ISOPERIMETRIC CONDITIONS AND

DIFFUSIONS IN RIEMANNIAN MANIFOLDS

Patrick McDonald

May 28, 1998

Abstract. We study diffusions in Riemannian manifolds and properties of their exit time moments from smoothly bounded domains with compact closure. For any smoothly bounded domain with compact closure, Ω, and for each positive integer k, we characterize the kth exit time moment of Brownian motion, averaged over the domain Ω with respect to the metric density, using a variational quotient. We prove that for Riemannian manifolds satisfying an isoperimetric condition, the averaged kth exit time moment of Brownian motion from domains of a fixed volume is bounded above. For the case of Euclidean space, we establish similar boundedness properties for a larger class of diffusions.

1. Introduction

In this note we study diffusions in a Riemannian manifold M and properties of their corresponding exit times from smoothly bounded, connected, open domains in M with compact closure. Our results relate boundedness properties of exit time moments and averaged exit time moments to isoperimetric properties of the underlying manifold. To formulate our results, we fix some notation.

Given a Riemannian manifold (M, g) we will denote by X_{t} a general diffusion on M with corresponding infinitesimal generator L. We let P_{x} be the probability associated to starting the process X_{t} at the point x and let E_{x} denote the corresponding expectation.

Our primary interest involves diffusions with associated infinitesimal generator arising as the Laplace operator with respect to some Riemannian metric on M. Given a Riemannian metric, g, we denote by Δ_{g} the Laplace operator with respect to g.

Let D be the space of smoothly bounded, connected, open domains in M with compact closure. Given Ω ∈ D, let τ = τ(ω) = inf{t ≥ 0 : X_{t}(ω) ∉ Ω} be the first exit time of X_{t} from Ω.

Typeset by AMS-TEX

Research at MSRI is supported in part by NSF grant DMS-9701755. The author was partially supported by a DSR grant from the University of South Florida

For k any positive integer, we will study properties of the maps S_{k}, A_{k} : D → IR defined by

(1.1)  S_{k}(Ω) = sup_{x∈Ω} E_{x}[τ^{k}]
(1.2)  A_{k}(Ω) = ∫_{Ω} E_{x}[τ^{k}]dg

where dg is the volume form induced by g on M. Our first theorem provides a description of the domain functionals A_{k} in terms of variational quotients:

Theorem 1.1. Let M be a Riemannian manifold with metric g. Let X_{t} be Brownian motion on M with respect to the metric g. Let Ω be a smoothly bounded open domain in M with the closure of Ω, denoted Ω̄, compact. Let A_{k} be defined as in (1.2). Let F_{k} be defined by

(1.3) F_{k} = {f ∈ C^{∞}(Ω̄); ∫_{Ω} fdx ≠ 0, f = Δ_{g}f = ⋯ = Δ_{g}^{k−1}f = 0 on ∂Ω} .

Then, for k odd,

(1.4)  A_{k}^{−1} = inf_{f∈F_{k}} \frac{∫_{Ω} |∇Δ[[\frac{k}{2}]]_{g}f|^{2}dg}{(∫_{Ω} fdg)^{2}}

and for k even,

(1.5)  A_{k}^{−1} = inf_{f∈F_{k}} \frac{∫_{Ω} |Δ[[\frac{k}{2}]]_{g}f|^{2}dg}{(∫_{Ω} fdg)^{2}} .

Variational quotients such as those appearing in the statement of Theorem (1.1) have found a number of applications in both physics and geometry (a very nice survey can be found in [O1]). For example; when M is two dimensional Euclidean space and k = 1, (A_{k}(Ω))^{−1} is perhaps better known as the torsional rigidity associated to a beam with cross-section Ω. In 1856, St-Venant conjectured that among domains of a given fixed area, torsional rigidity is maximized by a disk. The first complete proof of St-Venant’s conjecture was given by Polya in 1948. Polya’s approach to resolving the conjecture involved the study of the behavior of the variational quotient

\frac{∫_{Ω} |∇f|^{2}dx}{(∫_{Ω} fdx)^{2}}

under (Schwarz) symmetrization. This approach had been used by Faber [F1] and Krahn [K1] to resolve the conjecture of Rayleigh concerning the behavior of the first Dirichlet eigenvalue: Among smoothly bounded domains of fixed volume in Euclidean space, the principle Dirichlet eigenvalue is minimized by a ball. To resolve this conjecture, one studies the behavior of the variational quotient

\frac{∫_{Ω} |∇f|^{2}dx}{∫_{Ω} f^{2}dx}

under (Schwarz) symmetrization.

The results of Faber and Krahn establish more than the conjecture of Rayleigh: they relate the size of the principle Dirichlet eigenvalue on domains in a Riemannian manifold to the size of the principle Dirichlet eigenvalue in a constant curvature comparison space, provided the Riemannian manifold satisfies an isoperimetric condition (cf Definition (2.1) below). Faber and Krahn’s result can be concisely stated: If (M, g) is a Riemannian manifold satisfying an isoperimetric condition with constant curvature space form M_{κ}, then Volume_{g}(Ω) = v_{0} implies that λ_{1}(Ω) ≥ λ_{1}(B) where B is a geodesic ball in M_{κ} of volume v_{0} and λ_{1} is the first Dirichlet eigenvalue for the Laplace operator induced by the corresponding metric. Our second theorem establishes a similar result for the averaged moments A_{k}(Ω) and for the extremized moments S_{k}(Ω).

