Markov processes associated with L p -resolvents and applications to stochastic differential equations on Hilbert space

We give general conditions on a generator of a C₀-semigroup (resp. of a C₀-resolvent) on L^p(E,μ), p ≥ 1, where E is an arbitrary (Lusin) topological space and μ a σ-finite measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Time-dependent diffusion operators on <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>

We study the Cauchy problem for time-dependent diffusion operators with singular coefficients on L¹-spaces induced by infinitesimal invariant measures. We give sufficient conditions on the coefficients such that the Cauchy-Problem is well-posed. We construct associated diffusion processes with the help of the theory of generalized Dirichlet forms. We apply our results in particular to construct a large class of Nelson-diffusions that could not been constructed before.     

One-point extensions of Markov processes by darning

This paper is a continuation of the works by Fukushima–Tanaka (Ann Inst Henri Poincaré Probab Stat 41: 419–459, 2005) and Chen–Fukushima–Ying (Stochastic Analysis and Application, p.153–196. The Abel Symposium, Springer, Heidelberg) on the study of one-point extendability of a pair of standard Markov processes in weak duality. In this paper, general conditions to ensure such an extension are given. In the symmetric case, characterizations of the one-point extensions are given in terms of their Dirichlet forms and in terms of their L ²-infinitesimal generators. In particular, a generalized notion of flux is introduced and is used to characterize functions in the domain of the L ²-infinitesimal generator of the extended process. An important role in our investigation is played by the α-order approaching probability u _α . The research of Z.-Q. Chen is supported in part by NSF Grant DMS-0600206. The research of M. Fukushima is supported in part by Grant-in-Aid for Scientific Research of MEXT No.19540125.     

On a Poincaré-type Inequality for Energy Forms in L p

We consider Dirichlet spaces ( ) in L ² and more general energy forms in L ^p , . For the latter we introduce the notions of an extended ’Dirichlet’ space and a transient form. Under the assumption that , resp. , are compactly embedded in L ², resp. L ^p , we prove a Poincaré inequality for transient (Dirichlet) forms. If both and its adjoint are sub-Markovian semigroups, we show that the transience of T _t is independent of ) and that it is implied by the transience of the energy form of and the form belonging to .     

About a construction and some analysis of time inhomogeneous diffusions on monotonely moving domains

We construct and analyze in a very general way time inhomogeneous (possibly also degenerate or reflected) diffusions in monotonely moving domains E⊂R×R^d, i.e. if E_t?{x∈R^d|(t,x)∈E}, t∈R, then either E_s⊂E_t, ∀s?t, or E_s⊃E_t, ∀s?t, s,t∈R. Our major tool is a further developed L²(E,m)-analysis with well chosen reference measure m. Among few examples of completely different kinds, such as e.g. singular diffusions with reflection on moving Lipschitz domains in R^d, non-conservative and exponential time scale diffusions, degenerate time inhomogeneous diffusions, we present an application to what we name skew Bessel process on γ. Here γ is either a monotonic function or a continuous Sobolev function. These diffusions form a natural generalization of the classical Bessel processes and skew Brownian motions, where the local time refers to the constant function γ≡0.     

Estimates of transition densities for Brownian motion on nested fractals

Summary We obtain upper and lower bounds for the transition densities of Brownian motion on nested fractals. Compared with the estimate on the Sierpinski gasket, the results require the introduction of a new exponent,d _J, related to the shortest path metric and chemical exponent on nested fractals. Further, Hölder order of the resolvent densities, sample paths and local times are obtained. The results are obtained using the theory of multi-type branching processes.     

Dirichlet operators and the positive maximum principle

Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup onL ^p (m), p1 andm not necessarily -finite. We show under mild regularity conditions thatA is a Dirichlet operator in all spacesL ^q (m), qp. It turns out that, in the limitq,A satisfies the positive maximum principle. If the test functionsC _c D(A), then the positive maximum principle implies thatA is a pseudo-differential operator associated with a negative definite symbol, i.e., a Lévy-type operator. Conversely, we provide sufficient criteria for an operator (A, D(A)) onL ^p(m) satisfying the positive maximum principle to be a Dirichlet operator. If, in particular,A onL ² (m) is a symmetric integro-differential operator associated with a negative definite symbol, thenA extends to a generator of a regular (symmetric) Dirichlet form onL ² (m) with explicitly given Beurling-Deny formula.     

