Showing papers in "Potential Analysis in 2000"
TL;DR: In this paper, the authors generalize the definition of the first order Sobolev spaces with zero boundary values to an arbitrary metric space endowed with a Borel regular measure and show that many classical results extend to the metric setting.
Abstract: We generalize the definition of the first order Sobolev spaces with zero boundary values to an arbitrary metric space endowed with a Borel regular measure. We show that many classical results extend to the metric setting. These include completeness, lattice properties and removable sets.
116 citations
TL;DR: In this paper, Borel functions on Borel curves were integrated with the Itors formula for new classes of functions and defined a local time process of a linear Brownian motion on any borelian curve.
Abstract: Let
$$\left( {L_t^x ;x \in \mathbb{R},t \geqslant 0} \right)$$
be the local time process of a linear Brownian motion B. We integrate the Borel functions on
$$\mathbb{R}_ \times \mathbb{R}_ + $$
with respect to
$$\left( {L_t^x ;x \in \mathbb{R},t \geqslant 0} \right)$$
. This allows us to write Itors formula for new classes of functions, and to define a local time process of B on any borelian curve. Some results are extended from deterministic to random functions.
107 citations
TL;DR: In this article, generalized Mehler semigroups were studied in the non-Gaussian case, with a special emphasis on the nonGaussian cases, and a generalized generalized Mehraghdam semigroup was constructed in an appropriate state space.
Abstract: We study generalized Mehler semigroups, introduced in [7], with special emphasis on the non-Gaussian case. We review and simplify the method of construction. In the general (non-Gaussian) case we construct an associated cadlag Markov process in an appropriate state space obtained as a solution of a stochastic equation which can be solved ‘ω by ω’. We also show tightness of the associated (r, p)-capacities. Invariant measures, time regularity and a definition of the generator are also studied.
92 citations
TL;DR: In this article, the notion of regular (infinitesimal) invariant probability measures for second order elliptic integro-differential operators (Levy type operators) is posed.
Abstract: Second order elliptic integro-differential operators (Levy type operators) are investigated. The notion of regular (infinitesimal) invariant probability measures for such operators is posed. Sufficient conditions for the existence of such regular infinitesimal invariant probability measures are obtained and the symmetrization problem is discussed.
77 citations
TL;DR: In this article, Gibbs states on path spaces of the form C = √( √ √ R, √ Z √ D ) that infinite stochastic differential equations of gradient type or as thermodynamic limits constructed with help of the method of cluster expansions can be found.
Abstract: Gibbs states on path spaces of the form C
$$\left( {\mathbb{R},\mathbb{R}} \right)^{\mathbb{Z}^d }$$
infinite stochastic differential equations of gradient type or as thermodynamic limits constructed with help of the method of cluster expansions.
28 citations
TL;DR: In this paper, the problem of Brownian motion penetration in the Sierpinski gasket was studied with Dirichlet forms and function space theory, and the results obtained are somewhat different from, and from certain aspects more general than, the results in [8].
Abstract: Can a Brownian motion penetrate the two-dimensional Sierpinski gasket? This question was studied in [8], and an affirmative answer was given. In this paper, the problem is studied with a different approach, using Dirichlet forms and function space theory. The results obtained are somewhat different from, and from certain aspects more general than, the results in [8].
27 citations
TL;DR: In this article, the authors give a version of the 3G theorem for Dini-smooth Jordan domains in R2 and prove comparison theorems between the Green kernel of Δ and δ − δ, where δ is a nonnegative Radon measure.
Abstract: We establish inequalities for Green functions of Dini-smooth Jordan domains in R2.We give a version of the 3G theorem for these domains. With the help of these results, we prove comparison theorems between the Green kernel of Δ and the Green kernel of Δ − μ where μ is a nonnegative Radon measure.
25 citations
TL;DR: In this article, the authors considered a second order elliptic operator with complex coefficients on an open set of ℝ� N�, and obtained necessary and sufficient conditions on the coefficients for the contractivity in ℽℽ ∞ of the semigroup defined under different boundary conditions.
