Asymptotics for the heat kernel in multicone domains

0.31 MB·English

by  Pierre Collet

SubmittedtotheAnnalsofProbability
arXiv:1501.04595
ASYMPTOTICS FOR THE HEAT KERNEL IN
MULTICONE DOMAINS
By Pierre Collet , Mauricio Duarte , Servet Mart´ınez ,
∗ † ∗
5
Arturo Prat-Waldron and Jaime San Mart´ın
1 ∗
0
Ecole Polytechnique, Universidad Andres Bello, Centro de Modelamiento
2
Matem´atico and Max Planck Institute for Mathematics
n
a Abstract. AmulticonedomainΩ⊆Rn isanopen,connectedset
J
that resembles a finite collection of cones far away from the origin.
3
We study the rate of decay in time of the heat kernel p(t,x,y) of a
2
Brownian motionkilleduponexitingΩ,usingbothprobabilisticand

analytical techniques. We find that the decay is polynomial and we
]
R characterize limt→∞t1+αp(t,x,y) in terms of the Martin boundary
P ofΩatinfinity,whereα>0dependsonthegeometryofΩ.Wenext
. derive an analogous result for tκ/2Px(T >t), with κ=1+α−n/2,
h
whereT istheexit timeform Ω.Lastly,wededucetherenormalized
t
a Yaglom limit for the process conditioned on survival.
m
[

1. Introduction. Let O be a domain (open and connected set) in Rn,
2
v regular for the Dirichlet problem. Consider an n dimensional Brownian
5 motion B starting from the interior of O, with e−xit time TO. The heat
9 t
5 kernel pO(t,x,y) is the Radon-Nikodym derivative of the Borel measure
4 A P (B A,TO > t) with respect to the n dimensional Lebesgue
x t
0 7→ ∈ −
measure, and it is characterised to be the fundamental solution of the heat
.
1
equation with Dirichlet boundary condition, that is: as a function of (t,y)
0
it solves the heat equation ∂ u= 1∆u, it vanishes continuously on ∂O, and
5 t 2
1 it satisfies the initial condition u(0,y) = δ (y).
x
:
v It is well known that pO(t,x,y) tends to zero as time grows to infinity. A
i
X classicalproblemistofindtheexactasymptotic(intime)forthedecayofthe
r heatkernelandthesurvivalprobability.Thisiswellunderstoodforbounded
a
domains (see [10] and [11]). For results in some planar domains we refer the
∗Wethank theCenter for Mathematical Modeling (CMM) Basal CONICYT Program
PFB 03.
†ThanksforthesupportfromproyectFONDECYT3130724,andtheProgramaInicia-
tivaCientificaMileniograntnumberNC130062throughtheNucleusMilleniumStochastic
Models of Complex and Disordered Systems..
AMS 2000 subject classifications: Primary 60J65, 35K08, 35B40; secondary 60H30.
Keywords and phrases: Brownian motion, heat kernel, harmonic functions, renormal-
ized Yaglom limit
1
2 P. COLLET ET AL.
reader to [2]. The large time asymptotic problem is treated in [9] for a large
class of (non symmetric) diffusions under some integrability conditions on
the ground state. Exact asymptotic are computed for Benedicks domains in
[4], and for exterior domains in [5]. Our work focuses on finding the exact
asymptotic in time for pΩ(t,x,y) and P (TΩ > t) for a multicone domain Ω,
x
which we define next.
Let Sn 1 = x Rn : x = 1 be the unit sphere in Rn. Points in Rn will
−
{ ∈ | | }
be regarded as x = rθ, where r = x and θ Sn 1. Given a Lipschitz,
−
| | ∈
proper subdomain D of Sn 1, and a vector a Rn, a truncated cone with
−
∈
opening D and vertex a is the set
C(a,D,R)= a+x :x = rθ Rn : r > R, θ D ,
{ ∈ ∈ }
where R 0. When R > 0, the set S = a+RD will be called the base of
≥
the truncated cone. When R = 0, we will refer to the set in the previous
display as cone with vertex a.
