Revuz Correspondence Research Articles

Distribution flows associated with positivity preserving coercive forms

For a given quasi-regular positivity preserving coercive form, we construct a family of ($\sigma$-finite) distribution flows associated with the semigroup of the form. The canonical cadlag process equipped with the distribution flows behaves like a strong Markov process. Moreover, employing distribution flows we can construct optional measures and establish Revuz correspondence between additive functionals and smooth measures. The results obtained in this paper will enable us to perform a kind of stochastic analysis related to positivity preserving coercive forms.

Smooth measures and capacities associated with nonlocal parabolic operators

We consider a family {L_t,, tin [0,T]} of closed operators generated by a family of regular (non-symmetric) Dirichlet forms {(B^{(t)},V),tin [0,T]} on L^2(E;m). We show that a bounded (signed) measure mu on (0,T)times E is smooth, i.e. charges no set of zero parabolic capacity associated with frac{partial }{partial t}+L_t, if and only if mu is of the form mu =fcdot m_1+g_1+partial _tg_2 with fin L^1((0,T)times E;mathrm{d}totimes m), g_1in L^2(0,T;V'), g_2in L^2(0,T;V). We apply this decomposition to the study of the structure of additive functionals in the Revuz correspondence with smooth measures. As a by-product, we also give some existence and uniqueness results for solutions of semilinear equations involving the operator frac{partial }{partial t}+L_t and a functional from the dual {{mathcal {W}}}' of the space {{mathcal {W}}}={uin L^2(0,T;V):partial _t uin L^2(0,T;V')} on the right-hand side of the equation.

Large deviations for continuous additive functionals of symmetric Markov processes

Let $X$ be a locally compact separable metric space and $m$ a positive Radon measure on $X$ with full topological support. Let ${\bf{M}}=(P_x,X_t)$ be an $m$-symmetric Markov process on $X$. Let $(\mathcal{E},\mathcal{D}(\mathcal{E}))$ be the Dirichlet form on $L^2(X;m)$ generated by ${\bf{M}}$. Let $\mu$ be a positive Radon measure in the Green-tight Kato class and $A^\mu_t$ the positive continuous additive functional in the Revuz correspondence to $\mu$. Under certain conditions, we establish the large deviation principle for positive continuous additive functionals $A^\mu_t$ of symmetric Markov processes.

Markov processes on the Lipschitz boundary for the Neumann and Robin problems

We investigate the Markov process on the boundary of a bounded Lipschitz domain associated to the Neumann and Robin boundary value problems. We first construct Lp-semigroups of sub-Markovian contractions on the boundary, generated by the boundary conditions, and we show that they are induced by the transition functions of the forthcoming processes. As in the smooth boundary case the process on the boundary is obtained by the time change with the inverse of a continuous additive functional of the reflected Brownian motion. The Robin problem is treated with a Kato type Lp-perturbation method, using the Revuz correspondence. An exceptional (polar) set occurs on the boundary. We make the link with the Dirichlet forms approach.

Large time asymptotics of Feynman–Kac functionals for symmetric stable processes

Let μ be a positive Kato measure on associated with the Green kernel of the transient symmetric α‐stable process, the Markov process with generator (). Let be the positive continuous additive functional in the Revuz correspondence with μ. If, in addition, μ is of compact support, we give exact large time asymptotics of the expectation of the Feynman–Kac functional, .

Bivariate Revuz Measures and the Feynman-Kac Formula on Semi-Dirichlet Forms

In this paper we shall first establish the theory of bivariate Revuz correspondence of positive additive functionals under a semi-Dirichlet form which is associated with a right Markov process X satisfying the sector condition but without duality. We extend most of the classical results about the bivariate Revuz measures under the duality assumptions to the case of semi-Dirichlet forms. As the main results of this paper, we prove that for any exact multiplicative functional M of X, the subprocess X M of X killed by M also satisfies the sector condition and we then characterize the semi-Dirichlet form associated with X M by using the bivariate Revuz measure, which extends the classical Feynman-Kac formula.