Abstract: Soit un operateur elliptique du second ordre a coefficients complexes sur un ouvert de ℝ
N
. On obtient des conditions necessaires et suffisantes sur les coefficients, pour que le semi-groupe qu'il definit, suivant les conditions au bord considerees, contracte L
∞. On montre en particulier que cette propriete est assez specifique aux coefficients reels.
Abstract. Consider a second order elliptic operator with complex coefficients on an open set of ℝ
N
. We obtain necessary and sufficient conditions on the coefficients for the contractivity in L
∞ of the semigroup defined under different boundary conditions. In particular, we show that this property is closely related to the fact that the coefficients are actually real valued.
25 citations
TL;DR: In this paper, the transition semigroup of a Gaussian Mehler semigroup on a separable Banach space is shown to be strongly continuous on BUC(E) if and only if S(t) = Ifor all t⩽ 0
Abstract: We present sufficient conditions on a Gaussian Mehler semigroup on a reflexive Banach space Eto be induced by a single positive symmetric operator Q \in \(Q \in \mathcal{L}(E^* ,E)\), and give a counterexample which shows that this representation theorem is false in every nonreflexive Banach space with a Schauder basis. We also show that the transition semigroup of a Gaussian Mehler semigroup on a separable Banach space Eacts in a pointwise continuous way, uniformly on compact subsets of E, in the space BUC(E) of bounded uniformly continuous real-valued funtions on E. The transition semigroup is shown to be strongly continuous on BUC(E) if and only if S(t) = Ifor all t⩽ 0
21 citations
TL;DR: In this article, the authors consider the case of n points on a smooth surface with n points weighted extremal points and present sharp estimates for the difference of the potentials of these points.
Abstract: Consider the unit measure
$$\mu _{F_n }$$
associating the mass
$${{1{\kern 1pt} } \mathord{\left/ {\vphantom {{1{\kern 1pt} } n}} \right. \kern-
ulldelimiterspace} n}$$
with n points on a smooth surface in
$$\mathbb{R}^d ,d \geqslant 3$$
, minimizing discrete energy under the influence of an external field
$$f$$
. We call such points weighted extremal points. How well do the
$$\mu _{F_n }$$
approximate the
$$f$$
-weighted equilibrium distribution
$$\mu _f$$
of the surface? We answer this question by presenting sharp estimates for the difference of the potentials of
$$\mu _{F_n }$$
and
$$\mu _f$$
, for the discrete energy of
$$\mu _{F_n }$$
and for the discrepancy
$$\left| {\mu _{F_n } \left( B \right) - \mu _f \left( B \right)} \right|$$
, where the supremum is taken over a reasonable class of test sets B. In the unweighted case f=0, extremal points reduce to d-dimensional Fekete points, and, up to a logarithmic term, the presented discrepancy estimate solves a conjecture of J. Korevaar [13].
19 citations
TL;DR: The existence of horizontal lift to a line bundle of diffusion processes on some infinite-dimensional manifolds was proved in this article, where the corresponding loop spaces were shown to admit a horizontal lift.
Abstract: We prove existence of the horizontal lift to a line bundle of certain diffusion processes on some infinite-dimensional manifolds. We provide three classes of finite-dimensional manifolds for which the corresponding loop spaces have a line bundle and thus provide three classes of loop manifolds on which certain diffusion processes admit a horizontal lift. Applications to Quantum Field Theory are indicated.
TL;DR: In this paper, the authors prove a large deviation principle for the family of solutions of Volterra equations in the plane obtained by perturbation of the driving white noise, using the method developped by Azencott for diffusion processes.
Abstract: We prove a Large Deviation Principle for the family of solutions of Volterra equations in the plane obtained by perturbation of the driving white noise. One of the motivations for the study of such class of equations is provided by non-linear hyperbolic stochastic partial differential equations appearing in the construction of some path-valued processes on manifolds. The proof uses the method developped by Azencott for diffusion processes. The main ingredients are exponential inequalities for different classes of two-parameter stochastic integrals; these integrals are related to the representation of the stochastic term in the differential equation as a representable semimatringale.
