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27
Brownian motion and harmonic analysis on Sierpinski carpets
 MR MR1701339 (2000i:60083
, 1999
"... Abstract. We consider a class of fractal subsets of R d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to con ..."
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Cited by 66 (12 self)
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Abstract. We consider a class of fractal subsets of R d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting. 1
Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes
 Electron. J. Probab
"... In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discretetime Markov processes, following Kontoyiannis and Meyn [32]. We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically er ..."
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Cited by 23 (5 self)
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In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discretetime Markov processes, following Kontoyiannis and Meyn [32]. We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of DonskerVaradhan. For any such process Φ = {Φ(t)} with transition kernel P on a general state space X, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals F: X → C, the kernel ̂ P (x, dy) = e F (x) P (x, dy) has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a “maximal ” solution (λ, ˇ f) to the multiplicative Poisson equation, defined as the eigenvalue problem ̂ P ˇ f = λ ˇ f. The functional Λ(F) = log(λ) is convex, smooth, and its convex dual Λ ∗ is convex, with compact sublevel sets.
Sobolev Spaces on NonSmooth Domains and Dirichlet Forms Related to Subordinate Reflecting Diffusions
, 2001
"... Let Ω be a bounded domain with fractal boundary, for instance von Koch’s snowflake domain. First we determine the range and the kernel of the trace on ∂Ω of Sobolev spaces of fractional order defined on Ω. This extends some earlier results of H. Wallin and J. Marschall. Secondly we apply these resu ..."
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Cited by 12 (0 self)
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Let Ω be a bounded domain with fractal boundary, for instance von Koch’s snowflake domain. First we determine the range and the kernel of the trace on ∂Ω of Sobolev spaces of fractional order defined on Ω. This extends some earlier results of H. Wallin and J. Marschall. Secondly we apply these results in studying Dirichlet forms related to subordinate reflecting diffusions in nonsmooth domains.
F.: Coupling and Harnack inequalities for Sierpinski carpets
 Bull. Amer. Math. Soc. (N.S
, 1993
"... Abstract. Uniform Harnack inequalities for harmonic functions on the pre and graphical Sierpinski carpets are proved using a probabilistic coupling argument. Various results follow from this, including the construction of Brownian motion on Sierpinski carpets embedded in R d, d≥3, estimates on the ..."
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Cited by 11 (1 self)
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Abstract. Uniform Harnack inequalities for harmonic functions on the pre and graphical Sierpinski carpets are proved using a probabilistic coupling argument. Various results follow from this, including the construction of Brownian motion on Sierpinski carpets embedded in R d, d≥3, estimates on the fundamental solution of the heat equation, and Sobolev and Poincaré inequalities. The Sierpinski carpets (SCs) we will study are generalizations of the Cantor set. Let F0 = [0, 1] d be the unit cube in R d, d ≥ 2, centered at z0 = (1/2,..., 1/2). Let k, a be integers with 1 ≤ a < k and a + k even. Divide F0 into k d equal subcubes, remove a central block of a d subcubes, and let F1 be what remains: thus F1 = F0 − ((k − a)/2k, (k + a)/2k) d. Now repeat this operation on each of the k d − a d remaining subcubes to obtain F2. Iterating, we obtain a decreasing sequence of closed sets Fn; then F = ⋂ ∞ n=0 Fn is a Sierpinski carpet and has Hausdorff dimension df = df(F) = log(k d − a d) / log(k). (When d = 2, k = 3, and a = 1, we get the usual Sierpinski carpet.) Let ̂ Fn = k n Fn ⊂ [0, ∞) d, and define the preSierpinski carpet by ̂ F = ⋃ ∞ n=1 ̂ Fn (see [10]). The graphical Sierpinski carpet is the graph G = (V, E) with vertex set V = (z0 + Z d) ∩ ̂ F and edge set E = { {x, y} ∈ V: x − y  = 1}. Thus int ( ̂ F) is a domain in R d with a largescale structure which mimics the smallscale structure of F. We are interested in the behavior of solutions of the Laplace and heat equations on F, ̂ F, and G. One reason for this is applications to “transport phenomena ” in disordered media (see [6]); another is the new type of behavior of the heat kernel on these spaces. Let W be Brownian motion on ̂ F with normal reflection on ∂ ̂ F, and let q(t, x, y) be the transition density of W, so that q solves the heat equation on ̂ F with Neumann boundary conditions on ∂ ̂ F. 1991 Mathematics Subject Classification. Primary 60B99; Secondary 60J35. Key words and phrases. Harnack inequality, Sierpinski carpets, self–similar, fractals, Brownian
On Neumann eigenfunctions in lip domains
 J. Amer. Math. Soc
"... AplanarsetDwill be called a lip domain if it is Lipschitz, open, bounded, connected, and given by (1) D = {(x1,x2):f1(x1) <x2 <f2(x1)}, where f1, f2 are Lipschitz functions with constant 1. The assumption that D is a ..."
