Now, we consider an m-dimensional Brownian motionB(s) of a particle starting at the origin in Rm, and calculate the probability of the particle hitting a sphere{x∈Rm: kxk ≤r} of radius r
within timet. By a well known formula first derived in [Ken80], we see that such a probability is given as follows:
P( sup
0≤s≤tkB(s)k ≥r) = 1− 1 2ν−1Γ(ν+ 1)
∞
X
k=1
jν,kν−1 Jν+1(jν,k)e−
j2 ν,kt
2r2 , ν >−1, (5.24) where ν = m2−2 is the “order” of the Bessel process,Jν is the Bessel function of the first kind of orderν, and 0< jν,1< jν,2< ...is the sequence of positive zeros ofJν.
We will be interested in the regime whent is of the order of r2 (i.e. respecting the parabolic scaling). In this direction let us, for a large positive numberλ, chooset=λ−1, and letr=Cλ−1/2, whereC is a constant to be chosen later, independently ofλ. Plugging this in (5.24) then reads,
P( sup
0≤s≤λ−1kB(s)k ≥Cλ−1/2) = 1− 1 2ν−1Γ(ν+ 1)
∞
X
k=1
jν,kν−1 Jν+1(jν,k)e−
j2 ν,k
2C2, ν >−1. (5.25) We need to make a few comments about the asymptotic behaviour ofjν,k here. For notational convenience, we writeαk∼βk, ask→ ∞if we haveαk/βk →1 ask→ ∞. The result in [Wat44], p. 506, gives the asymptotic expansion
jν,k= (k+ν/2 + 1/4)π+o(1) ask→ ∞, (5.26) which tells us thatjν,k∼kπ. Also, from [Wat44], p. 505, we have that
Jν+1(jν,k)∼(−1)k−1
√2 π
√1
k. (5.27)
These asymptotic estimates, in conjunction with (5.25), tell us that keepingν bounded, and given a smallη >0, one can choose the constantC small enough (depending onη) such that
P( sup
0≤s≤λ−1kB(s)k ≥Cλ−1/2)>1−η. (5.28) In this context, see also refer to Proposition 5.1.4 of [Hsu02].
Chapter 6
Estimates on nodal domains
In this Chapter we study some aspects of the geometry of nodal domains of Laplacian eigenfunctions.
First, we address the following loosely-formulated question:
Question 6.0.1. Can a nodal domain be ”thin and straight”, thus resembling a long thin cylinder?
The more precise form of the question we will address is the following:
Question 6.0.2. Suppose thatΣ⊂M is sufficiently flat (where flattness is defined in an appropriate way) submanifold. Can a nodal domain be contained in sufficiently small wavelength neighbourhood of Σ?
We will apply Brownian motion techniques to address these last questions.
We will afterwards address the width of nodal domains. To this end we will study the inner radius of nodal domains (i.e. the radius of the largest inscribed geodesic ball inside of a nodal domain). We will be interested in the following
Question 6.0.3. Given a nodal domainΩλcorresponding to an eigenfunctionφλ, how isinrad(Ωλ) compared to wavelength (i.e. 1/√
λ)?
Simple examples (such as tori) suggest that nodal domains possess approximately wavelength inner radius. In order to approach such a question, one could try to simplify the discussion and focus only on a particular nodal domain Ωλ with the corresponding eigenfunctionφλbeing restricted on Ωλ. In fact, sinceφλ does not vanish on Ωλ, it is the first Dirichlet eigenfunction when restricted to Ωλ. However, using only this information (i.e. forgetting about the complement of Ωλ) is not sufficient to obtain inner radius bounds. It turns out that in dimension n ≥3 that one can introduce thin spikes which do not change the first Dirichlet eigenvalue, but have a significant impact on the the domain’s inner radius (cf. [Hay78]). From this point of view nodal domains seem to be sensitive objects and sharper inner radius bounds would require some further ”global”
information (i.e. outside of the particular nodal domain).