TL;DR: In this paper, a convergence theorem for the invariants of the deterministic motion of a Hamiltonian system perturbed by a white noise force is proved, which corresponds to convergence of the system to a stationary distribution.
Abstract: Hamiltonian systems perturbed by a white noise force are discussed in several dimensions. By using an appropriate scaling of the stochastic force a convergence theorem for the invariants of the deterministic motion is proved. This corresponds to convergence of the system to a stationary distribution. Especially motion in a central force field is considered; the energy and angular momentum processes are investigated.
TL;DR: In this article, the authors discuss recurrence, transience and other potential theoretic aspects based on symmetric regular Dirichlet space and show that recurrence automatically provides us with an exhaustion function which is usable to verify Liouville property on subharmonic functions.
Abstract: In this paper, we will discuss recurrence, transience and other potential theoretic aspects based on symmetric regular Dirichlet space. We will first deal with Dirichlet space with the strong local property and give a recurrence criterion in terms of exhaustion function. This criterion shows that recurrence automatically provides us with an exhaustion function which is usable to verify a Liouville property on subharmonic functions. Secondly, a recurrence criterion and a transience criterion for a Nonlocal Dirichlet space will be presented. Those criteria can be applied to Albeverio–Karwowski"s random walks on p-adic number field. Lastly, we will prove the assertions which cover other potential theoretic aspect of p-adic number field such as Liouville property on harmonic functions.
TL;DR: In this article, it was shown that Ω ∖ E is a p-fine domain whenever Ω ⊂ Rn is a P-fine topology, and E ⊆ Rn ⊈ E is p-polar, and 1 < p ≤ n.
Abstract: We prove that Ω ∖ E is a p-fine domain whenever Ω ⊂ Rn is a p-fine domain, E ⊂ Rn is p-polar, and 1 < p ≤ n. By a p-fine domain we understand an open connected set in the p-fine topology, i.e. in the coarsest topology making all p-superharmonic functions continuous. As an application of our main result, we establish a general version of minimum principle.
TL;DR: In this paper, a sufficient integral criterion for the existence of a maximal number of minimal Martin boundary points at 0 in certain domains has been given, where the Schrodinger operator 1/2Δ-µ is considered.
Abstract: Cranston and Salisbury have obtained an integral test for the existence of a maximal number of minimal Martin boundary points at 0 in certain domains (cf. [8]). This paper will extend the result as follows: let D ⊂ R
d
be an open Greenian set (with respect to the Laplacian) consisting of n disjoint open connected cones with Lipschitz boundary and a subset of the boundary of these cones. Let μ be some local Kato measure supported by the boundary of the cones and consider the Schrodinger operator 1/2Δ-µ. We will assume a boundary Harnack principle and give a sufficient integral criterion for the existence of exactly n minimal Martin boundary points at 0. In certain cases there is a necessary criterion, too. When the sufficient integral criterion holds we will give a necessary and a sufficient condition for the existence of a certain process related to the Schrodinger operator that connects two different admissible boundary points. In the paper of Cranston and Salisbury the case μ = 0, d = 2 is treated, but many of the arguments work as well in the general situation.
TL;DR: The singularities of harmonic maps from a domain D in the plane to S 1 minimizing a renomalized energy tend to go to the boundary when their number becomes large as mentioned in this paper.
Abstract: We prove that the singularities of harmonic maps from a domain D in the plane to S1 minimizing a renomalized energy tend to go to the boundary when their number becomes large.
TL;DR: In this article, the generalized Schrodinger operator was shown to generate a unique semigroup of class (Co) in L 1 under mild conditions on the modulus φ = ∣ ψ∣ of the time independent wave function ψ.
Abstract: Under mild condition on the modulus φ = ∣ ψ∣ of the time independent wave function ψ, we prove that the generalized Schrodinger operator ℒφ= ℒ + 2 Γ (φ, ·)/φ (or the generator of Nelson's diffusion) defined on a good space of test-functions \(D\)on a general Polish space, generates a unique semigroup of class (Co) in L1. This result reinforces the known results on the essential Markovian self-adjointness in different contexts and extends our previous works in the finite dimensional Euclidean space setting. In particular it can be applied to the ground or excited state diffusion associated with an usual Schr\"odinger operator \( - L + V\), and to stochastic quantization of several Euclidean quantum fields.