Theorem 1.2. Let (M, g) be a Riemannian manifold satisfying an isoperimetric condition with constant curvature comparison space form M_{κ}. Then for each positive integer k, Volume_{g}(Ω) = v_{0} implies that A_{k}(Ω) ≤ A_{k}(B) where B is a geodesic ball in M_{κ} of volume v_{0} and A_{k} is the kth averaged exit time moment for Brownian motion the corresponding metric. The same result holds for the functions S_{k} defined by (1.1).

As an immediate corollary of Theorem (1.2), we get a result concerning the uniform boundedness of the maps A_{k} : If D_{v} = {Ω ∈ D : Vol_{g}(Ω) = v}, then for each positive integer k the map A_{k} : D_{v} → IR is bounded. This result also holds for the maps S_{k} (cf Corollary 4.1).

In the case where M is Euclidean space IR^{n}, we use the techniques employed to prove Theorem 1.2 to obtain a uniform boundedness result for more general diffusions (cf Theorem 5.1).

This paper is organized as follows. In section 2 we recall basic facts and definitions which will be used throughout. In particular, we provide a representation for the functions, S_{k} and A_{k} in terms of a heirarchy of boundary value problems on the underlying domain. We also provide necessary information on Schwarz symmetrization, including proofs of several elementary results which we were unable to locate in the literature. In section 3 we prove Theorem 1.1. In section 4 we prove Theorem 1.2. In section 5 we establish a sufficient condition for a diffusion on Euclidean space to admit uniformly bounded moments and uniformly bounded average moments of all orders, on the space of domains constrained to be of given volume.

Before proceeding a few words concerning background are in order. The literature concerning isoperimetric conditions and their various analytic and geometric implications is vast. Those looking for a nice introduction to the topic are directed to the survey article [O1], to the texts [B] and [C], and to the references therein. Much of the present work is inspired by investigations of the Euclidean case. In particular, the formulae for the variational quotients given in Theorem 1.1 where first discovered for IR^{n} and will appear in [KM2]. Similarly, an early version of the rearrangement argument which appears in the proof of Theorem 1.2 first appears for the special case of IR^{n} and k = 2 in [KM1]. The author would like to thank Kimberly Kinatader for a number of helpful conversations.

2. Basic results and definitions

Throughout this section, let (M, g) be a Riemannian manifold.

Let (Y, B) be a measurable space and {P_{x}}_{x∈M} a family of probability measures on (Y, B). Let {X_{t}}_{t≥0} denote a diffusion on M with generator L, a uniformly elliptic operator, and for which P_{x}{X_{0} = x} = 1, for x ∈ M.

Let Ω ⊂ M be a smoothly bounded, connected, open domain with compact closure. As in the introduction, we define the first exit time for a particle driven by X_{t} from Ω by τ = τ(ω) = inf{t : X_{t}(ω) ∉ Ω}. For each x ∈ M, we will denote the expected value of a random variable Z under the probability measure P_{x} by E_{x}(Z).

There is a useful relationship between the solution of a certain Poisson problem on the domain Ω ∈ D and the expected value of the kth power of the first exit time of a particle driven by X_{t} from Ω starting at x ∈ Ω. Suppose u_{k} solves the problem

\begin{aligned}L^{k}u_{k} + (−1)^{k−1}k! & = 0 on Ω \\ u_{k} = Lu_{k} = ⋯ L^{k−1}u_{k} & = 0 on ∂Ω. \end{aligned}

Note that u_{k} can be defined inductively by

(2.1)  \begin{aligned}Lu_{1} + 1 & = 0 on Ω \\ u_{1} & = 0 on ∂Ω \end{aligned}

and

(2.2)  \begin{aligned}Lu_{k} + ku_{k−1} & = 0 on Ω \\ u_{k} & = 0 on ∂Ω. \end{aligned}

Using the generalized Dynkin formula [H] (cf also [AK]), we have

E_{x}[u_{k}(X_{0})] − E_{x}[u_{k}(X_{τ})] = ∑_{j=1}^{k−1} \frac{(−1)^{j}}{j!} E_{x}[τ^{j}L^{j}u_{k}(X_{τ})]

+ \frac{(−1)^{k}}{(k − 1)!}E_{x} [∫_{0}^{τ} s^{k−1}L^{k}u_{k}(X_{s})ds] .

Using the definition of u_{k} and τ, we obtain u_{k}(x) = E_{x} [τ^{k}] .

Suppose that L = Δ_{g} is the Laplace operator, Lf = div(∇f). Let u_{k} be as defined in (2.2). Then, from the above, we have

(2.3)  S_{k}(Ω) = sup_{x∈Ω} u_{k}(x)
(2.4)  A_{k}(Ω) = ∫_{Ω} u_{k}(x)dg

where dg is the volume form induced by the metric g.

We now focus on those aspects of geometric measure theory that we will need in the sequel. Let Ω ⊂ M be a smoothly bounded, connected, open domain with compact closure. Suppose that w ∈ C^{0}(Ω̄)∩C^{∞}(Ω) satisfies w|_{∂Ω} = 0. Let Reg(w) denote the regular values of w ( Reg(w) is an open set in IR).

For μ ≥ 0, define

(2.5)  Ω(μ) = {x ∈ Ω : |w(x)| > μ}
(2.6)  Γ(μ) = {x ∈ Ω : |w(x)| = μ} = ∂Ω(μ).

