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On Sullivan's construction of eigenfunctions via exit times of Brownian motion | | Kingshook Biswas
; | | Date: |
1 Aug 2019 | | 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 $ au =
au_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 $phi : partial D o mathbb{C}$, the
function $$ h(x) = mathbb{E}_x(e^{-lambda au} phi(B_{ au})) , x in
D, $$ is an eigenfunction of the Laplacian on $D$ with eigenvalue $lambda$ and
boundary value $phi$. | | Source: | arXiv, 1908.0465 | | Services: | Forum | Review | PDF | Favorites | |
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