidean Harmonic Measure

Mark A. PINSKY

Department ^of Mathematics, Northwestern University,

Lunt Hall Evanston, ^Ill 60201, ^U.S.A.

ABSTRACT. ^- Let

HE

^{be the harmonic measure of the}

s-sphere

^{in a}

Riemannian manifold. Riemannlan We compute

lim

^{in terms of the}

^gradient

.

manifold. W e

compute lim ~~0

^{8 3 In terms 0 t e gra len}

of the Gaussian curvature. ^~~

°

RESUME. ^- Soit

HE

^{la mesure}

harmonique

^{de la}

sphere

^du rayon ⁸ d’une variété riemannienne de dimension deux. On

exprime

^lim

(HE -

en fonction de la derivee de la courbure

gaussienne.

1. INTRODUCTION

Let

(M, g)

^{be an} n-dimensional Riemannian manifold with

Brownian

motion

process {Xt,

^t &#x3E;

0 }. According

^{to the}

Onsager-Machlup

^for-

mula

[3 ],

^{the « most}

probable path »

^{is that of a classical}

particle

^{in a}

conservative force field whose

potential

^{energy is} one-twelfth the scalar curvature. This suggests ^{that the « most}

probable path »

^follows the nega- tive

gradient

^of the scalar curvature. The purpose of this note is to make this

precise

^{in terms} of the harmonic measure of a small

sphere.

^{For tech-}

nical reasons we restrict the discussion to surfaces. The

technique

^follows

the method used to

study

^{the mean exit time}

[1 ].

Annales de l’Institut Henri Poincaré - Probabilités

40 ^{M. A. PINSKY}

2. NOTATIONS AND DEFINITIONS

Let

(M, g)

^{be an} n-dimensional Riemannian manifold. We use the follow-

ing

notations :

M~

^{is the} tangent space ^{at m E} M.

Bm(e)

^{is the} ball of radius ^E in M with center at m E M.

is the ball of radius e in

M~

^{with center at}

6

E

M~.

exp~ is the

exponential mapping (which

^{is defined on all}

of Mm

^{in case M}

is

complete;

^{otherwise it is a}

mapping)

^{from to}

Bm(E)

^for

sufficiently

^{small e} &#x3E; 0. ,

4Y~

^{is the}

mapping

^on functions defined

by

D,

maps from to for

sufficiently

^{small e} &#x3E; 0.

A is the

Laplace-Beltrami

operator ^{of the} Riemannian manifold :

The

following result,

which will be used

repeatedly,

^was

proved

ⁱⁿ

[1].

PROPOSITION 2. O. ^- There exist second order

differential

operators

(Ll - 2, 03940, 03941, ...)

^{on such that}

for

^{each N} &#x3E;_ 0 and each

fe

l1j

^maps

polynomials of degree

^{k to}

polynomials of degree

^{k +}

j.

^In any ^nor- mal coordinate chart

(x 1, - - - , xn)

^{we have}

Here

Riajb

^is the Riemann tensor

and j

^{is the Ricci tensor at}

mEM.

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

Let

(Xt, Px)

be the Brownian motion process with infinitesimal _gene-

rator A. For each ~ ~ M let

Tg

be the exit time from the

geodesic

^ball

B~(e).

In case n ⁼ 2 we introduce

geodesic polar

coordinates

(r, 8) by

^{Xl = r cos}

8,

x2 ^{= r} sin 8. The coordinate formulas for the first three operators ^become

~K where K is the Gaussian curvature and

Ko = K(m), Ki

⁼

- ^(m),

~K .

~’

K2 = - ^(m).

^This

computation

^{is carried out in the}

Appendix,

^{Sec. 5.}

3. STATEMENT OF RESULTS

To

study

the harmonic measure we let

f

^{E a} smooth function

on the

circle. Extend f to R~B { 0 } by making f constant along

rays

through

the

origin. Equivalently f

^{can be}

thought

^{of as a function on}

MmB ~

⁰

}.

Then is a smooth function on

M B ~ m ~

^{which is constant}

along

geo- desics

eminating

^{from m.} The harmonic measure operator ^is

THEOREM 3.1. ^- When

8 ~

^{0 we have}

where ~

is normalized

Lebesgue

^measure

and ~ VK,

⁼

Kl

^{cos o +}

K2

^sin

o,

--71 ~ 0 71.

Proof

- We follow the

perturbation

method introduced in

[1 ].

^Let

Ui ^E

(i

⁼

^0,2, 3)

be defined

by

42 ^{M. A. PINSKY}

The

proof depends

^{on the}

following

three lemmas : LEMMA 1. ^-

LEMMA 2. ^-

LEMMA 3. ^-

Proof of Lemma

^{1. uo is}

’the

solution of the classical Dirichlet

problem

in the unit disc. thus

In

particular

ProofofLemma

^2.