TL;DR: In this article, the L∞ norm of the Dirichlet Laplace operator on a conic sector over a geodesic disc was obtained for an open, bounded and convex set D with inradius ρ and diameter d.
Abstract: We obtain the asymptotic behaviour for the L∞ norm of the first eigenfunction φ of the Dirichlet Laplace operator on a conic sector over a geodesic disc \({B_{\eta} }\) in \(\mathbb{S}^{m - 1}\) as \({\eta} \to {0}\) We are led to conjecture that for an open, bounded and convex set D with inradius ρ and diameter d, \(\left\| \phi \right\|_\infty \leqslant c_m \rho ^{{{\left( {1 - 3m} \right)} \mathord{\left/ {\vphantom {{\left( {1 - 3m} \right)} 6}} \right \kern-
ulldelimiterspace} 6}} d^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 6}} \right \kern-
ulldelimiterspace} 6}} \) where \(\left\| \phi \right\|_2\) and \(c_m\)
TL;DR: In this article, the Lp(Ω)-norm of the Skorohod and the forward integrals of the Volterra equations was deduced, and the existence of a unique solution to the problem was studied.
Abstract: In this paper we deduce some estimates of the Lp(Ω)-norm of the Skorohod and the forward integrals. These estimates allow us to study the existence of a unique solution to anticipating Volterra equations of the Skorohod and forward type. The coefficients Fi(t,s,x),t≥s, are \(F_t \)-measurable and satisfy some differentiability conditions (in the sense of the stochastic calculus of variations).
TL;DR: The concept of complex Dirichlet forms is introduced in this article, which is a form of first-order differential operators in complex weighted L2-spaces. But the complexity of the form is not fixed.
Abstract: The concept of complex Dirichlet forms ec resp. operators Lc in complex weighted L2-spaces is introduced. Perturbations of classical Dirichlet forms by forms associated with complex first-order differential operators provide examples of complex Dirichlet forms.
TL;DR: In this article, the authors prove the existence of a deformation transforming an arbitrary open set into the ball, which has the following properties: it keeps constant the measure, the kth eigenvalue of Laplace-Dirichlet operator is continuous from the left and the first eigen value is decreasing.
Abstract: In this paper we prove the existence of a deformation transforming an arbitrary open set into the ball, which has the following properties: it keeps constant the measure, the kth eigenvalue of Laplace–Dirichlet operator is continuous from the left and the first eigenvalue is decreasing. The deformation is given by a sequence of continuous Steiner symmetrizations, and the behavior of the eigenvalues is related to the stability of the Dirichlet problem.
TL;DR: In this article, it was shown that μ ≤ ν on the contact set {u − v = 0} of the δ-superharmonic function u − v, if this set is properly interpreted as the set of those minimal superharmonic functions s which satisfy lim sup Tsv/u = 1 for the co-fine neighborhood filter T s associated with s.
Abstract: Let u ≥ v be positive superharmonic functions in a general potential-theoretic setting, where these functions have a Choquet-type integral representation by minimal such functions with Choquet charges (i.e. representing measures) μ and ν, respectively. We show that μ ≤ ν on the contact set {u − v = 0} of the δ-superharmonic function u − v, if this set is properly interpreted as the set of those minimal superharmonic functions s which satisfy lim sup Tsv/u = 1 for the co-fine neighborhood filter T s associated with s. In the setting of classical potential theory for Laplace's equation this result improves on results obtained by Fuglede in 1992.
TL;DR: In this paper, the authors consider potentials Gk μ associated with the Weinstein equation with parameter k in ℝ, Σj = 1n (∂ 2u/∂ x2j) + (k/xn) (∆ u/∆ xn) = 0, on the upper half space in ↝n.