6.1 Thin and straight nodal domains
Problems, similar in spirit to Question 6.0.2, have been addressed in several results appearing in the works of Jerison-Grieser, Steinerberger, etc (cf. [GJ98], [Ste14] and the references therein).
By considering Brownian motion and applying the Feynman-Kac formula, followed by suitable hitting probability estimates, we further extend the results in these directions. In terms of presentation, we partly follow our work in [GM18b].
We consider a closedn-dimensional smooth Riemannian manifold (M, g). For an eigenvalueλ of the Laplacian ∆ and a corresponding eigenfunction φλ, we consider a nodal domain Ωλ (recall Definition 1.3.1).
We start by discussing the problem of whether a nodal domain can be squeezed in a tubular neighbourhood around a certain subset Σ ⊆ M. A result of Steinerberger (see Theorem 2 of [Ste14]) states that for some constant r0 >0 a nodal domain Ωλ cannot be contained in a √r0
λ -tubular neighbourhood of hypersurface Σ, provided that Σ is sufficiently flat in the following sense:
the hypersurface Σ must admit a unique metric projection in a wavelength (i.e. ∼ √1λ) tubular neighbourhood. The proof involves the study of a heat process associated to the nodal domain, where one also uses estimates for Brownian motion.
We relax the conditions imposed on Σ. Our first result is a direct extension of Theorem 2 of [Ste14]. Before stating the result, we begin with the following definition:
Definition 6.1.1 (Admissible Collections). For each fixed eigenvalue λ, we consider a natural number mλ ∈ N and a collection Σλ := ∪mi=1λΣiλ, where Σiλ is an embedded smooth submanifold (without boundary) of dimensionk, (1≤k≤n−1).
We callΣλ admissible up to a distance r if the following property is satisfied: for any x∈M with dist(x,Σλ) ≤ r there exists a unique index 1 ≤ ix(λ) ≤ mλ and a unique point y ∈ Σiλx(λ) realizing dist(x,Σλ)- that is, dist(x, y) =dist(x,Σλ).
We note that if Σλ consists of one submanifold which is admissible up to distance r, then Definition 6.1.1 means that ris smaller than the normal injectivity radius of Σλ. Moreover, if Σλ
consists of more submanifolds, then these submanifolds must be disjoint and the distance between every two of them must be greater thanr.
Let us also remark that, in contrast to Theorem 2 of [Ste14], we also allow Σλ to vary with respect toλin a controlled way, which is made precise by Definition 6.1.1. With that clarification in place, we have the following result:
Theorem 6.1.1. There is a constant r0 depending only on (M, g) such that if a submanifold Σλ ⊂ M is admissible up to distance √1
λ, then no nodal domain Ωλ can be contained in a √r0 λ -tubular neighbourhood of Σλ.
Proof. We begin by outlining the strategy.
First, one considers a pointx0∈Ωλ where the eigenfunction achieves a maximum on the nodal domain (w.l.o.g. we assume that the eigenfunction is positive on Ωλ). One then considers the quantity p(t, x0) - i.e. the probability that a Brownian motion started at x0 escapes the nodal domain within timet.
The main strategy is to obtain two-sided bounds forp(t, x0).
On one hand, we have the Feynman-Kac formula (see Section 5.2) which provides a straightforward upper bound only in terms oft.
On the other hand, we provide a lower bound forp(t, x0) in terms of some geometric data. To this end, we take advantage of various tools some of which are: formulas for hitting probabilities of spheres and the parabolic scaling between the space and time variables (cf. Section 5.4);
comparability of Brownian motions on manifolds with similar geometry (cf. Section 5.3).
Step 1 - An associated diffusion process and the Feynman-Kac formula.
Given an open subsetV ⊂M, we consider the solutionpt(x) to the following diffusion process:
(∂t−∆)pt(x) = 0, x∈V, (6.1)
pt(x) = 1, x∈∂V, (6.2)
p0(x) = 0, x∈V. (6.3)
By the Feynman-Kac formula (see Section 5.2), this diffusion process can be understood as the probability that a Brownian motion particle started in xwill hit the boundary within time t.
