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A subharmonicity property of harmonic measures

01 Oct 2015-Vol. 144, Iss: 5, pp 2073-2079
TL;DR: In this article, it was shown that for compact sets F lying on a sphere S in Rn, the harmonic measure in the complement of F with respect to any point a ∈ S \ F has convex density on any arc of F.
Abstract: Recently it has been established that for compact sets F lying on a circle S, the harmonic measure in the complement of F with respect to any point a ∈ S \ F has convex density on any arc of F . In this note we give an alternative proof of this fact which is based on random walks, and which also yields an analogue in higher dimensions: for compact sets F lying on a sphere S in Rn, the harmonic measure in the complement of F with respect to any point a ∈ S \F is subharmonic in the interior of F . 1 The result in two dimensions Let G be a domain G ⊂ R with compact boundary. In what follows, we denote the n-dimensional harmonic measure for a point z ∈ G by ω(·, z, G) (if it exists). So this is a measure on the boundary of G and it is the reproducing measure for harmonic functions in G: if u is harmonic in G (including at infinity if G is unbounded) and continuous on the closure G, then

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A subharmonicity property of harmonic measures
Vilmos Totik
August 5, 2015
Abstract
Recently it has been established that for compact sets F lying on a
circle S, the harmonic measure in the complement of F with respect to
any point a S \ F has convex density on any arc of F . In this note we
give an alternative proof of this fact which is based on random walks, and
which also yields an analogue in higher dimensions: for compact sets F
lying on a sphere S in R
n
, the harmonic measure in the complement of
F with respect to any point a S \ F is subharmonic in the interior of F .
1 The result in two dimensions
Let G be a domain G R
n
with compact boundary. In what follows, we denote
the n-dimensional harmonic measure for a point z G by ω(·, z, G) (if it exists).
So this is a measure on the boundary of G and it is the repr oducing measure
for harmonic functions in G: if u is harmonic in G (including at infinity if G is
unbounded) and continuous on the closure
G, then
u(z) =
Z
G
u(t)(t, z, G).
See [4], [7], [9] or [10] for the notion of harmonic measures.
The following convexity result was proved in [2].
Theorem 1 Let S be a circl e on the plane, and F S a closed subs et of S.
If a S \ F and I F is an arc, then the dens ity of the harmonic measure
ω(·, a, R
2
\ F ) with respect to arc-measure on S is convex on I.
Actually, the s tronger log-convexity was established (i.e. even the logarithm
of the density is convex), and that is what we shall also prove below.
The theorem implies that if S is a circle and F S is a closed set with non-
empty (one dimensional) interior, then the equilibriu m measure of F is convex
on any subarc of the interior of F , see [2, Theorem 1.5].
AMS Classification: 31C12, 31A15, 60J45, Keywords: harmonic measures, convexity,
subharmonicity, Brownian motion
1

G
G
-
+
F
G
Figure 1: The domain G and its boundary
The circle S in these statements can also be a line and the F S a com-
pact subset of that line (just apply the theorem to circles of radius R with
R ). In particular, if F R is a compact set with non-empty (one dimen-
sional) interior, then the density of the equilibrium measu r e of F with respect
to Lebesgue-measure on R is log-convex on any interval I that lies in F .
We also mention that Theorem 1 was e xt en d ed to Riesz potentials in [3].
In [2] the proof of Theorem 1 was given by an iterated balayage technique. In
this paper first we reprove Theorem 1 using the c onn ect ion b etween harmonic
measures and random walks. This proof will allow us in the next section to
prove an analogue in higher dimensions.
Proof of Theorem 1. Let S
1
be the unit circle and let D
resp. D
+
be the
inner resp. exterior domains complement to S
1
(i.e D
is the open unit disk
and D
+
is the exterior of the closed unit disk). For a set E S
1
let
E
= {t [0, 2π)
e
it
E}.
Let F S
1
be a closed set and J a closed subarc in the (one dimensional)
interior of F . For some small ε > 0 set
G = {z
dist(z, S
1
\ F ) < min{
1
2
dist(z, F ), ε}}.
Then G is a small neighborhood of S
1
\ F , s ee Figure 1. Let Γ be the boundary
of G and Γ
±
= Γ D
±
the part of t hat boundary that lies in the uni t disk and
in its exterior, respectively.
Let a S
1
\F be fixed. What we want to show is that the harmonic measure
ω(·, a, R
2
\F ) has log-convex de ns ity on J. We can formulate the claim as there
is a convex function σ
a
(t) (convexity in the variable t) on J
such for any Borel
2

