a note on self adjoint extensions of the laplacian on weighted graphs abstract we study the uniqueness of self adjoint and markovian extensions of the laplacian on weighted graphs we first show that for locally finite graphs and a certain family of metrics completeness of the graph implies uniqueness of these extensions moreover in the case when the graph is not metrically complete and the cauchy boundary has finite capacity we characterize the uniqueness of the markovian extensions

on the lp spectrum of laplacians on graphs abstract we study the p independence of spectra of laplace operators on graphs arising from regular dirichlet forms on discrete spaces here a sufficient criterion is given solely by a uniform subexponential growth condition moreover under a mild assumption on the measure we show a one sided spectral inclusion without any further assumptions we study applications to normalized laplacians including symmetries of the spectrum and a characterization for positivity of the cheeger constant furthermore we consider laplacians on planar tessellations for which we relate the spectral p independence to assumptions on the curvature

volume growth spectrum and stochastic completeness of infinite graphs we study the connections between volume growth spectral properties and stochastic completeness of locally finite weighted graphs for a class of graphs with a very weak spherical symmetry we give a condition which implies both stochastic incompleteness and discreteness of the spectrum we then use these graphs to give some comparison results for both stochastic completeness and estimates on the bottom of the spectrum for general locally finite weighted graphs

invariance principle for the random conductance model with unbounded conductances we study a continuous time random walk x in an environment of i i d random conductances e 1 we obtain heat kernel bounds and prove a quenched invariance principle for x this holds even when e e

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

random walks on supercritical percolation clusters we obtain gaussian upper and lower bounds on the transition density qt x y of the continuous time simple random walk on a supercritical percolation cluster c1 in the euclidean lattice the bounds analogous to aronsen s bounds for uniformly elliptic divergence form diusions hold with constants ci depending only on p the percolation probability and d the irregular nature of the medium means that the bound for qt x only holds for t sx where the constant sx depends on the percolation congura tion

ollivier ricci curvature for general graph laplacians heat equation laplacian comparison non explosion and diameter bounds abstract discrete time random walks on a finite set naturally translate via a one to one correspondence to discrete laplace operators typically ollivier curvature has been investigated via random walks we first extend the definition of ollivier curvature to general weighted graphs and then give a strikingly simple representation of ollivier curvature using the graph laplacian using the laplacian as a generator of a continuous time markov chain we connect ollivier curvature with the heat equation which is strongly related to continuous time random walks in particular we prove that a lower bound on the ollivier curvature is equivalent to a certain lipschitz decay of solutions to the heat equation this is a discrete analogue to a celebrated ricci curvature lower bound characterization by renesse and sturm our representation of ollivier curvature via the laplacian allows us to deduce a laplacian comparison principle by which we prove non explosion and improved diameter bounds

volume growth and spectrum for general graph laplacians we prove estimates relating exponential or sub exponential volume growth of weighted graphs to the bottom of the essential spectrum for general graph laplacians the volume growth is computed with respect to a metric adapted to the laplacian and use of these metrics produces better results than those obtained from consideration of the graph metric only conditions for absence of the essential spectrum are also discussed these estimates are shown to be optimal or near optimal in certain settings and apply even if the laplacian under consideration is an unbounded operator

a generalized conservation property for the heat semigroup on weighted manifolds in this text we study a generalized conservation property for the heat semigroup generated by a schrodinger operator with nonnegative potential on a weighted manifold we establish khasminskii s criterion for the generalized conservation property and discuss several applications

compressed algebraic cubature over polygons with applications to optical design abstract in this paper we propose an algorithm to determine cubature rules of algebraic degree of exactness on general polygons p by means of matlab polyshape objects and near minimal rules on triangles obtaining by caratheodory tchakaloff subsampling a pi positive interior final formula with cardinality at most n 1 2 2 we test our algorithm on polygons with different shape and we also discuss an application to the computation of the rmswe root mean square wavefront error on obscured and vignetted pupils in the framework of optical design by numerical ray tracing for the lsst large synoptic survey telescope

stochastic completeness for graphs with curvature dimension conditions abstract we prove pointwise gradient bounds for heat semigroups associated to general possibly unbounded laplacians on infinite graphs satisfying the curvature dimension condition cd k using gradient bounds we show stochastic completeness for graphs satisfying the curvature dimension condition

stability of parabolic harnack inequalities let g e be a graph with weights a xy for which a parabolic harnack inequality holds with space time scaling exponent 2 suppose xy is another set of weights that are comparable to a xy we prove that this parabolic harnack inequality also holds for g e with the weights a xy we also give stable necessary and sufficient conditions for this parabolic harnack inequality to hold

intrinsic metrics for non local symmetric dirichlet forms and applications to spectral theory we present a study of what may be called an intrinsic metric for a general regular dirichlet form for such forms we then prove a rademacher type theorem for strongly local forms we show existence of a maximal intrinsic metric under a weak continuity condition and for dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals we then turn to applications on spectral theory and provide for measure perturbation of general regular dirichlet forms an allegretto piepenbrinck type theorem which is based on a ground state transform and a shnol type theorem our setting includes laplacian on manifolds on graphs and stable processes

on stochastic completeness of jump processes we prove the following sufficient condition for stochastic completeness of symmetric jump processes on metric measure spaces if the volume of the metric balls grows at most exponentially with radius and if the distance function is adapted in a certain sense to the jump kernel then the process is stochastically complete we use this theorem to prove the following criterion for stochastic completeness of a continuous time random walk on a graph with a counting measure if the volume growth with respect to the graph distance is at most cubic then the random walk is stochastically complete where the cubic volume growth is sharp

construction of right processes from excursions in a previous work the author obtained the strong markov property of a process from conditions on its excursion process the ray and right properties are obtained here under similar conditions using the ray knight compactification

analysis on graphs and noncommutative geometry abstract we study the form of the continuous time heat kernel for a second order discrete laplacian on a weighted graph the analysis is shown to be closely related to the theory of symmetric markov semigroups on noncommutative l p spaces and to the noncommutative geometry of connes the paper obtains better pointwise upper bounds on the heat kernels than those previously known by the use of a novel metric on the graph in certain cases it is shown that the new estimates are optimal of their type the metric is investigated at some length and compared with other known metrics on graphs

on uniqueness class for a heat equation on graphs abstract we study the uniqueness class problem for a class of heat equations defined on weighted graphs as an application we recover the volume growth criterion for stochastic completeness of grigor yan huang and masamune in the case of weighted graphs

stochastic incompleteness for graphs and weak omori yau maximum principle abstract we prove an analogue of the weak omori yau maximum principle and khas minskii s criterion for graphs in the general setting of keller and lenz our approach naturally gives the stability of stochastic incompleteness under certain surgeries of graphs it allows to develop a unified approach to all known criteria of stochastic completeness incompleteness as well as to obtain new criteria

brownian motions on metric graphs brownian motions on a metric graph are defined their generators are characterized as laplace operators subject to wentzell boundary at every vertex conversely given a set of wentzell boundary conditions at the vertices of a metric graph a brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary conditions

sufficient fuzzy information systems in this paper we suggest and study a new selection criterion in order to compare probabilistic information systems when the available information from them is vague in the sense that it might be considered as fuzzy information tanaka okuda and asai this criterion is based on the concept of sufficient probabilistic information system established by blackwell

analysis of the physical laplacian and the heat flow on a locally finite graph we study the physical laplacian and the corresponding heat flow on an infinite locally finite graph with possibly unbounded valence