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 ndimensional 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)dω(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 arcmeasure on S is convex on I.
Actually, the s tronger logconvexity 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 nonempty (one dimen
sional) interior, then the density of the equilibrium measu r e of F with respect
to Lebesguemeasure on R is logconvex 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 logconvex 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 2dimensional 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
−
)
dµ
a,−
(ζ) +
Z
Γ
+
ω(E, ζ, D
+
)
ω(F, ζ, D
+
)
dµ
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
−
)
dµ
a,−
(ζ) +
Z
Γ
+
P
ζ
(e
it
)
ω(F, ζ, D
+
)
dµ
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
−
)
dµ
a,−
(ζ) +
Z
Γ
+
P
ζ
(e
it
)
ω(F, ζ, D
+
)
dµ
a,+
(ζ) . (2)
Now it is easy to see that the sum, and hence the integral of logconvex func
tions is again logconvex (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 logconvex on J
∗
. But that is simple: the function
1
1 − 2 cos v + 1
=
1
4 sin
2
(v/2)
is strictly logconvex on the open interval (0, 2π), hen ce
1
1 − 2 cos(θ − t) + 1
is strictly logconvex 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 logconvex for t ∈ J
∗
. But if ζ = re
iθ
∈ 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 logconvexi 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 (n−1)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
n−1
, 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
LaplaceBeltrami operator on S (see e.g. [1, Section 3.1]), which is the rest r ic 
tion to S
n−1
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)
dω(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 submeanvalue 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 submeanvalue 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
2−n
) of a nonpolar 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