set E J we have
ω(E, a, R
2
\ F ) =
Z
E
σ
a
(t)dt.
By simple approximation from the outside we may assume that F consists
of finitely many arcs on S
1
.
Start a 2-dimensional Brownian motion X(t), t 0, at a: X(0) = a. We
are going to use Kakutani’s theorem that for E J the har monic measure
ω(E, a, R
2
\ F ) is the probability that X leaves the domain R
2
\ F first at a
point of E (see [4, Theorem F6, (F.10)], [6], or [9, Section 3.4]). Let
T
X
= min{t
X(t) F }.
Note that, by the recurrence of the two dimensional Brownian motion, we h ave
T
X
< almost surely (because we assumed that F contains an d arc), although
we are not going to use that. Let
T
0
X
= sup{t < T
X
X(t) S
1
\ F }
and
T
1
X
= min{t t T
0
X
, X(t) Γ}.
Again, T
1
X
< T
X
< almost surely, and X(T
1
X
) Γ. Suppose that, say,
X(T
1
X
) Γ
. Then {X(t)
t T
1
X
} is a Brownian motion starting at X(T
1
X
)
D
that leaves the domain D
in a point of F , so the probability that it leaves
D
at a point of a given Borel set E J is (conditional probability)
ω(E, X(T
1
X
), D
)
ω(F, X(T
1
X
), D
)
.
Note that here the denominator is positive since F contains an arc of the unit
circle. In a similar fashion, if X(T
1
X
) Γ
+
, then the probability that the
Brownian motion {X(t)
t T
1
X
} starting at X(T
1
X
) D
+
leaves the domain
D
+
at a point of E J is
ω(E, X(T
1
X
), D
+
)
ω(F, X(T
1
X
), D
+
)
.
Now
µ
a,±
(A) = P(X(T
1
X
) A), A Γ
±
, A Borel,
are two positive Borel measures on Γ
±
, respectively, and, according to what we
have just explained, we have the formula
ω(E, a, R
2
\ F ) =
Z
Γ
ω(E, ζ, D
)
ω(F, ζ, D
)
a,
(ζ) +
Z
Γ
+
ω(E, ζ, D
+
)
ω(F, ζ, D
+
)
a,+
(ζ) . (1)
But here ω(E, ζ, D
) is given (see [4, Sec. 1.1] or [9, Theorem 3.44]) by the
Poisson integral
ω(E, ζ, D
) =
Z
E
P
ζ
(e
it
)dt
3

with the Poisson kernel
P
ζ
(e
it
) =
1
2π
1 |ζ|
2
|ζ e
it
|
2
, ζ D
,
and similarly
ω(E, ζ, D
+
) =
Z
E
P
ζ
(e
it
)dt
with the exterior Poisson kernel
P
ζ
(e
it
) =
1
2π
|ζ|
2
1
|ζ e
it
|
2
, ζ D
+
.
Thus,
ω(E, a, R
2
\F ) =
Z
E
Z
Γ
P
ζ
(e
it
)
ω(F, ζ, D
)
a,
(ζ) +
Z
Γ
+
P
ζ
(e
it
)
ω(F, ζ, D
+
)
a,+
(ζ)
!
dt,
i.e. the density σ
a
in question is
σ
a
(t) =
Z
Γ
P
ζ
(e
it
)
ω(F, ζ, D
)
a,
(ζ) +
Z
Γ
+
P
ζ
(e
it
)
ω(F, ζ, D
+
)
a,+
(ζ) . (2)
Now it is easy to see that the sum, and hence the integral of log-convex func-
tions is again log-convex (see [2]), so it is sufficient to sh ow that if ε (appearing
in the definition of Γ) is sufficiently small, then for all ζ Γ the function P
ζ
(e
it
)
is log-convex on J
. But that is simple: the function
1
1 2 cos v + 1
=
1
4 sin
2
(v/2)
is strictly log-convex on the open interval (0, 2π), hen ce
1
1 2 cos(θ t) + 1
is strictly log-convex on any interval not containing θ (mod2π). This and s imp le
compactness implies that if θ (S
1
\ F )
, then for r sufficiently close to 1 the
functions (in t)
1
1 2r cos(θ t) + r
2
are log-convex for t J
. But if ζ = re
Γ, then θ (S
1
\ F )
and r is to 1
(1 ε r 1), furthermore
P
ζ
(e
it
) =
1
2π
|1 r
2
|
1 2r cos(θ t) + r
2
,
so the log-convexi ty of P
ζ
(e
it
) on J
for all ζ Γ follows.
4