In the same context as above, given a base S = a+RD, let 0 < λ1 <
λ2 λ3 be the eigenvalues of the Laplace-Beltrami operator on D,
≤ ≤ ···
with corresponding orthonormal basis m1,m2,m3,... of L2(D,σ), where
{ }
σ is the surface measure on Sn 1. Let αi = λi+(n 1)2 1/2. We define
− 2 −
the character of the base S as the number α(cid:0)= α(D) = α1.(cid:1)The character
of the truncated cone C(a,D,R) is also defined as α.
A multicone domain Ω Rn is a connected, open set such that there
⊆
exists a bounded domain Ω Ω and finitely many truncated cones Ω =
0 j
⊆
C(a ,D ,R ), with j = 1,...N, such that Ω Ω = for 1 j < i N,
j j j j i
∩ ∅ ≤ ≤
and
N
Ω Ω = Ω .
0 j
\
j[=1
Here Ω is the closure of the set Ω . The set Ω will be called the core, and
0 0 0
for j 1, the sets Ω are called branches of the multi-cone set. Notice that
j
≥
byconstruction,thebranchesaredisjointfromthecore.Also,wewilldenote
the base of the truncated cone Ω by S . Without loss of generality, we can
j j
assume that R = 1, which makes the exposition that follows much easier.
j
Thecharacter of thetruncated cone Ω will bedenoted by α . We definethe
j j
character of the multicone Ω as the number α = min α :j = 1,...,N .
j
{ }
An index l such that α = α will be called maximal. We denote by M the
l
set of maximal indices.
To state the main results of this article, we need to introduce the Martin
boundary at infinity for Ω.
HEATKERNEL IN MULTICONE DOMAINS 3
It is well known that there is a unique minimal harmonic function w
on a cone with vertex C = C(a,D,0) that vanishes continuously on ∂C .
0 0
Actually, there is only one positive harmonic function in C that vanishes
0
continuously onitsboundary(Theorem1.1in[1]).For x = a+ x a θ C
0
| − | ∈
this function is given by:
(1.1) v(x) = x a α−(n2−1)m1(θ),
| − |
whereαisthecharacterofDandm1 isthefirsteigenfunctionoftheLaplace-
Beltrami operator on D.Notice how wehavechosen tonormalize w interms
of the normalization of m1 in L2(D,σ). In order to simplify our exposition,
we set κ = 1+α n/2, so that v(x) = x α κm1(θ).
− | − |
Similarly, if C = C(a,D,R) is a truncated cone, there is a unique (min-
imal) positive harmonic function w in C that vanishes continuously on ∂C,
C
which is defined as follows: let T be the exit time of a Brownian motion B
t
from the cone C. Then
(1.2) w(x) = v(x)−Ex(v(BTC)), x ∈ C.
Let w be the unique minimal harmonic function in Ω . By a standard
j j
balayage argument [7], one can extend w to a minimal harmonic function
j
in Ω. Such extension is given by
1
(1.3) u (x) = w (x) (x)+ G(x,y)∂ w (y)σ (dy), x Ω,
j j 1Ωj 2 ZS n j j ∈
j
where ∂ denotes the (inward) normal derivative on S , and σ is the trans-
n j j
lation of σ by a , and G is the Green function of the domain Ω:
j
∞
G(x,y) = p(t,x,y)dt.
Z
0
Reciprocally, we have that
(1.4) w (x) = u (x) E u(B ), x Ω ,
j j x Tj j
− ∈
where B is an n dimensional Brownian motion, stopped at its exit time Tj
−
from Ω .
j
It is direct to verify from the last two equations that the function u is
j
bounded in Ω Ω , and satisfies that for x = a +rθ,
j j
\
u (a +rθ)
j j
(1.5) lim = 1,
r wj(aj +rθ)
→∞
4 P. COLLET ET AL.
for fixed θ D .
j
∈
We are ready to state the main results of this paper.
