Resolvent Of Kernels Research Articles

Markov Processes Associated to Resistance Forms

We give a method to associate a right Markov process to a resistance form, that is different from the one based on the regularity assumption for the associated Dirichlet form. We use tools from potential theory associated to resolvents of kernels and cones of potentials (H-cones). The induced capacity and processes obtained by time change with the inverses of continuous additive functionals also occur.

Construction of ${\mathcal L}^{{p}}$ -strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvents of kernels having the $\mathcal L^p$-strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients and singular drifts on general open sets with absorption at the boundary. In this application elliptic regularity results imply the desired regularity assumptions.

From resolvents to càdlàg processes through compact excessive functions and applications to singular SDE on Hilbert spaces

We present an approach for constructing càdlàg strong Markov processes given a resolvent of kernels. The conditions imposed on the resolvent are checkable in applications and allow the control of the set of admissible starting points of the process. The main application is to singular SDEs on Hilbert spaces.

Sub-Markovian resolvents under weak duality hypothesis

We study the weak duality between two sub-Markovian resolvents of kernels on a Lusin topological space with respect to a given measure m. Our frame covers the probabilistic context of two Borel Markov processes in weak duality. The main results are related to: the coincidence of the m-semipolar and m-cosemipolar sets, the Revuz correspondence between the measures charging no cofinely open m-polar set and the potential kernels associated with the homogeneous random measures of the process, the equivalence between smoothness and cosmoothness (a smooth measure is the Revuz measure of a continuous additive functional). We extend and improve results of R.M. Blumenthal-R.K. Getoor, J. Azema, D. Revuz, J.B. Walsh, R.K. Getoor-M.J. Sharpe.

Additive Kernels and Integral Representation of Potentials

Let P=(P t ) t>0 be a submarkovian semigroup of kernels on a measurable space (X,ℬ). An additive kernel of P is a kernel K from X into ]0,∞[ such that P t K(x,A)=K(x,A+t) for every t>0,x∋X and every Borel subset A of ]0,∞[. It is proved in this paper that for every potential f of P, there exits an additive kernel K of P, unique (up to equivalence) such that f=K1=∫0 ∞ K(⋅,dt). This result is already well known for regular potentials of right processes. If U=(U p ) p>0 is a sub-Markovian resolvent of kernels on (X,ℬ), we give a notion of additive kernel of U and we prove a similar integral representation of potentials of U.

Excessive kernels and Revuz measures

We consider a proper submarkovian resolvent of kernels on a Lusin measurable space and a given excessive measure ξ. With every quasi bounded excessive function we associate an excessive kernel and the corresponding Revuz measure. Every finite measure charging no ξ–polar set is such a Revuz measure, provided the hypothesis (B) of Hunt holds. Under a weak duality hypothesis, we prove the Revuz formula and characterize the quasi boundedness and the regularity in terms of Revuz measures. We improve results of Azema [2] and Getoor and Sharpe [20] for the natural additive functionals of a Borel right process.

Absorbent, parabolic, elliptic and quasielliptic balayages in potential theory; relationships with the Green function

In the frame of standard H-cones of functions (the cone of all excessive functions with respect to a submarkovian resolvent of kernels with reference measure on a measurable space) on a Green set we show that the cofine closure of the complement of an absorbent set in coabsorbent. We obtain different characterizations concerning the parabolicity, ellipticity and quasiellipticity in terms of the Green function. We also show that these notions are the same in the direct and the dual theory.

Contribution to the nonlinear theory of viscoelasticity

The author presents a generalization of certain results obtained in [3] relating to the region of rigid behavior of plastics for which the nonlinearity of the viscoelastic properties is important even in the area of small strains. Certain questions connected with the derivation of the basic equations of viscoelasticity are considered for small strains. Formulas are obtained for the resolvents of kernels of arbitrary order. The general equations of the "principal quadratic theory of viscoelasticity" are derived.