- Equivalently

^{we write} uo ^{as a} Fourier series

Thus

.

00

This is a sum

of homogeneous polynomials ^{hn. By}

^{Lemma 6.3 of}

^[7]

^{we have}

u2(0)

⁼⁼

03A3 1 (n+2)2 ~ 03C0 -03C0 hn(03B8)03C9(d ^0).

^{But from the}

orthogonality

relations for

i

the functions

(cos

^sin

~0),

^{all of these}

integrals

^are zero, hence

M~(0)

^{= 0.}

D

(Note :

^{The above}

depends

^{on the fact that}

03940u0

^{is a harmonic function}

which vanishes at the

origin.

This may also be seen from the relation

0394-203940 - AoA-2

^{= -}

applied

^to

uo.)

3

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

Proof of

Lemma 3. We have

Referring

^{to the formula}

(2.6),

^{we see that}

00 .

a sum

of homogeneous polynomials I ^{hk. By}

Lemma 6 . 3 of

[1] we

^have

u3(0) == I ^{(k + 2)} hk(03B8)03C9(d03B8). But from

^the

2

of Fourier series all of these

integrals

^{are zero} except ^{for k ==}

2,

^{I. e. n == I.}

This

gives

^.

Ct)

On the other hand

/(0)

⁼

Ao + I

₁

^{(An cos}

^{n03B8 + and thus}

1 We have

proved

^that

u3(0)

.

== - - 1 fn

^{cos 6 +}

^K2

^sin

03B8)03C9(d03B8)

^as

required.

^{D 32 -n}

44 M. A. PINSKY

To

complete

^the

proof

^{of the theorem we}

appeal

^to

Proposition

^2.0

with f =

uo +

E2u2

⁺

E~u~ ;

^thus

where we have used the

defining

^relations

(3.1)-(3.3).

^{Hence AU =}

0(E2)

where U ⁼ +

E2u2

⁺

E3u3). Noting

^{that U = on we have}

that for _{any x} E

by Dynkin’s

^formula

Setting

^{x = m we have}

U(m)

⁼

uo(O)

⁺

e2u2(O)

⁺ from which the result follows. D

4. COMPARISON

WITH THE NON-STOCHASTIC MEAN VALUE

The mean-value operator ^{of a} Riemannian manifold is defined

by

where _6E is the surface measure on the

e-sphere Se and fis

^{a smooth function}

on the unit

sphere.

^{We have the}

following

^result.

THEOREM 4 . 1. ^- When e

1

⁰

Proof

^From the method of

Gray

and Willmore

[2 ],

^{we have}

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

where

Using

^the

expansion

we have

In

particular

Thus

which was to be

proved.

COROLLARY. ^{- =} every ^E &#x3E;

then

(M, g)

^has constant curvature.

Proof

^{We have}

If this is zero for

aUf;

^then

K2

⁼

0,

^{i. e.}

VK(m)

⁼

0,

^{and hence K}

is constant.

, 5. APPENDIX. COMPUTATION

Ao, Ai

In

geodesic polar

coordinates we have

The Jacobi

equation connecting

^{the metric with the curvature is}

Vol. 21, ^n° 1-1985.

46 M. A. PINSKY

In ^a

neighborhood

^{of m, we can write}

This

yields

Performing

^{the indicated}

operations,

^{we have}

Substituting

ⁱⁿ

(5.1)

^and

equating

coefficients of r

yields (2.4)-(2.6).

ACKNOWLEDGEMENT

I would like to thank Victor Mizel for some

helpful suggestions.

REFERENCES

[1] ^{A. GRAY and M.} PINSKY, The mean exit time from a small geodesic ^{ball in a Riemannian} manifold, Bulletin des Sciences Mathématiques, ^t. 107, 1983, p. 345-370.

[2] ^{A. GRAY and T. J.} WILLMORE, Mean-value ^theorems for Riemannian manifolds, ^Pro- ceedings of ^the Royal Society of Edinburgh, ^{t. 92} A, 1982, p. 343-364.

[3] ^Y. TAKAHASKI and S. WATANABE, ^The Onsager-Machlup ^{function of diffusion} processes, Stochastic Integrals, Springer-Verlag Lecture Notes in Mathematics, t. 851, 1981,

p. 433-463.

(Manuscrit reçu le 10 Avril 1984) (Corrigé ^{le 25} Septembre 1984)

Annales de l’Institut Henri Poincare - Probabilités et Statistiques