Theorem 1.1. Let Ω be a multicone domain with branches Ω ,...,Ω .
1 N
Let α > 0 be the character of Ω, and let M be the set of maximal indices.
Then,
1
(1.6) lim t1+αp(t,x,y) = u (x)u (y),
t 2αΓ(1+α) l l
→∞ lXM
∈
The limit is in the topology of uniform convergence on compact sets.
Theorem 1.2. Let Ω be a multicone domain with branches Ω ,...,Ω .
1 N
Let α > 0 be the character of Ω, and let M be the set of maximal indices.
Set κ = 1+α n/2. Then
−
Γ κ+n
(1.7) lim tκ/2P (T > t) = 2 m1(θ)σ(dθ) u (x).
t→∞ x 2κ/2Γ(cid:0) κ+(cid:1)n2 lXM(cid:18)ZDl l (cid:19) l
(cid:0) (cid:1) ∈
The limit is in the topology of uniform convergence on compact sets.
Theorem 1.3. Let Ω be a multicone domain with character α > 0, and
set β = 1+α+n/2. Fix x Ω, and 1 j N. For each y = y θ, with
∈ ≤ ≤ | |
θ D , we have that a +√ty Ω , for large enough values of t, and
j j j
∈ ∈
(1.8) tlim tβ/2p(t,x,aj +√ty)= 1M(j)2uαjΓ(x(1)v+j(yα))e−|y|2/2.
→∞
The limit is in the sense of uniform convergence on compact sets on the
variables x and y.
The paper is organized as follows. Section 2 lists some key results that we
take from the literature on heat kernels for killed diffusions, in particular,
subsection 2.1includesourmaintheorems forthecaseof acone withvertex.
Section 3 deals with the asymptotics for truncated cones, and Section 4
includes some lemmas leading up to the proofs of the main theorems, which
are contained at the end of Section 4 for the decay of the heat kernel, and in
Section 5forthedecay of thesurvivalprobability.Finally, Section 6includes
the proof of Theorem 1.3 and discusses a renormalized Yaglom limit for the
killed Brownian motion.
HEATKERNEL IN MULTICONE DOMAINS 5
2. Preliminary results. In what follows we make the following sim-
plifications, in order to keep the exposition clear. We set T = TΩ, Tj = TΩj
and denote by p and pj the respective heat kernels. In some of the formulas
below, integrals over S are understoodto be with respect to the translated
j
measure σj, but we will omit the index since the dependence on j is clear
from the domain of integration. Also, we will abuse the notation by omit-
ting the vector a form all the formulas involving functions in cones, since
j
its inclusion affects all such functions by a simple translation of coordinates.
In particular, we will write pj(t,x,y) for x = x θ,y = y η for θ,η D
j
| | | | ∈
instead of pj(t,x + a ,y + a ) in order to simplify our exposition. In this
j j
spirit, we will often say that x radially in Ω to mean that x = a +rθ,
j j
→ ∞
and r .
→ ∞
We start by listing some general properties of heat kernels in unbounded
domains.
Lemma2.1(Lemma2.1in[5]). LetO bearegulardomainfortheDirich-
let problem. Let u(t,x) be a positive solution of the heat equation in R O,
+
×
and consider a function a :R R such that
+ +
→
a(t+s)
(2.1) sup < ,
a(t) ∞
t t0,s 2
≥ | |≤
forsomet > 0.Further, assumethatthefamilyoffunctions a(t)u(t, ) :t t
0 0
{ · ≥ }
is bounded on compact sets. Then, the family a(t)u(t, ) :t t +1 is
0
{ · ≥ }
equicontinuous on compact sets of O.