Using the metric density, dg, define

(2.8)  V (μ) = Vol_{g}(Ω(μ))

where Vol_{g} denotes the volume with respect to the density dg. It is well known (cf [F])that V (μ) is decreasing, has discontinuities only at those t for which Γ(μ) has positive g-volume, and is a smooth function on Reg(w). For μ ∈ Reg(w), the derivative of V (μ) is given by the co-area formula:

(2.9)  V′(μ) = − ∫_{Γ(μ)} |∇w|^{−1}dA_{μ}

where ∇ denotes the gradient operator and dA_{μ} is the induced Riemannian density on Γ(μ). We will use the co-area formula to study the behavior of solutions to Poisson problems for smoothly bounded domains in a large class of Riemannian manifolds. We begin by recalling a number of standard definitions and constructions.

Let M_{κ} be a constant curvature space form of curvature κ and dimension n. Recall that for κ > 0, M_{κ} is given by the Euclidean sphere of radius κ centered at the origin, that for κ = 0, M_{κ} is n-dimensional Euclidean space, and that for κ < 0, M_{κ} is hyperbolic n space, which we model using the Poincaré disk. For each value of κ we fix a basepoint p ∈ M_{κ} : for κ > 0, let p be the south pole, for κ ≥ 0, let p be the origin. For r ∈ IR, r > 0, we designate by B_{κ}(r) the open geodesic ball in M_{κ} of radius r, centered at p ∈ M_{κ}. Given x ∈ M_{κ}, we will denote by |x|_{κ} the distance from x to p ∈ M_{κ}. If Ω ⊂ M_{κ}, we will denote by Vol_{κ}(Ω) the volume of Ω with respect to the natural density dg_{κ}.

We will be particularly interested in those manifolds which satisfy an isoperimetric condition:

Definition 2.1. We say that a Riemannian manifold (M, g) satisfies an isoperimetric condition if there is a constant curvature space form, M_{κ}, such for all smoothly bounded domains with compact closure, Ω ⊂ M, we have

Vol_{g}(Ω) = v_{0} implies that Area_{g}(∂Ω) ≥ Area_{κ}(∂B)

where B is a geodesic ball in M_{κ} of volume v_{0} and Area is the surface area induced by the corresponding metric. If (M, g) satisfies an isoperimetric condition, we call the associated constant curvature space form, M_{κ}, the comparison space form.

We note that any constant curvature space form, M_{κ}, satisfies an isoperimetric condition with comparison space form M_{κ}.

Extensive work in the case of Euclidean space has provided direction as to how to use the existence of an isoperimetric condition to investigate properties of solutions to various elliptic boundary value problems for domains in IR^{n}. We will restrict our attention to an approach established by Schwarz in his study of the classical isoperimetric inequality (cf [O1], [B], and references therin). The approach associates to a domain Ω ⊂ IR^{n}, and a nonnegative function w : Ω → IR, a rearranged domain, Ω^{∗} ⊂ IR^{n} (a ball centered at the origin), and a symmetric decreasing function, w^{∗} : Ω^{∗} → IR, which is equimeasurable with w. Extensions of Schwarz symmetrization have been used by a number of authors to study the properties of Riemannian manifolds satisfying isoperimetric conditions (cf [O1], [B], [C]). We recall the definitions and basic properties of Schwarz symmetrization below and follow the methods of Faber and Krahn to obtain results which will be needed to establish Theorem 1.2.

Definition 2.2. Let (M, g) be a Riemannian manifold satisfying an isoperimetric condition with comparison space form M_{κ}. Let Ω ⊂ M be a bounded, connected open domain. The symmetrization of Ω, denoted Ω^{∗} ⊂ M_{κ}, is the geodesic ball B_{κ}(r) satisfying Vol_{g}(Ω) = Vol_{κ}(B_{κ}(r)).

Suppose that w : Ω → IR is nonnegative. For μ > 0, let

Ω(μ) = {x ∈ Ω : w(x) ≥ μ}.

The symmetrization of w, denoted w^{∗}, is the function w^{∗} : Ω^{∗} → IR given by

(2.10)  w^{∗}(x) = sup{μ : x ∈ Ω(μ)^{∗}}.

The following properties of symmetric rearrangement follow easily from the definition:

(1) The rearrangement w^{∗} depends only on the geodesic distance to p ∈ M_{κ} and w^{∗} is nonincreasing.

(2) If w_{1} ≤ w_{2}, then w_{1}^{∗} ≤ w_{2}^{∗}.

(3) If w is continuous on Ω̄,

(2.11)  sup_{x∈Ω} w(x) = sup_{x∈Ω^{∗}} w^{∗}(x).

(4) The functions w and w^{∗} are equimeasurable in the sense that Vol_{g}({x ∈ Ω : w(x) ≥ μ})) = Vol_{κ}({x ∈ Ω^{∗} : w^{∗}(x) ≥ μ}) for all nonnegative real μ. In particular, for μ > 0,

(2.12)  ∫_{Ω(μ)} wdg ≤ ∫_{Ω(μ)^{∗}} w^{∗}dg_{κ}

where dg is the volume form induced by g and dg_{κ} is the natural volume form on M_{κ}.

Suppose that Ω ⊂ M and that w : Ω → IR is nonnegative. Let T = sup{t : w(x) = t for some x ∈ D} and define a function r : [0, T] → IR by

(2.13) r(μ) = sup{s ∈ IR; Vol_{κ}(B_{κ}(s)) < V (μ)}

where V (μ) is as in (2.8). Then r is continuous on [0, T], smooth on Reg(w)∩(0, T), and is strictly decreasing. Let R be the inverse function for r. Then, if t = r(μ) where V (μ) = Vol_{κ}(B_{κ}(r(μ))), we have

(2.14)  R(t) = w^{∗}(x) for x satisfying |x|_{κ} = r(μ).

