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Invariant Wedges for a Two-Point Reflecting Brownian Motion and the ``Hot Spots'' Problem

   
@article{EJP91,
	author = {Rami Atar},
	title = {Invariant Wedges for a Two-Point Reflecting Brownian Motion and the ``Hot Spots'' Problem},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {6},
	year = {2001},
	keywords = {Reflecting Brownian motion, Neumann eigenvalue problem, convex domains.},
	abstract = {We consider domains $D$ of $R^d$, $d\ge 2$ with the property that there is a wedge $V\subset R^d$ which is left invariant under all tangential projections at smooth portions of $\partial D$. It is shown that the difference between two solutions of the Skorokhod equation in $D$ with normal reflection, driven by the same Brownian motion, remains in $V$ if it is initially in $V$. The heat equation on $D$ with Neumann boundary conditions is considered next. It is shown that the cone of elements $u$ of $L^2(D)$ satisfying $u(x)-u(y)\ge0$ whenever $x-y\in V$ is left invariant by the corresponding heat semigroup. Positivity considerations identify an eigenfunction corresponding to the second Neumann eigenvalue as an element of this cone. For $d=2$ and under further assumptions, especially convexity of the domain, this eigenvalue is simple.},
	pages = {no. 18, 1-19},
	issn = {1083-6489},
	doi = {10.1214/EJP.v6-91},    
        url = {http://ejp.ejpecp.org/article/view/91}}