2 Harmonic measures in higher dimensions
In this section, we prove the following higher dimensional analogue of Theorem
1.
Theorem 2 Let S be an (n1)-dimensional sphere in R
n
, n 3, and F S a
closed set. If a S \F , then the density of the harmonic measure ω(·, a, R
n
\F )
is subharmonic on the ((n 1)-dimensional) interior of F.
An explanation is needed for the statement. We may assume S to be
S
n1
, the unit sphere in R
n
. Let Int(F ) be the (n 1)-dimensional inte-
rior of F . On S we consider the geodesic distant d and geodesic spheres
S
ρ
(P ) = {Q S
d(Q, P ) = ρ} (of dimension n 2) and geodesic balls
B
ρ
(P ) = {Q S d(Q, P ) = ρ} (of dimension n 1). Let also be the
Laplace-Beltrami operator on S (see e.g. [1, Section 3.1]), which is the rest r ic -
tion to S
n1
of the angular part of the Laplacian on R
n
. If dS denotes the
surface element on S, then, in the interior of F , the harmonic measure has the
form (see the proof below)
(E, a, R
n
\ F ) =
Z
E
σ
a
(x)dS(x),
where σ
a
(x) is the density func ti on in the theorem. Now the subharmonicity of
σ
a
means either of the following:
(i) σ
a
has the submean-value property on every geodesic sphere S
ρ
(P ) lying in
Int(F ) together with its interior (i.e. the value σ
a
(P ) is at most as large
as the average of σ
a
over S
ρ
(P )),
(ii) σ
a
has the submean-value pr operty on every geodesic ball B
ρ
(P ) lying in
Int(F ) (i.e. the value σ
a
(P ) is at most as large as the average of σ
a
over
B
ρ
(P )),
(iii) σ
a
(x) 0 for all x Int(F ).
Naturally, the averages in question are taken with respect to the corresponding
surface or volume elements on S
ρ
(P ) or B
ρ
(P ), respectively.
We are going to prove Theore m 2 in the form ( iii ). From there (ii) and (i)
follow from the spherical version of Green’s formulae, see e.g. [8, Proposition
2.1, (2.1), (2.2)]. An alternative approach is to apply that ∆-harmonic functions
have the mean value property over geodesic balls (see e.g. [5, Corollary X.7.3] as
well as the Remark there, or see [11]) and hence also over geodesic spheres, and
then showing that (iii) implies that if σ
a
agrees with a ∆-harmonic function on
a geodesic sphere, then it is below that function inside that sphere (otherwise,
if we add to σ
a
a small multiple of an appropriate function with st r ic tly positive
spherical Laplacian, at a local maximum point (iii) would be violated).
Consider now the eq ui lib r iu m measure µ
F
in the Newtonian c ase (the kernel
is |z|
2n
) of a non-polar compact set F R
n
in R
n
(see e.g. [9 , Sec . 4.3] where
it is call ed harmonic measure from infinity, or [7, Section II.1], where a different
normalization is used). Let us record the following
5

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"A subharmonicity property of harmon..." refers background in this paper

  • ...See [4], [7], [9] or [10] for the notion of harmonic measures....

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TL;DR: Potential theory is the broad area of mathematical analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, harmonic measure, Green's functions, potentials and capacity.
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"A subharmonicity property of harmon..." refers background in this paper

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TL;DR: In this article, the authors propose a differentiation and integration of spherical harmonics over the sphere for Spectral Methods, and apply it to Spectral methods in a variety of applications.
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395 citations

Journal ArticleDOI
01 Jan 1944

199 citations

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Q1. What are the contributions in "A subharmonicity property of harmonic measures" ?

In this paper, it was shown that for compact sets F lying on a sphere S in Rn, the harmonic measure in the complement of F with respect to any point a ∈ S \\ F has convex density on any arc of F.