We will need the following lemma, due to Faber-Krahn (cf [C]), whose proof we include for completeness:

Lemma 2.2. Suppose that (M, g) satisfies an isoperimetric condition with comparison space M_{κ}. Let w : Ω → IR be as above. Then, for μ ∈ Reg(w),

(2.15)  ∫_{Γ(μ)} |∇w|dA_{μ} ≥ −Area_{κ}(∂B_{κ}(r(μ)))\frac{1}{r′(μ)}.

Proof. From the definition of r we have

V′(μ) = \frac{d}{dμ}Vol_{κ}(B_{κ}(r(μ)))
= Area_{κ}(∂B_{κ}(r(μ)))r′(μ)

and hence

r′(μ) = \frac{V′(μ)}{Area_{κ}(∂B_{κ}(r(μ)))}.

Using (2.9), we obtain

r′(μ) = −\frac{1}{Area_{κ}(∂B_{κ}(r(μ)))} ∫_{Γ(μ)} |∇w|^{−1}dA_{μ}.

By Cauchy-Schwarz,

(2.16)  −r′(μ) ∫_{Γ(μ)} |∇w|dA_{μ} ≤ \frac{A^{2}(μ)}{Area_{κ}(∂B_{κ}(r(μ)))}.

The lemma follows from (2.16) and the isoperimetric condition.

The following theorem, due to Szegö for Euclidean space (cf [PS1] and also [B]), will be crucial in the proof of Theorem 1.2. The proof is very similar to the argument given in the Euclidean case in [B].

Theorem 2.1. Let (M, g) be a Riemannian manifold satisfying an isoperimetric condition with comparison space form M_{κ}. Let Ω ⊂ M be a bounded, connected open domain and suppose that f : Ω → IR is a positive Lipschitz function. Suppose that u solves

(2.17)  \begin{aligned}Δ_{g}u + f & = 0 on Ω \\ u & = 0 on ∂Ω. \end{aligned}

Let Ω^{∗} be the symmetrization of Ω and let u^{∗}, f^{∗} : Ω^{∗} → IR be the symmetrizations of u and f, respectively. Let v be the solution of the symmetrized problem:

(2.18)  \begin{aligned}Δ_{κ}v + f^{∗} & = 0 on Ω^{∗} \\ v & = 0 on ∂Ω^{∗} \end{aligned}

where Δ_{κ} is the Laplace operator on M_{κ}. Then, u^{∗}(x) ≤ v(x) for all x ∈ Ω^{∗}.

Proof. Let u be as above. Then, for all μ, u is constant on the boundary of Ω(μ) and the divergence theorem gives

(2.19)  − ∫_{Ω(μ)} Δ_{g}udg = ∫_{Γ(μ)} |∇u|dA_{μ}.

Since u satisfies Δ_{g}u = −f,

∫_{Γ(μ)} |∇u|dA_{μ} = ∫_{Ω(μ)} fdg
≤ ∫_{Ω(μ)^{∗}} f^{∗}dg_{κ}

where we have used (2.12). Applying Lemma 2.2 to u, we conclude, using (2.15), that

−\frac{1}{r′(μ)} ≤ \frac{∫_{Ω(μ)^{∗}} f^{∗}dg_{κ}}{Area_{κ}(∂B_{κ}(r(μ)))}.

We rewrite this as

(2.20)  −R′(μ) ≤ \frac{∫_{Ω(μ)^{∗}} f^{∗}dg_{κ}}{Area_{κ}(∂B_{κ}(r(μ)))}.

Note that Ω(μ)^{∗} is a ball in the constant curvature comparison space M_{κ} and that f^{∗} depends only on the geodesic distance from the center of Ω(μ)^{∗}. Using (2.18), the Divergence Theorem, and the fact that v depends only on geodesic distance to the center of Ω(μ)^{∗}, the right hand side of (2.20) can be integrated. If we integrate both sides of the inequality (2.20) with respect to μ from μ = T and use (2.14), we obtain

u^{∗}(x) ≤ v(x)

which concludes the proof of the proposition.

3. Variational Characterizations and the proof of Theorem (1.1)

Throughout this section let M be a Riemannian manifold with metric g, let Δ_{g} denote the Laplace operator, let Ω be as above and let F_{k} be given as in Theorem 1.1.

Let u_{1} be as defined in (2.1) and let f ∈ F_{k}. Let ν be the outward pointing unit normal vector to ∂D. By the Divergence Theorem,

∫_{Ω}(fΔ_{g}u_{1} − u_{1}Δ_{g}f)dg = ∫_{∂D} f(⟨∇u_{1}, ν⟩ − u_{1}⟨∇f, ν⟩)dA_{g} = 0

where the pairing is given by the metric g and dA_{g} denotes the measure induced by g on ∂D. We conclude

(3.1)  ∫_{Ω} fdg = − ∫_{Ω} u_{1}Δ_{g}fdg.

If X is a vectorfield on Ω and f ∈ F_{k}, then div(fX) = fdiv(X) + ⟨∇f, X⟩. By the Divergence Theorem,

∫_{Ω} div(fX)dg = ∫_{∂D} f⟨X, ν⟩dA_{g} = 0

and we conclude that

∫_{Ω} fdiv(X)dg = − ∫_{Ω}⟨∇f, X⟩dg.

