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On Sullivan's construction of eigenfunctions via exit times of Brownian motion

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Pith's one-line read The expected value of e^{-λτ}φ(B_τ) over Brownian exit times solves Δh = λh with boundary value φ on negatively curved manifolds.

arxiv 1908.00465 v1 pith:GHLR7IOV submitted 2019-08-01 math.DG math.PR

Kingshook Biswas This is my paper · ORCID
classification math.DGmath.PR
keywords lambdabrownianmotionexitlaplacianmathbbboundaryeigenfunctions
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The pith

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The reading

Imagine a random walker moving on a curved, infinitely large space, with a bounded region D that has a smooth edge. The walker starts inside D and moves randomly until it first touches the edge; that moment is the exit time, call it τ. Sullivan's recipe for building special solutions of the Laplacian is to take the expected value of the boundary data φ, multiplied by an exponential weight e^{-λτ}, over all random paths. This paper proves that this recipe works in full detail for spaces with pinched negative curvature, meaning the curvature lies between two negative constants, like a saddle that curves everywhere. The proof uses standard machinery. The heat kernel describes how random motion spreads out over time. The Dirichlet heat kernel for D absorbs the walker at the edge. A series of estimates controls how likely the walker is to linger near the boundary. The key technical step shows that the function h defined by the random-path average is smooth, satisfies Δh = λh inside D, and approaches the prescribed boundary value φ as the starting point gets close to the edge. The case λ = 0 reduces to the classical solution of the Dirichlet problem. This is a 'details' paper: the result was stated by Dennis Sullivan in 1987 without a proof, and experts believed it to be true. This note makes the argument available as a readable, checkable proof, and notes that it should extend to any setting where Brownian motion is transient and the manifold is stochastically complete.
Extended reading notes

Core claim

For any λ ∈ ℂ with Re λ > λ1, and for any continuous function φ : ∂D → ℂ, the function h(x) := E_x(e^{-λτ}φ(Bτ)) is C∞ on D and is an eigenfunction of Δ on D with eigenvalue λ and boundary value φ, meaning Δh = λh and h(x) → φ(ξ) as x ∈ D → ξ ∈ ∂D (Theorem 1.1). If the paper is correct, this theorem is true.

Load-bearing premise

The proof depends on the long-time heat kernel upper bound (equation (2)): p(t,x,y) ≤ K(1 + d(x,y)^2/t)^{(1+n)/2} exp(λ1 t - d(x,y)^2/(4t)) for t > 1, from Grigoryan [Gri94]. This estimate provides the exponential decay with the negative spectral bottom λ1 < 0, and is used in Proposition 6.1 to prove that e^{-λτ} is integrable for Re λ > λ1, and in Lemma 6.5 to control boundary limits. If λ1 were 0, this representation would fail for negative real λ, so the negative spectral gap is load-bearing.

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Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. It relies on standard results from spectral geometry and stochastic analysis, all cited. The most load-bearing external input is Grigoryan's heat kernel upper bound (equation (2)), which supplies the exponential decay with the negative spectral bottom λ1; without it, the integrability of the exit-time weight and the boundary convergence argument would collapse.

assumptions (8)
  • standard math Heat kernel upper bounds of Grigoryan: p(t,x,y) ≤ C t^{-n/2} exp(c λ1 t - d^2/(D t)) for all t>0, and p(t,x,y) ≤ K(1 + d^2/t)^{(1+n)/2} exp(λ1 t - d^2/(4t)) for t>1.
    Used throughout: Lemma 2.1, Proposition 6.1, Lemma 6.5. Provides the exponential decay with λ1 that makes the exit-time weight integrable.
  • standard math Yau's theorem: a complete Riemannian manifold with Ricci curvature bounded below is stochastically complete, so the heat semigroup satisfies e^{tΔ}1 = 1.
    Used in Section 3.2 to verify the Feller property (e) and ensure Brownian motion is conservative.
  • standard math Feller-Dynkin construction: a Feller semigroup on a locally compact separable metric space gives rise to a Markov process with cadlag paths, and with continuous paths if condition (7) holds (Ethier-Kurtz Theorem 2.7, Chapter 4).
    Used in Section 3 to construct Brownian motion from the heat semigroup.
  • standard math Spectral expansion of the Dirichlet heat kernel on a precompact domain with smooth boundary: p_D(t,x,y) = Σ e^{λ_k t} φ_k(x) φ_k(y), converging absolutely and uniformly for t>0.
    Used in Section 5 to establish the Dirichlet heat kernel and its properties, including p_D ≤ p.
  • standard math Parabolic maximum principle for the heat equation on a cylinder (Chavel, Section VIII.1).
    Used to prove g ≤ 0, hence p_D ≤ p (equation (10)), and Lemma 5.2.
  • standard math Transience criterion: Brownian motion is transient iff ∫_1^∞ p(t,x,x) dt < ∞.
    Used in Section 6 to show P_x(τ < ∞) = 1, so the exit time is finite almost surely.
  • standard math Gaffney's theorem: the Laplacian on C_c^∞(X) is essentially self-adjoint on L^2(X), so the heat semigroup e^{tΔ} is defined by functional calculus.
    Used in Section 2 to define the heat semigroup and the spectral parameter λ1.
  • standard math Elliptic regularity: if a distribution u satisfies Δu = λu on a domain, then u is smooth.
    Used in Proposition 6.4 to conclude h is C∞ on D from the distributional equation.