The next lemma corresponds to Lemmas 2.1-2.4 in [4], which are proved
for Benedicks domains in Rn. Nonetheless, the proofs work in a much more
generalsetting, aslongasthedomainO isaregular domainfortheDirichlet
problem, with infinite interior radius.
Lemma 2.2 (Lemmas 2.1-2.4 in [4]). In the same setting of Lemma 2.1,
for x,y O and s R we have
∈ ∈
p(t+s,x,y)
(2.2) lim = 1.
t p(t,x,y)
→∞
The limit is uniform in compact sets of O. Also, the map t p(t,x,x) is
7→
decreasing.
6 P. COLLET ET AL.
Lemma 2.3. In the same setting as in Lemma 2.2, further assume that
for all s R,
∈
a(t+s)
(2.3) lim =1.
t a(t)
→∞
1+y
Ifa(t)p(t,x,y) Cx | | forlargeenought,thenanylimitpointofa(t)p(t, , )
≤ · ·
(in the topology of uniform convergence on compact sets) has the following
properties:
(i) is a symmetric, non-negative function;
(ii) is harmonic in each component;
(iii) and vanishes continuously on ∂O.
Proof. For the sake of simplicity we denote h (x,y) =a(t)p(t,x,y). Let
t
t be a sequence such that h converges uniformly on compact sets of
k → ∞ tk
O to a function h. It is clear that h is symmetric and non-negative. Notice
that for any s R, the sequence h also converges uniformly on compact
∈ tk+s
sets of O. This is direct from Lemma 2.2 and the hypothesis.
By the Chapman-Kolmogorov equation, for any s R and large enough
∈
k N,
∈
a(t +s)
k
h (x,y) = h (x,z)p(s,z,y)dz.
tk+s a(t ) Z tk
k Ω
1+z
By assumption, htk(x,z) ≤ Cx | |, which is p(s,z,y)dz-integrable as it can
be checked by comparing p with the free Brownian motion’s kernel. Thus,
we can apply the Dominated Convergence Theorem to obtain
h(x,y) = h(x,z)p(s,z,y)dz = E (h(x,X )).
y s
Z
Ω
Itis standardtoshow thath(x,X )is amartingale, fromwhereits standard
s
to deduce that y h(x,y) is harmonic by means of the optional sampling
7→
theorem.
Consider a sequence y O, with y y ∂O. By using once again
n n
∈ → ∈
the Gaussian upper bound on p, and applying the Dominated Convergence
Theoremtoh(x,z)p(1,z,y ),itisdeducedthath(x, )vanishescontinuously
n
·
on ∂O.
HEATKERNEL IN MULTICONE DOMAINS 7
Lemma 2.4. Let U and O be domains in Rn that are regular for the
Dirichlet problem. For ξ ∂U, x U
∈ ∈
1
(2.4) P (B σ(dξ),TU ds)= ∂ pU(s,x,ξ)σ(dξ)ds.
x TU ∈ ∈ 2 n
Here, ∂ represents the inward normal derivative at ξ ∂U.
n
∈
Also, if U O, then
⊆
(2.5)
t 1
pO(t,z,y) = pU(t,z,y)+ ∂ pU(s,x,ξ)pO(t s,ξ,y)σ(dξ)ds.
n
Z Z 2 −
0 ∂U
Proof. These results are well known so we only are going to comment
their proofs. The proof of (2.4) uses Green’s theorem and the heat equa-
tion, and it is very straightforward carry out. Equation (2.5) follows as an
elementary application of the strong Markov property at time TU.
The following lemma characterizes all positive, harmonic functions van-
ishing on ∂Ω. In other words, we characterise the Martin boundary of Ω.
We use the notation from the Introduction.