In particular, if u_{1} is as defined in (2.1) and X = ∇u_{1}, then

(3.2)  ∫_{Ω} fdg = ∫_{Ω}⟨∇u_{1}, ∇f⟩dg.

Lemma 3.1. Let u_{n} be as defined by (2.2), let k be a positive integer, and let f ∈ F_{k}. If k = 2n, then

(3.3)  ∫_{Ω} fdg = \frac{(−1)^{n}}{n!} ∫_{Ω} u_{n}Δ_{g}^{n}fdg.

If k = 2n + 1, then

(3.4)  ∫_{Ω} fdg = \frac{(−1)^{n}}{(n + 1)!} ∫_{Ω}⟨∇u_{n+1}, ∇Δ_{g}^{n}f⟩dg

where the pairing is given by the metric g.

Proof. Suppose that k = 2n and for 0 ≤ l ≤ n − 1, define

(3.5)  P_{l} = (Δ_{g}^{l}u_{n})(Δ_{g}^{n−l}f) − (Δ_{g}^{l+1}u_{n})(Δ_{g}^{n−(l+1)}f).

Then

∑_{l=0}^{n−1} P_{l} = u_{n}Δ_{g}^{n}f − fΔ_{g}^{n}u_{n}.

Let ν be the outward pointing unit normal vector along ∂D. By the Divergence Theorem and the fact that Δ_{g}^{l}u_{n} = 0 on ∂Ω, for l = 0, …, n − 1,

(3.6) ∫_{Ω} P_{l}dg = ∫_{∂D}(Δ_{g}^{l}u_{n})⟨∇Δ_{g}^{n−l−1}f, ν⟩ − (Δ_{g}^{n−(l+1)}f)⟨∇Δ_{g}^{l}u_{n}, ν⟩dA_{g} = 0.

Combining (3.5) and (3.6) and using that Δ_{g}^{n}u_{n} = (−1)^{n}n!, we have established (3.3).

Suppose k = 2n + 1 and for 0 ≤ l ≤ n, define

R_{l} = (Δ_{g}^{l}u_{n+1})(Δ_{g}^{n+1−l}f) − (Δ_{g}^{l+1}u_{n+1})(Δ_{g}^{n+1−(l+1)}f).

Then

∑_{l=0}^{n} R_{l} = u_{n+1}Δ_{g}^{n+1}f − fΔ_{g}^{n+1}u_{n+1}.

As above, we use the Divergence Theorem to see that

∫_{Ω} R_{l}dg = ∫_{∂D}(Δ_{g}^{l}u_{n+1})⟨∇Δ_{g}^{n−l}f, ν⟩ − (Δ_{g}^{n−l}f)⟨∇Δ_{g}^{l}u_{n+1}, ν⟩dA_{g} = 0.

Since L^{n+1}u_{n+1} = (−1)^{n+1}(n + 1)!, we conclude

∫_{Ω} fdg = \frac{(−1)^{n+1}}{(n + 1)!} ∫_{D} u_{n+1}Δ_{g}(Δ_{g}^{n}f).

By the Divergence Theorem,

∫_{Ω} u_{n+1}Δ_{g}(Δ_{g}^{n}f)dg + ∫_{Ω}⟨∇u_{n+1}, Δ_{g}^{n}f⟩dg = ∫_{Ω} div(u_{n+1}∇Δ_{g}^{n}f)dg
= 0.

We conclude that

∫_{Ω} u_{n+1}Δ_{g}(Δ_{g}^{n}f)dg = − ∫_{Ω}⟨∇u_{n+1}, ∇Δ_{g}^{n}f⟩dg

and we have established (3.4).

Proof of Theorem 1.1. The proof of Theorem 1.1 now procedes along the lines of the argument given in [KM2]. Suppose k = 2n and, for f ∈ F_{k}, consider the quotient

Q_{k}(f) = \frac{(∫_{Ω} fdg)^{2}}{∫_{Ω} |Δ_{g}^{n}f|^{2}dg} .

From (3.3)

Q_{k}(f) = \frac{(\frac{1}{n!})^{2} (∫_{Ω} u_{n}Δ_{g}^{n}fdg)^{2}}{∫_{Ω} |Δ_{g}^{n}f|^{2}dg} .

Let G_{k} = {h ∈ F_{n} : h = Δ_{g}^{n}f for some f ∈ F_{k}}. Let H_{k} be the completion of G_{k} in W^{2,2}, the Hilbert space of square integrable functions on Ω. If we denote the inner product of h and w ∈ W^{2,2} by ⟨h, w⟩ and by ∥h∥ the W^{2,2} norm of h, then we can view Q_{k} as a map Q_{k} : G_{k} ⊂ H_{k} → IR,

Q_{k}(g) = (\frac{1}{n!})^{2} (\frac{⟨u_{n}, h⟩}{∥h∥})^{2} .