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Pith. "Pith review of On Sullivan's construction of eigenfunctions via exit times of Brownian motion." pith.science (2026). https://pith.science/paper/GHLR7IOV

@misc{pith2026190800465,
  author       = {Pith},
  title        = {Pith review of: On Sullivan's construction of eigenfunctions via exit times of Brownian motion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GHLR7IOV}},
  note         = {Machine review of arXiv:1908.00465}
}
abstract

The purpose of this note is to give details for an argument of Sullivan to construct eigenfunctions of the Laplacian on a Riemannian manifold using exit times of Brownian motion \cite{sullivanpos}. Let $X$ be a complete, simply connected Riemannian manifold of pinched negative sectional curvature. Let $\lambda_1 = \lambda_1(X) < 0$ be the supremum of the spectrum of the Laplacian on $L^2(X)$, and let $D \subset X$ be a bounded domain in $X$ with smooth boundary. Let $(B_t)_{t \geq 0}$ be Brownian motion on $X$ and let $\tau = \tau_D$ be the first exit time of Brownian motion from $D$. For each $\lambda \in \mathbb{C}$ with $\hbox{Re } \ \lambda > \lambda_1$ and $x \in D$, we show that for any continuous function $\phi : \partial D \to \mathbb{C}$, the function $$ h(x) = \mathbb{E}_x(e^{-\lambda \tau} \phi(B_{\tau})) \ , \ x \in D, $$ is an eigenfunction of the Laplacian on $D$ with eigenvalue $\lambda$ and boundary value $\phi$.

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Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    I. Chavel. Eigenvalues in R iemannian geometry. Academic Press, New York , 1984

    work page 1984

  2. [2]

    J. Dodziuk. Maximum principle for parabolic inequalities and the heat flow on open manifolds. Indiana Univ. Math J., 32 , no.5 , pages 703--716, 1983

    work page 1983

  3. [3]

    S. N. Ethier and T. G. Kurtz. Markov processes: characterization and convergence. Wiley series in probability and mathematical statistics , 1986

    work page 1986

  4. [4]

    M. P. Gaffney. A special S tokes' theorem for complete R iemannian manifolds. Ann. of Math (2) 60 , pages 140--145, 1954

    work page 1954

  5. [5]

    Grigoryan

    A. Grigoryan. Heat kernel upper bounds on a complete non-compact manifold. Revista Matematica Iberoamericana, 10 no.2 , pages 395--452, 1994

    work page 1994

  6. [6]

    Grigoryan

    A. Grigoryan. Analytic and geometric background of recurrence and non-explosion of the B rownian motion on R iemannian manifolds. Bull. Amer. Math. Soc., 36 , pages 135--249, 1999

    work page 1999

  7. [7]

    H. P. Mckean. An upper bound to the spectrum of the L aplacian on a manifold of negative curvature. J. Differential Geometry, 4 , pages 359--366, 1970

    work page 1970

  8. [8]

    Sullivan

    D. Sullivan. Related aspects of positivity in R iemannian geometry. J. Differential Geometry, 25 , pages 327--351, 1987

    work page 1987

Show all 9 references
  1. [9]

    S. T. Yau. On the heat kernel of a complete R iemannian manifold. J. Math Pures Appl., ser. 9, 57 , pages 191--201, 1978

    1978

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