Indeed, the Feynman-Kac formula yields
pt(x) = 1−Ex(ψV(ω, t)), t >0, (6.4) where we recall thatω(t) denotes an element of the probability space of Brownian motions starting atx,Exis the expectation with regards to the measure on that probability space, and whereψV is the cut-off function
ψV(ω, t) =
(1, ifω([0, t])⊂V
0, otherwise. (6.5)
Now, we adopt this construction to the eigenfunctionφλ (corresponding to the eigenvalue λ) and the nodal domain Ωλ(replacing the open setV) upon which, without loss of generality,φλ>0.
Setting Φ(t, x) :=et∆φλ(x), we see that Φ solves
(∂t−∆)Φ(t, x) = 0, x∈Ωλ, (6.6)
Φ(t, x) = 0, x∈∂Ωλ⊆ {φλ= 0}, (6.7)
Φ(0, x) =φλ(x), x∈Ωλ. (6.8)
Using the Feynman-Kac formula given by Theorem 5.2.1, we have,
et∆φλ(x) =Ex(φλ(ω(t))ψΩλ(ω, t)), t >0, (6.9) where the cut-off function is given by
ψΩλ(ω, t) =
(1, ifω([0, t])⊂Ωλ
0, otherwise. (6.10)
Now, let us specify x0 ∈ Ωλ such that φλ(x0) = kφλkL∞(Ωλ). We have the following direct bounds:
Φ(t, x) =e−λtφλ(x) =Ex(φλ(ω(t))ψΩλ(ω, t)) (6.11)
≤ kφλkL∞(Ωλ)Ex(ψΩλ(ω, t)) =kφλkL∞(Ωλ)(1−pt(x)).
Setting t=t0λ−1 for a positive numbert0 and x=x0, elementary algebraic manipulations imply that the probabilitypt(x) of the Brownian motion starting at an extremal point x0 and leaving Ω within timeλ−1is bounded as:
pt(x)≤1−e−t0. (6.12)
x
Ω λ
Brownian motion inn−k“bad directions”
Sn−k−1
Σλ
Figure 6.1: The nodal domain ”squeezed” in a thin tubular neighbourhood around Σλ.
A rough interpretation is that maximal points xare situated deeply into the nodal domain Ωλ. Using the notation for hitting probabilities introduced in Section 5.3, the last derived upper estimate translates to
ψM\Ωλ(t0λ−1, x)≤1−e−t0. (6.13)
Step 2 - A lower bound for the hitting probability.
In order to prove the claimed result in Theorem 6.1.1, we proceed by assuming the contrary. In order words, we assume that Ωλ is contained in an (r0/√
λ)-neighbourhood of Σλ, where we have the freedom to choose the number r0 as small as we wish.
First, by the admissibility condition on Σλ in Definition 6.1.1 we know that the point of maximumx0has a unique metric projection on one and only one Σiλx0 from the collection Σλ.
Further on, let us choose a suitable small radiusR and small time parametert0 such that the Brownian motion comparability result in Theorem 5.3.1 holds atx0. Below we will specify further how smallR, t0 should be taken.
In this direction (as we assumed the contrary) we can chooser0to be sufficiently smaller than R (again determined below) and assert that Ωλ is contained in a √r0λ-tubular neighbourhood of Σλ
- for convenience, we denote this tubular neighbourhood byNr0λ−1/2(Σλ).
Note that from the remarks after Definition 6.1.1, it follows that Ωλ⊆Nr0λ−1/2(Σiλx0).
Now, we start a Brownian motion atx0and, roughly speaking, we see that locally the particle has freedom to wander in n−k“bad directions”, namely the directions normal to Σiλx0, before it hits∂Ωλ. That means, we may consider an (n−k)-dimensional Brownian motionB(t) starting at x0- cf. Figure 6.1.