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Cited by 10 (3 self)
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AplanarsetDwill be called a lip domain if it is Lipschitz, open, bounded, connected, and given by (1) D = {(x1,x2):f1(x1) <x2 <f2(x1)}, where f1, f2 are Lipschitz functions with constant 1. The assumption that D is a
WEAK CONVERGENCE OF REFLECTING BROWNIAN MOTIONS
 ELECTRONIC COMMUNICATIONS IN PROBABILITY
, 1998
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Synchronous couplings of reflected Brownian motions in smooth domains
 Illinois J. Math
, 2006
"... Abstract. For every bounded planar domain D with a smooth boundary, we define a “Lyapunov exponent ” Λ(D) using a fairly explicit formula. We consider two reflected Brownian motions in D, driven by the same Brownian motion (i.e., a “synchronous coupling”). If Λ(D)> 0 then the distance between the ..."
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Cited by 9 (7 self)
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Abstract. For every bounded planar domain D with a smooth boundary, we define a “Lyapunov exponent ” Λ(D) using a fairly explicit formula. We consider two reflected Brownian motions in D, driven by the same Brownian motion (i.e., a “synchronous coupling”). If Λ(D)> 0 then the distance between the two Brownian particles goes to 0 exponentially fast with rate Λ(D)/(2D) as time goes to infinity. The exponent Λ(D) is strictly positive if the domain has at most one hole. It is an open problem whether there exists a domain with Λ(D) < 0. 1. Introduction and
Fiber Brownian Motion And The "Hot Spots" Problem
 Duke Math. J
, 2000
"... . We show that in some planar domains both extrema of the second Neumann eigenfunction lie strictly inside the domain. The main technical innovation is the use of "fiber Brownian motion," a process which switches between twodimensional and onedimensional evolution. 1. Introduction. The m ..."
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Cited by 9 (5 self)
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. We show that in some planar domains both extrema of the second Neumann eigenfunction lie strictly inside the domain. The main technical innovation is the use of "fiber Brownian motion," a process which switches between twodimensional and onedimensional evolution. 1. Introduction. The main purpose of this article is to give a stronger counterexample to the "hot spots" conjecture than the one presented in Burdzy and Werner (1999). Along the way we will define and partly analyze a new process, which we call "fiber Brownian motion," and which may have some interest of its own. Consider a Euclidean domain which has a discrete spectrum for the Laplacian with Neumann boundary conditions, for example, a bounded domain with Lipschitz boundary. Recall that the first Neumann eigenfunction is constant. The "hot spots" conjecture says that the maximum of the second Neumann eigenfunction is attained at a boundary point. Burdzy and Werner (1999) constructed a domain where the second eigenfunction...
Invariant wedges for a two point reflecting Brownian motion and the "hot spots" problem
, 2001
"... We consider domains D of IR d , d 2 with the property that there is a wedge V IR d which is left invariant under all tangential projections at smooth portions of @D. It is shown that the dierence between two solutions of the Skorokhod equation in D with normal reection, driven by the same Brow ..."
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Cited by 8 (3 self)
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We consider domains D of IR d , d 2 with the property that there is a wedge V IR d which is left invariant under all tangential projections at smooth portions of @D. It is shown that the dierence between two solutions of the Skorokhod equation in D with normal reection, 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 of L 2 (D) with increments 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. Under further assumptions, especially convexity of the domain, this eigenvalue is simple. 1 Introduction The \hot spots" property of a bounded connected open domain D IR d refers to the location of the extrema of eigenfunctions corresponding to the second eigenvalue of the Laplacian on D with Neumann boundary...