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18
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 46 (9 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 16 (6 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.
Heat Equation And Reflected Brownian Motion In Time Dependent Domains
 J. Funct. Anal
"... This article is mostly devoted to questions of analytic nature; somewhat paradoxically in view of the original motivation, probability mostly plays here the role of a tool and not an end. The analytic literature on the heat equation and related problems is enormous and we would rather let the reader ..."
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Cited by 10 (4 self)
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This article is mostly devoted to questions of analytic nature; somewhat paradoxically in view of the original motivation, probability mostly plays here the role of a tool and not an end. The analytic literature on the heat equation and related problems is enormous and we would rather let the reader search the library than provide an exceedingly imperfect review. Crank (1984) provides an excellent review of various problems related to free and 1. Research partially supported by NSF grant DMS9700721. 2. Research partially supported by NSA grant MDA9049910104. 3. Research partially supported by NSF grant DMS{9801068 and ONR grants N0001493 {0295 and N000149810675. 1 moving bondaries. Although one can see obvious general similarities between our problem and the classical Stefan's problem, it remains to be seen if there exist any connections at the technical level. We will be much more specic on the probabilistic side as we feel that we can provide an eective guide to an uninitiated reader who wants to learn more about the reected Brownian motion. Brownian motion in time dependent domains belongs to \classical" subjects in probability. The model appears in the context of a problem often referred to as \boundary crossing." The literature on the problem is huge; we suggest Anderson and Pitt (1997) and Durbin (1992) as starting points. The boundary crossing problem was mainly motivated by statistical questions but the estimates derived in this area have been also applied to study Brownian path properties, see, e.g., Bass and Burdzy (1996) or Greenwood and Perkins (1983). In the context of our article, this classical model may be described as a Brownian motion killed on the boundary of a timedependent domain. The corresponding analytic problem may be called the heat...
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 tw ..."
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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 main purpos ..."
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Cited by 8 (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...
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 7 (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
Traces of symmetric Markov processes and their characterizations
 ANN. PROBAB
, 2006
"... Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the Revuz measure used in the time change. In this paper we give a complete characterization of time changed ..."
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Cited by 6 (1 self)
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Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the Revuz measure used in the time change. In this paper we give a complete characterization of time changed processes of an arbitrary symmetric Markov process, in terms of the Beurling–Deny decomposition of their associated Dirichlet forms and of Feller measures of the process. In particular, we determine the jumping and killing measure (or, equivalently, the Lévy system) for the timechanged process. We further discuss when the trace Dirichlet form for the time changed process can be characterized as the space of finite Douglas integrals defined by Feller measures. Finally, we give a probabilistic characterization of Feller measures in terms of the excursions of the base process.
Boundary trace of reflecting Brownian motions. Probab. Theory Relat
 Fields
"... We establish a uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of nonsmooth domains. Exact Hausdorff dimensions for the boundary occupation time and the boundary trace of RBM are given. Extensions to stablelike jump processes and to symmetric reflecting dif ..."
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Cited by 4 (2 self)
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We establish a uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of nonsmooth domains. Exact Hausdorff dimensions for the boundary occupation time and the boundary trace of RBM are given. Extensions to stablelike jump processes and to symmetric reflecting diffusions are also mentioned.