Clearly, the domain of Q_{k} can be extended to nonzero elements of H_{k} and Q_{k}(ch) = Q_{k}(h) for every nonzero scalar c. It follows that Q_{k} is maximized when h ∈ H_{k} is in the direction of u_{n} ∈ H_{k}. If h = cu_{n} we have Δ_{g}^{n}(c′u_{2n}) = h, and computing Q_{k}(cu_{n}) we see that

sup_{h∈H_{k}} Q_{k}(h) = Q_{k}(cu_{n})

= \begin{matrix}(∫_{Ω} u_{2n}dg) & 2 \\ ∫ & Ω & (Δ_{g}^{n}u_{2n}dg) & 2 \end{matrix}

where we have applied (3.3) of Lemma 3.1 to the numerator. Note that (Δ_{g}^{n}u_{2n})^{2} = \frac{(−1)^{n}}{n!} (2n)!u_{n}Δ_{g}u_{2n}. Applying Lemma 3.1 to the denominator we obtain

sup_{h∈H_{k}} Q_{k}(h) = \frac{(∫_{Ω} u_{2n}dg)^{2}}{(2n)! ∫_{Ω} u_{2n}dg}

= \frac{1}{k!}A_{k}(Ω)

which establishes (1.3) of Theorem 1.1.

The proof of (1.4) of Theorem 1.1 is similar. Suppose k = 2n+1 and, for f ∈ F_{k}, consider the quotient

Q̃_{k}(f) = \frac{(∫_{Ω} fdg)^{2}}{∫_{Ω} |∇Δ_{g}^{n}f|^{2}dg} .

From (3.4) of Lemma 3.1,

Q̃_{k}(f) = \frac{(\frac{1}{(n+1)!})^{2} (∫_{Ω}⟨∇u_{n+1}, ∇Δ_{g}^{n}f⟩dg)^{2}}{∫_{Ω} |∇Δ_{g}^{n}f|^{2}dg} .

Let C^{∞}(Ω̄, IR^{n}) be the space of smooth vectorfields on Ω̄. Let

G̃_{k} = {X ∈ C^{∞}(Ω̄, IR^{n}) : there is f ∈ F_{k}, and h ∈ F_{n+1} with h = Δ_{g}^{n}f, X = ∇h}.

Let H̃_{k} be the completion of G̃_{k} in the space of vectorfields square integrable with respect to the inner product determined by the metric g. We can view Q̃_{k} as a map Q̃_{k} : G̃_{k} ⊂ H̃_{k} → IR,

Q̃_{k}(h) = (\frac{1}{(n + 1)!})^{2} (\frac{⟨∇u_{n+1}, h⟩}{∥h∥})^{2} .

It is clear that the domain of Q̃_{k} extends to nonzero vectors in the space H̃_{k} and that for all nonzero scalars c, Q̃_{k}(ch) = Q̃_{k}(h). It follows that Q̃_{k} is maximized when when h = c∇u_{n+1} where c is some nonzero constant. In particular, the maximizer is a smooth vectorfield h where h = ∇Δ_{g}^{n}(cu_{2n+1}) for some constant c, and we get that h ∈ F_{n}. Computing Q̃_{k}(∇u_{n+1}) we see that

sup_{h∈H̃_{k}} Q̃_{k}(h) = Q̃_{k}(c∇u_{n+1})

= \frac{(∫_{Ω} u_{2n+1}dg)^{2}}{∫_{Ω} ∥∇Δ_{g}^{n}u_{2n+1}∥^{2}dg}

where we have used (3.4) on the numerator.

Note that ∥∇Δ_{g}^{n}u_{2n+1}∥^{2} = \frac{(−1)^{n}}{(n+1)!}(2n + 1)!⟨∇u_{n+1}, ∇Δ_{g}^{n}u_{2n+1}⟩. Applying (3.4) of Lemma 3.1 to the denominator we obtain

sup_{h∈H̃_{k}} Q̃_{k}(h) = \frac{(∫_{Ω} u_{2n+1}dg)^{2}}{(2n + 1)! ∫_{Ω} u_{2n+1}dg}

= \frac{1}{k!}A_{k}(Ω)

which establishes (1.4) of Theorem 1.1.

4: Isoperimetric-type bounds and the proof of Theorem (1.2)

Proof of Theorem 1.2. For the case k = 1, we could proceed along the lines of the classical Faber-Krahn result using Theorem 1.1 (a nice presentation of the Faber-Krahn argument can be found in [C]). Unfortunately, the Faber-Krahn argument breaks down for k > 1. As an alternative, we use Theorem 2.1 and induction.

Let (M, g) be a Riemannian manifold satisfying an isoperimetric condition with comparison space form M_{κ}. Let Ω ⊂ M be a smoothly bounded domain with compact closure. Let u_{1} be the solution of

\begin{aligned}Δ_{g}u_{1} + 1 & = 0 on Ω \\ u_{1} & = 0 on ∂Ω. \end{aligned}

Let Ω^{∗} be the rearrangement of Ω and let u_{1}^{∗} be the rearrangement of u_{1}^{∗}. Let v_{1} be the solution of the symmetrized problem:

(4.1)  \begin{aligned}Δ_{κ}v_{1} + 1 & = 0 on Ω^{∗} \\ v_{1} & = 0 on ∂Ω^{∗}. \end{aligned}

By Theorem 2.1, u_{1}^{∗}(x) ≤ v_{1}(x). Since u_{1} and u_{1}^{∗} are equimeasurable,

∫_{Ω} u_{1}dg ≤ ∫_{Ω^{∗}} v_{1}dg_{κ}

which proves the theorem for the case k = 1.

Assume that the claim is true for all k ≤ m − 1. We need a lemma, which follows directly from the maximum principle:

Lemma 4.1. Let Ω be a smoothly bounded domain in M and suppose f_{1} and f_{2} are nonnegative Hölder continuous functions on Ω satisfying f_{1} ≥ f_{2}. Suppose w_{1} and w_{2} satisfy

Δw_{i} + f_{i} = 0 on Ω
w_{i} = 0 on ∂Ω.