Lemma 2.5. Let u ,...,u be the minimal harmonic functions given
1 N
by (1.3). For every nonnegative harmonic function u in Ω, vanishing con-
tinuously on ∂Ω, there are unique nonnegative coefficients γ , ,γ such
1 N
···
that
N
(2.6) u(x) = γ u (x), x Ω.
j j
∈
Xj=1
Proof. For x Ω , consider the harmonic function w˜ (x) = u(x)
j j
∈ −
E (u(B )). It is standard to check that w˜ is harmonic in Ω , and that
x Tj j j
j
vanishes continuously on ∂Ω . For m > R, let T be the exit time from
j m
the set Ω B(a ,m). By Itˆo’s formula, the process u(B ) is a bounded
martingalje∩underj P , for x Ω . Therefore, t∧Tmj
x j
∈
u(x)= E u(B ) = E u(B ) +E u(B ) )
x(cid:16) Tmj (cid:17) x(cid:16) Tmj 1{Tmj<Tj}(cid:17) x(cid:16) Tj 1{Tmj=Tj} (cid:17)
E (u(B )) E u(B ) ) .
≥ x Tj − x(cid:16) Tj 1{Tmj<Tj} (cid:17)
8 P. COLLET ET AL.
Since Tj Tj, monotone convergence shows that u(x) E (u(B )),that
m x Tj
ր ≥
is, w˜ is nonnegative. Thus, w˜ (x) = γ w (x) by uniqueness. For z Ω, set
j j j j
∈
N
u˜(z) = γ u (z) u(z),
j j
−
Xj=1
whichisharmonicinΩ,andvanishescontinuouslyon∂O.Wewillnextshow
that u˜ is bounded, for which it is enough to show that it is bounded in each
branch of Ω.
Fix i 1,...,N , and consider x Ω . We have
i
∈ { } ∈
N
u˜(x)= E (u(B ))+γ E (u (B ))+ γ u (x)
x Ti i x i Ti j j
−
j=X1,j=i
6
The first term on the right hand side is bounded by sup u(x) , and the
x∈Γi| |
second one by γ sup u (x) . The summation is bounded as each term
i x∈Γi| i |
u (x) is bounded in Ω . We conclude that u˜ is harmonic and bounded in Ω,
j i
and vanishes continuously on ∂Ω. It follows that u˜(B ) is a martingale,
t T
∧
and so
u˜(z) = E (u˜(B )) 0, as t .
z t T
∧ → → ∞
Uniquenessfollowsfromtheboundednessofu inΩ Ω ,anditsunbound-
j j
\
edness in Ω .
j
2.1. Asymptotics in a cone with vertex. In what follows we consider a
cone V, with opening D and vertex a = 0, that is, V = C(0,D,0). Let pV
be the heat kernel in V. Let 0 < λ1 < λ2 λ3 be the eigenvalues of
≤ ≤ ···
the Laplace-Beltrami operator on D, with corresponding orthonormal basis
m1,m2,m3,... of L2(D,σ). We also denote by αi = λi+(n 1)2 1/2.
{ } 2 −
(cid:0) (cid:1)
The behaviour of the heat kernel with Dirichlet boundary conditions is
well known for a cone with vertex. The following results are taken from [3].
Theorem 2.6. For x = rθ,y = ρω V, with θ,ω D and r = x ,
∈ ∈ | |
ρ = y , the heat kernel with Dirichlet boundary conditions in V is given by:
| |
exp r2+ρ2
(2.7) pV(t,x,y) = (cid:16)− 2t (cid:17) ∞ J rρ mi(θ)mi(ω)
t(rρ)n2−1 Xi=1 αi(cid:16) t (cid:17)
HEATKERNEL IN MULTICONE DOMAINS 9
where J is the modified Bessel function of first kind of order ν, that is, the
ν
solution of
z2J (z)+zJ (z2 +ν2)J = 0,
ν′′ ν′ − ν
satisfying the growing conditions:
zν zν
(2.8) J (z) ez,
2νΓ(1+ν) ≤ ν ≤ 2νΓ(1+ν)
for z > 0, and ν 0.