Abstract: We consider potentials Gk μ associated with the Weinstein equation with parameter k in ℝ, Σj=1n (∂2u/∂ x2j) + (k/xn) (∂ u/∂ xn) = 0, on the upper half space in ℝn. We show that if the representing measure μ satisfies the growth condition ∫ ynω/(1+|y|)n-k < ∞, where max(k, 2 − n) < ω ≤ 1, then Gk μ has a minimal fine limit of 0 at every boundary point except for a subset of vanishing (n − 2 + ω) dimensional Hausdorff measure. We also prove this exceptional set is best possible.
TL;DR: In this article, a class of parabolic equations with a nonlinear gradient term is studied and the unique solution of this problem can be represented as the Wick product between a normalized random variable of exponential form and the solution of a non-linear parabolic equation.
Abstract: In this paper we study a class of parabolic equations with a nonlinear gradient term. The system is disturbed by white noise in time. We show that the unique solution of this problem can be represented as the Wick product between a normalized random variable of exponential form and the solution of a nonlinear parabolic equation. We allow random initial data which might be anticipating. A relation between the Wick product with a normalized exponential and translation is proved in order to establish our results.
TL;DR: In this paper, a new proof of the Tanaka formula in analysis is given, based on probabilistic arguments, using Brownian motion, stochastic calculus and Burkholder-Gundy inequalities for martingales.
Abstract: In our previous paper [5], we have obtained a decomposition of ∣f∣, where f is a function defined on Rd, that is analogous to the one proved by H. Tanaka in the early sixties for Brownian martingales (the so-called ‘Tanaka formula’). The original proofs use purely analytic methods (e.g. the Calderon–Zygmund theory, etc.). In this paper, we give a new proof of our `Tanaka formula in analysis’, that is based on probabilistic arguments. The main tools here are Brownian motion, stochastic calculus and Burkholder–Gundy inequalities for martingales. These methods allow us to improve somewhat our previous results, by proving that some significant constants do not depend on the dimension d.
TL;DR: In this article, a relation between the Bessel potential spaces and Riesz potential spaces is presented, where the authors characterize each potential space and give a correspondence between individual Bessel terms and their corresponding Riez potentials.
Abstract: We present a relation between the Bessel potential spaces and the Riesz potential spaces. The ideas of the proof are to characterize each potential spaces and to give a correspondence between individual Bessel potentials and Riesz potentials.
TL;DR: In this article, Nevanlinna et al. define conditions necessaires and suffisantes for l'ordre des solutions, non bornees au voisinage d'une singularite, soit infini.
Abstract: Dans ce travail, nous etudions l"ordre de Nevanlinna des solutions de l'equation deSchrodinger Δ u = u μ, ou μ est une mesure de Kato, positive et radiale Nous montrons que si μ ne verifie pas le principe de Picard, alors toute solution non bornee de Δ u = u μ est d'ordre infini En utilisant la theorie de perturbation, l'effilement minimal de M Brelot et la notion d'ensembles essentiels, nous donnons des conditions necessaires et des conditions suffisantes pour que l'ordre des solutions, non bornees au voisinage d'une singularite, soit infini Enfin nous prouvons que le comportement des solutions d'une telle equation ne permet pas de caracteriser la validite ou non du principe de Picard
TL;DR: In this paper, the authors generalise some of the results of Dellacherie to the case where the complete maximum principle does not hold and show that nonlinear kernels can have a resolvent associated with them.
Abstract: In [1], Dellacherie showed that also nonlinear kernels can have a resolvent associated with them. In this text, we will generalise some of his results to the case where the complete maximum principle does not hold.
TL;DR: In this article, it was shown that f(x, y) is a complex function defined on D × G such that the harmonic analogue of the local polynomial condition of Leja can be verified at some point in D.
Abstract: Given D a domain in \(\mathbb{R}^M\), G an open set in \(\mathbb{R}^N \) and E a subset of D verifying the harmonic analogue of ‘Local Polynomial Condition of Leja’ at some point in D. We prove that if f(x, y) is a complex function defined on D × G such that