Then w_{1} ≥ w_{2} on Ω.

Let u_{k} denote the solution of

\begin{aligned}Δ_{g}u_{k} + ku_{k−1} & = 0 on Ω \\ u_{k} & = 0 on ∂Ω. \end{aligned}

Let Ω^{∗} be the rearrangement of Ω and let u_{k−1}^{∗} be the rearrangement of u_{k−1}. Define v_{j} inductively as follows: for j = 1, let v_{j} be the solution of (4.1). Having defined v_{k−1} such that u_{k−1}^{∗} ≤ v_{k−1}, define v_{k} as the solution of

\begin{aligned}Δ_{κ}v_{k} + kv_{k−1} & = 0 on Ω^{∗} \\ v_{k} & = 0 on ∂Ω^{∗}. \end{aligned}

By Lemma 4.1, u_{k}^{∗} ≤ v_{k} and the inductive definition of the family v_{j} is complete. Since u_{k} and u_{k}^{∗} are equimeasurable, we have

∫_{Ω} u_{k}dg ≤ ∫_{Ω^{∗}} v_{k}dg_{κ}

which completes the proof of that portion of Theorem 1.2 which addresses the functional A_{k}. The proof of the portion of Theorem 1.2 addressing the functional S_{k} follows from the above construction of v_{k}(x) and (2.2). This concludes the proof of Theorem 1.2

Remark 4.1 There should be a convexity argument to establish Theorem 1.2. Such an argument was given in the case of Euclidean space by Aizenmann and Simon [AS]. For k ≥ 2, it would be interesting to find an argument which used the variational formulae given in Theorem 1.1. I was unable to find such an argument that worked, in part due to the continuity properties of symmetric rearrangement.

As an immediate corollary to Theorem 1.2 we have the following:

Corollary 4.1. Suppose that (M, g) is a Riemannian manifold which satisfies an isoperimetric condition with comparison space form M_{κ}. Suppose that v ∈ IR is fixed and let D_{v} = {Ω ∈ D : Vol_{g}(Ω) = v}. Then, for each positive integer k, the maps

S_{k} : D_{v} → IR
A_{k} : D_{v} → IR

are uniformly bounded.

5: Uniform bounds for exit time moments for a general class of diffusions

In this section we examine results concerning the existence of uniform bounds for general classes of diffusions on spaces which satisfy an isoperimetric conditions. We begin with some notation:

Definition 5.1. Suppose that X_{t} is a diffusion on M and that V ⊂ D. We say that X_{t} has bounded kth moment on V if there is a real number N such that S_{k}(Ω) < N for all Ω ∈ V. Similarly, we say that X_{t} has bounded average kth moment on V if there is a real number N such that A_{k}(Ω) < N for all Ω ∈ V. When V = D_{v} = {Ω ⊂ IR^{n} : Vol(Ω) = v}, and X_{t} has bounded k-th moment on V for every choice of v, we say that X_{t} has bounded k-th moments.

Let

C_{+}^{∞}(M) = {f ∈ C^{∞}(M) : inf_{x∈M} f(x) > 0}

Our first result is that C_{+}^{∞}(M) acts on the space of diffusions admitting bounded exit time moments and on the space of diffusions admitting bounded average exit time moments.

Proposition 5.1. Suppose that X_{t} is a diffusion on M with infintesimal generator L a uniformly elliptic operator. Suppose that X_{t} has bounded jth moments on V ⊂ D, j ≤ k. Suppose that M = fL where f ∈ C_{+}^{∞}(M), and that Y_{t} is the diffusion corresponding to M. Then M has bounded jth moments on V, j ≤ k. Similarly, Suppose if X_{t} has bounded jth average moments on V ⊂ D, j ≤ k, then M has bounded average jth moments on V, j ≤ k.

Proof. Suppose that inf f(x) = μ, μ > 0. Let w_{1} be the solution of

(5.1)  \begin{aligned}Mw_{1} & = −1 on Ω \\ w_{1} & = 0 on ∂Ω \end{aligned}

and let v_{1} be the solution of

(5.2)  \begin{aligned}μLv_{1} & = −1 on Ω \\ v_{1} & = 0 on ∂Ω. \end{aligned}

Then w_{1} is the solution of

Lw_{1} = −\frac{1}{f} on Ω
w_{1} = 0 on ∂Ω

and hence,

(5.3)  \begin{aligned}L(w_{1} − v_{1}) & = \frac{1}{μ} − \frac{1}{f} on Ω \\ w_{1} − v_{1} & = 0 on ∂Ω. \end{aligned}

But \frac{1}{μ} ≥ \frac{1}{f} , and hence the right hand side of (5.3) is nonnegative. Since L is elliptic and w_{1} − v_{1} is zero on the boundary, we obtain w_{1} − v_{1} ≤ 0. Since X_{t} has bounded first moment, we have proved the result for k = 1.

Having defined w_{1} by (5.1) and v_{1} by (5.2), define w_{k} and v_{k} inductively by

\begin{aligned}Mw_{k} + kw_{k−1} & = on Ω \\ w_{k} & = 0 on ∂Ω \end{aligned}

and

\begin{aligned}μLv_{k} + kv_{k−1} & = 0 on Ω \\ v_{k} & = 0 on ∂Ω. \end{aligned}

Assume that w_{j} − v_{j} ≤ 0 for all j < k. Then

\begin{aligned}L(w_{k} − v_{k}) & = \frac{kv_{k−1}}{μ} − \frac{kw_{k−1}}{f} on Ω \\ w_{k} − v_{k} & = 0 on ∂Ω. \end{aligned}

It follows from the maximum principle that w_{k} ≤ v_{k}, which proves the first part of the proposition. The second statement of the proposition follows from the first and the proof of the proposition is complete.