More formally, we choose a normal coordinate chart (U, φ) around Σiλx0, where the metric is
comparable to the Euclidean metric and where we have that
φ(Σiλx0) =φ(U)∩ {Rk× {0}n−k}, (6.14) φ(N2r0λ−1/2(Σiλx0)) =φ(U)∩
( Rk×
−2r0
√λ,2r0
√λ n−k)
. (6.15)
We take a geodesic ballB⊂U ⊂M atx0 of radius √Rλ. Using the hitting probability notation from Section 5.3 and monotonicity with respect to set inclusion we have
ψM\Ωλ
t0
λ, x0
≥ψB\Ωλ
t0
λ, x0
≥ψB
\N2r
0λ−1/2(Σixλ0)
t0
λ, x0
, (6.16)
and the comparability Theorem 5.3.1 implies that, if c = Rt02, then there exists a constant C, depending oncandM, such that
ψB
\N2r
0λ−1/2(Σixλ0)
t0
λ, x0
≥Cψe
φ(B\N2r
0λ−1/2(Σixλ0))
t0
λ, φ(x0)
, (6.17)
where, as before, ψe denotes the hitting probability in Euclidean space. We denoteNre0λ−1/2 :=
φ(Nr0λ−1/2(Σiλx0)).
Let us consider the “solid cylinder”S=B(k)√R0 λ
×B(n−k)√r0 λ
=:B1×B2, a product ofkdimensional Euclidean ball of radiusR0/√
λandn−kdimensional Euclidean ball of radiusr0/√
λ, respectively, both centered at φ(x0). By construction, we can appropriately choose R0 with respect to R, so that S is a cylinder contained inN2re
0λ−1/2∩B.
IfB(t) = (B1(t), ..., Bn(t)) is an n-dimensional Brownian motion, the components Bi(t)’s are independent Brownian motions (see, for example, Chapter 2 of [MP10]). Denoting by Bk(t) and Bn−k(t) the projections ofB(t) onto the first kand last n−k components respectively, it follows that the following lower bound in terms of hitting theB2-side ofS holds:
ψe
φ(B\N2r
0λ−1/2(Σixλ0))
t0
λ, φ(x0)
≥P sup
0≤s≤t0λ−1kBk(t)k ≤ R0
√λ
!
P sup
0≤s≤t0λ−1kBn−k(t)k ≥ r0
√λ
!
≥ckP sup
0≤s≤t0λ−1kBn−k(t)k ≥ r0
√λ
! ,
whereckis a constant depending onkand the ratiot0/R20; moreover,ck can be calculated explicitly from (5.25).
Using the estimate on hitting probabilities of spheres (5.28), we may taker0 ≤R0 sufficiently small so that
P sup
0≤s≤t0λ−1kBn−k(t)k ≥ r0
√λ
!
>1−ε, (6.18)
whereεis a fixed sufficiently small number.
To conclude, let us specify the ”smallness” of the parameters as announced at the beginning.
Assume for a moment thatt0is selected. By adjustingR according tot0 we keep the ratioc=Rt02
and, hence,C and Rt02
0 fixed. By takingr0≤Rappropriately (much smaller than t0 so that (6.18) holds), the above arguments yield
ψM\Ωλ(t0λ−1, x)≥Cck(1−), (6.19) where we emphasize that the constants on the right hand side depend only on the ratio tR0 which is kept fixed.
Combining this with the estimate from the previous Step 1, we obtain
1−e−t0 ≥Cck(1−). (6.20)
Finally, ift0is a priori selected small, the left hand side will become less than the expression on the right (which depends on the ratio t0/Rand not ont0), thus yielding a contradiction.
Remark 6.1.1. Note that the constant r0 above is independent of Σλ (it is selected so that the above Euclidean Brownian motion hitting probabilities hold); in other words, the same constantr0
will work for Theorem 6.1.1 as long as the surface is admissible up to a wavelength distance. Indeed, this results from the fact that r0 depends only on the diffusion process associated to the Brownian motion, and is an inherent property of the manifold itself.