Large deviations for symmetric stable processes with Feynman-Kac functionals and its application to pinned polymers

Let $\nu$ and $\mu$ be positive Radon measures on ${\boldsymbol R} ^d$ in Green-tight Kato class associated with a symmetric $\alpha$-stable process $(X_t , P_x)$ on ${\boldsymbol R}^d$, and $A_t ^\nu$ and $A_t ^\mu$ the positive continuous additive functionals under the Revuz correspondence to $\nu$ and $\mu$. For a non-negative $\beta$, let $P_{x,t} ^{\beta \mu}$ be the law $X_t$ weighted by the Feynman-Kac functional $\exp(\beta A_t ^\mu)$, i.e., $P_{x,t} ^\mu =(Z_{x,t} ^\mu)^{-1}\exp(\beta A_t ^\mu)P_x$, where $Z_{x,t} ^\mu$ is a normalizing constant. We show that $A_t ^\nu /t$ obeys the large deviation principle under $P_{x,t}^{\beta \mu}$. We apply it to a polymer model to identify the critical value $\beta _{\rm cr}$ such that the polymer is pinned under the law $P^{\beta \mu} _{x,t} $ if and only if $\beta$ is greater than $\beta_{\rm cr}$. The value $\beta _{\rm cr} $ is characterized by the rate function.

Measures not charging polar sets and Schrödinger equations in L p

We study the Schrodinger equation (q − ℒ)u + µu = f, where ℒ is the generator of a Borel right process and µ is a signed measure on the state space. We prove the existence and uniqueness results in Lp, 1 ⩽ p < ∞. Since we consider measures µ charging no polar set, we have to use new tools: the Revuz formula with fine versions and the appropriate Revuz correspondence, the perturbation (subordination) operators (in the sense of G. Mokobodzki) induced by the regular strongly supermedian kernels. We extend the results on the Schrodinger equation to the case of a strongly continuous sub-Markovian resolvent of contractions on Lp. If the measure µ is positive then the perturbed process solves the martingale problem for ℒ − µ and its transition semigroup is given by the Feynman-Kac formula associated with the left continuous additive functional having µ as Revuz measure.

On the Strongly Supermedian Functions and Kernels

We answer to a question of P.J. Fitzsimmons and R.K. Getoor concerning the characterization of a strongly supermedian function by the property of being “universally” supermedian. Based on the Revuz correspondence we also prove (alternatively to the probabilistic approach of Fitzsimmons and Getoor) that the Radon–Nikodym derivative between the Revuz measures of a semiregular excessive kernel and the regular strongly supermedian kernel generating it, does not depend on the kernel.

Fine Densities for Excessive Measures and the Revuz Correspondence

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If ξ is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure η with η≪ξ the Radon–Nikodym derivative dη/dξ possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the σ-finite measures charging no set that is both ξ-polar and ρ-negligible (ρ○U being the potential component of ξ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azema, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.

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.

Strongly supermedian kernels and Revuz measures

In the frame of Borel right Markov processes, we investigate, following an analytical point of view, the Revuz correspondence between classes of potential kernels and their associated measures, improving upon the results of Revuz, Azéma, Getoor and Sharpe, Fitzsimmons, Fitzsimmons and Getoor and Dellacherie, Maisonneuve and Meyer. In the probabilistic approach of the problem, the kernels that occur are the potential operators of different types of homogeneous random measures. We completely characterize the hypothesis (B) of Hunt in terms of Revuz measures.

Smooth Measures and Regular Strongly Supermedian Kernels Generating Sub-Markovian Resolvents

In the context of a transient Borel right Markov process with a fixed excessive measure ξ, we characterize the regular strongly supermedian kernels, producing smooth measures by the Revuz correspondence. In the case of the measures charging no ξ-semipolar sets, this is the analytical counterpart of a probabilistic result of Revuz, Fukushima, and Getoor and Fitzsimmons, concerning the positive continuous additive functionals. We also consider the case of the measures charging no set that is both ξ-polar and ρ-negligible (ρ○U being the potential part of ξ), answering to a problem of Revuz.