Proposition 5.1 and Theorem 1.2 have a number of corollaries. For example

Corollary 5.2. Let (M, g) be a Riemannian manifold satisfying an isoperimetric condition and suppose that M is two dimensional. Let h be a metric conformally equivalent to g with conformal factor α ∈ C_{+}^{∞}(M). Suppose that Y_{t} is Brownian motion on M with respect to h. Then Y_{t} has bounded exit time moments and bounded average exit time moments of all orders.

We now restrict our attention to diffusions in Euclidean space IR^{n} with the standard metric. We will be interested in classifying those diffusions which admit bounded kth moments.

Let X_{t} be a diffusion in Euclidean space with infinitesimal generator L, a second order differential operator with smooth coefficient. We introduce some notation:

Definition 5.2. Let L be a second order differential operator with smooth coefficients on IR^{n}. We say that L is self-adjoint if, for every smoothly bounded domain Ω ⊂ IR^{n}, with compact closure, L is self-adjoint with respect to Dirichlet conditions on ∂Ω.

It is easy to see that L is self-adjoint if and only if L is of divergence form.

Theorem 5.1. Let Y_{t} be a diffusion on Euclidean space with infinitesimal generator L, a second order differential operator with smooth coefficients. Suppose that L is self-adjoint and uniformly elliptic. Then Y_{t} admits bounded moments of all orders.

Proof. Suppose that L = div(a_{ij}∇) where a_{ij} is smooth and symmetric. Let 0 < λ be an ellipticity constant; for 0 < |ξ|,

λ|ξ|^{2} ≤ ∑ a_{ij}ξ_{i}ξ_{j} ≤ λ^{−1}|ξ|^{2}.

Let Ω ⊂ IR^{n} be a smoothly bounded domain with compact closure. Let G_{L}(x, y) be the Green’s function for L on Ω. Let u_{1}(x) be the solution of

Lu_{1} + 1 = 0 on Ω
u_{1} = 0 on ∂Ω.

Then u_{1} can be expressed in terms of G_{L} :

u_{1}(x) = ∫_{Ω} G_{L}(x, y)dy.

For n > 2, it is a theorem of [GW] that G_{L}(x, y) can be estimated in terms of the Newtonian potential and the ellipticity constant. More precisely, there is a constant, K, depending only on dimension, such that

(5.4)  G_{L}(x, y) ≤ Kλ^{−1}|x − y|^{2−n}.

Let Ω^{∗} be the rearrangement of Ω (Ω^{∗} is the ball of volume Vol(Ω) centered at the origin). Then, we have

∫_{Ω} |x − y|^{2−n}dy ≤ ∫_{Ω^{∗}} |y|^{2−n}dy

≤ CR^{2} (5.5)

where R is the radius of the ball of volume Vol(Ω) centered at the origin and C is a constant depending only on dimension. Let τ be the first exit time of Y_{t}. Combining (5.4), (5.5), and (2.3), we conclude

sup_{x∈Ω} E_{x}[τ] ≤ K′λ^{−1}

with constant K′ depending only on the volume of Ω and dimension. This proves that Y_{t} has bounded first moments. As in (2.2), let u_{k} be the solution of

Lu_{k} + ku_{k−1} = 0 on Ω
u_{k} = 0 on ∂Ω.

Then u_{k} can be expressed in terms of G_{L} and u_{k−1} :

u_{k}(x) = ∫_{Ω} ku_{k−1}G_{L}(x, y)dy

and the theorem follows by induction.

It is clear from the proof that any process for which there is a kernel which can be bounded in terms of the Newtonian potential will have bounded first moments. A class of processes (of importance in modeling both physical and economic systems) for which this might be the case is the class of symmetric α-stable processes. To carry out a study of the boundedness properties for exit time moments for such a process on a domain D ∈ D, one needs an estimate for the Green function, G_{D}(x, y), associated to the process of the form

G_{D}(x, y) ≤ C(α, n)|x − y|^{α−n}.

Such estimates will be developed elsewhere.

References

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[C] Chavel, I. (1984), Eigenvalues in Riemannian Geometry, Academic Press, Orlando, USA.

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[H] Ha’sminskii, R. Z. (1960), Probabilistic representation of the solution of some differential equations,, Proc. 6th All Union Conference on Theor. Prob. and Math. Stat. (Vilnius 190).

[KM1] Kinateder, K. K. J. and McDonald, P. (1997), Hypersurfaces in IR^{n} and the variance of exit times for Brownian motion, Proc. Amer. Math. Soc. 125, 2453–2462.

[KM2] Kinateder, K. K. J. and McDonald, P. (To appear), Variational principles for average exit time moments for diffusions in Euclidean space, Proc. Amer. Math. Soc..

[K1] Krahn, E. (1925), Über eine von Rayleigh formulierte Minmalegenschaft des Kreises, Math. Ann. 94, 97–100.

[O1] Osserman, R. (1978), The isoperimetric inequality, Bull. A.M.S. 84, 1182–1238.

[PS] Polya, G. and Szego, G. (1951), Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, N.J..

Department of Mathematics, New College of the University of South Florida 5700 N. Tamiami Trail Sarasota, USA

   E-mail address: ptm@virtu.sar.usf.edu