≥
Recall that the unique minimal positive harmonic function in V is given
by v(x) = v( x θ)= x α1−(n2−1)m1(θ).
| | | |
Corollary 2.7. For each x,y V, we have
∈
v(x)v(y)
(2.9) lim t1+α1pV(t,x,y) =
t 2α1Γ(1+α1)
→∞
pV(t,x,y) v(x)v(y)
(2.10) lim =
t pV(t,w,z) v(w)v(z)
→∞
Both limits are uniform in compact sets.
Proof. Clearly, (2.10) follows from (2.9), so we only prove the latter.
From Theorem 2.6, we get the bound
t1+α1pV(t,x,y) v(x)v(y) C tα1e−r22+ts2 J rs (rs)α1−(n2−1) +
(cid:12)(cid:12)(cid:12)(cid:12) − 2α1Γ(1+α1)(cid:12)(cid:12)(cid:12)(cid:12)≤ +(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t(−rs(α)2n2−−α11) ∞α1(cid:16)(rst)α(cid:17)k−−(2n2α−11Γ)(,1+α1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
kX=22αkΓ(1+αk)
where C = sup m1(θ)2. The uniform convergence on compact sets for
θ D
∈
the first term is easily deduced from (2.8). The series on the right hand side
converges uniformly in compact sets, so the whole term converges to zero,
as t , since α2 > α1.
→ ∞
3. Asymptotics in a truncated cone. The main goal of this section
is to extend Corollary 2.7 to a truncated cone C = C(a,D,R). As before,
we assume that R = 1 and a = 0.
We will often use the following version of the Harnack inequality up to
the boundary.
10 P. COLLET ET AL.
Theorem 3.1 (From [12], see also [8]). Let O be a precompact, regular
domain for the Dirichlet problem, and let u 0 be a solution of the heat
≥
equation on O [0,T) with Dirichlet boundary condition. Then, givenx O,
× ∈
there is C > 0 such that u(t,z) C u(T,x), for all (t,z) [0,T) O where
1 1
≤ ∈ ×
the constant C depends only on x and T t.
1
−
Corollary 3.2. Let V be a cone with vertex. For any x V there is a
∈
constant C > 0, only dependent on x, such that for all y V the following
x
∈
inequality holds for all t > 1:
(3.1) pV(t,x,y) C1+y pV(t,x,x).
≤ x | |
Proof. Assume x = 1, otherwise the corollary follows by scaling. The
| |
inequality holds for small y , by a direct application of the boundary Har-
| |
nack inequality (Theorem 3.1), so we assume that y > 2.
| |
Let r be positive, but small enough so that B(x,r) V. It follows
⊆
by scaling that B(νx,r) V for all ν 1. Thus, applying the standard
⊆ ≥
parabolicHarnack inequality several timesintheballB(0,r)tothefunction
u(s,z) = pV(t+s,νx+z,y) for fixed, but arbitrary ν > 1,y V, we get
∈
pV(t,νx,y) C1+rνpV(t+1+rν,x,y) C2+2rνpV(t+2+2rν,νx,y),
≤ 2 ≤ 2
for a positive constant C that only depends on x.
2
The heat kernel in V has the following scaling property:
t x y
pV(t,x,y) = λ npV , , , λ > 0.
−
(cid:18)λ2 λ λ(cid:19)
From all the inequalities above, it follows that
pV(t,x,y) C1+|y|pV(t+1+r y ,x y ,y)
≤ 3 | | | |
t+1+r y y
= C31+|y||y|−npV (cid:18) y 2 | |,x, y (cid:19)
| | | |
t+1+r y
≤ C1C31+|y||y|−npV (cid:18) y 2 | | +1,x,x(cid:19)
| |
t
≤ C1C31+|y||y|−npV (cid:18) y 2,x,x(cid:19),
| |
where the second to last line comes from the boundary Harnack inequality,
whereas the last one comes form the fact that t pV(t,x,x) is decreasing
7→