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177
Heat kernel estimates for jump processes of mixed types on metric measure spaces
 FIELDS
"... In this paper, we investigate symmetric jumptype processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors dregular sets, which is a class of fractal sets that contains ge ..."
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Cited by 50 (30 self)
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In this paper, we investigate symmetric jumptype processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors dregular sets, which is a class of fractal sets that contains geometrically selfsimilar sets. A typical example of our jumptype processes is the symmetric jump process with jumping intensity e −c0(x,y)x−y � α2 α1 c(α, x, y) ν(dα) x − y  d+α where ν is a probability measure on [α1, α2] ⊂ (0, 2), c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c0(x, y) is a
Nonlocal Dirichlet forms and symmetric jump processes
 Transactions of the American Mathematical Society
, 1999
"... We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x ..."
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Cited by 31 (16 self)
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We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x  −d−α ≤ J(x, y) ≤ κ2y − x  −d−β for 0 < α < β < 2, x − y  < 1. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E, F). We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.
Harnack inequalities and subGaussian estimates for random walks
 Math. Annalen
, 2002
"... We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the m ..."
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Cited by 30 (6 self)
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We show that a fiparabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fiGaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R . The latter condition can be replaced by a certain estimate of a resistance of annuli.
Construction of Diffusions on Configuration Spaces
"... We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Ga ..."
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Cited by 27 (3 self)
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We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Gamma E ; ¯) for a large class of measures ¯ on \Gamma E (without assuming an integration by parts formula) generalizing all corresponding results known so far. Subsequently, we prove that under mild conditions the Dirichlet forms E \Gamma ¯ are quasiregular, and that hence there exist associated diffusions on \Gamma E , provided E is a complete separable metric space and \Gamma E is equipped with a suitable topology, which is the vague topology if E is locally compact. We discuss applications to the case where E is a finite dimensional manifold yielding an existence result on diffusions on \Gamma E which was already announced in [AKR96a, AKR96b], resp. used in [AKR98, AKR97b]. Furthermore...
Subtree prune and regraft: A reversible realtree valued Markov chain
 Ann. Prob
"... Abstract. We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis. A key technic ..."
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Cited by 25 (5 self)
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Abstract. We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis. A key technical ingredient in this work is the use of a novel Gromov– Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion. 1.
Harnack inequality for some classes of Markov processes
"... In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. ..."
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Cited by 25 (13 self)
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In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps.
Twosided heat kernel estimates for censored stablelike processes
, 2008
"... In this paper we study the precise behavior of the transition density functions of censored (resurrected) αstablelike processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored αstablelike process in any bounded Lipschitz open set is intrins ..."
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Cited by 24 (17 self)
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In this paper we study the precise behavior of the transition density functions of censored (resurrected) αstablelike processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored αstablelike process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp twosided estimates for the transition density functions of a large class of censored αstablelike processes in C 1,1 open sets. We further obtain sharp twosided estimates for the Green functions of these censored αstablelike processes in bounded C 1,1 open sets.
Potential theory of truncated stable processes
 MATHEMATISCHE ZEITSCHRIFT
, 2007
"... For any α ∈ (0, 2), a truncated symmetric αstable process is a symmetric Lévy process in R d with a Lévy density given by cx  −d−α 1{x<1} for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonneg ..."
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Cited by 22 (18 self)
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For any α ∈ (0, 2), a truncated symmetric αstable process is a symmetric Lévy process in R d with a Lévy density given by cx  −d−α 1{x<1} for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic functions of these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains. We give an example of a nonconvex domain for which the boundary Harnack principle fails.
Boundary Harnack principle for subordinate Brownian motions
"... We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κfat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the mini ..."
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Cited by 21 (18 self)
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We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κfat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded κfat open sets with respect to these processes with their Euclidean boundaries.