Dynkin's formula and large coupling convergence

Let H?1 be a selfadjoint operator in H, let J be a linear and bounded operator from (D(H^1/2),∥H^1/2·∥) to H_aux and for β>0 let be the nonnegative selfadjoint operator in H satisfying

     

Homogenization of a parabolic model of ferromagnetism

This work deals with the homogenization of hysteresis-free processes in ferromagnetic composites. A degenerate, quasilinear, parabolic equation is derived by coupling the Maxwell-Ohm system without displacement current with a nonlinear constitutive law:

     

The short-cut test

The short-cut test detects existence and uniqueness of “Laplacians” on finitely ramified, graph-directed fractals. Previous results by Sabot, Nussbaum and the author are improved and extended. It opens up the way for further studies because it combines well established spectral, dynamical and analytic techniques. Its algorithmic and recursive structure is designed to provide computable and flexible criteria.     

A class of elliptic pseudo differential operators generating symmetric Dirichlet forms

It is shown that a special class of symmetric elliptic pseudo differential operators do generate a Feller semigroup and therefore a non-local Dirichlet form.     

Heat kernel estimates for jump processes of mixed types on metric measure spaces

In this paper, we investigate symmetric jump-type 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 d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity where ν is a probability measure on , c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c ₀(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ₁ and γ₂, where either γ₂ ≥ γ₁ > 0 or γ₁ = γ₂ = 0. This example contains mixed symmetric stable processes on as well as mixed relativistic symmetric stable processes on . We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes. Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday. The research of Zhen-Qing Chen is supported in part by NSF Grants DMS-0303310 and DMS-06000206. The research of Takashi Kumagai is supported in part by the Grant-in-Aid for Scientific Research (B) 18340027.     

Perturbation semi-linéaire des résolvantes et des semi-groupes

Résumé La perturbation semi-linéaire des résolvantes et des semi-groupes linéaires, nous donne des résolvantes et des semi-groupes non linéaires. Nous étudions alors les propriétés de ces opérateurs non linéaires et en particulier les fonctions surmédianes et excessives associées.

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We are concerned with nonlinear resolvents and semi-groups. They are obtained by perturbing linear ones. Properties of these nonlinear operators are investigated, particularly supermedian and excessive functions.

Ce travail est soutenu par la fondation nationale pour la recherche scientifique. Projet MA4-89-FST.     

A property of nonlinear elliptic equations when the right-hand side is a measure

LetA(u)=–diva(x, u, Du) be a Leray-Lions operator defined onW ₀ ^1,p () and be a bounded Radon measure. For anyu SOLA (Solution Obtained as Limit of Approximations) ofA(u)= in ,u=0 on , we prove that the truncationsT _k(u) at heightk satisfyA(T _k(u)) A(u) in the weak * topology of measures whenk + .

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Résumé SoitA(u)=–diva(x, u, Du) un opérateur de Leray-Lions défini surW ₀ ^1,p () et une mesure de Radon bornée. Pour toutu SOLA (Solution Obtenue comme Limite d'Approximations) deA(u)= dans ,u=0 sur , nous démontrons que les troncaturesT _k(u) à la hauteurk vérifientA(T _k(u)) A(u) dans la topologie faible * des mesures quandk + .

     

Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps

Summary. We consider a -dimensional Euclidean domain whose boundary is Lipschitz continuous but admits locally finite number of outward or inward H?lder cusp points. Using a method of Stampacchia and Moser for PDE, we first construct a conservative diffusion process on the Euclidean closure of possessing a strong Feller resolvent and associated with a second order uniformly elliptic differential operator of divergence form with measurable coefficients . The sample path of the constructed diffusion can be uniquely decomposed as a sum of a martingale additive functional and an additive functional locally of zero energy. The second additive functional will be proved to be of bounded variation with a Skorohod type expression whenever is weakly differentiable and the H?lder exponent at each outward cusp boundary point is greater than regardless the dimension . Received: 4 October 1995     

Choquet-type integral representation of polysupermedian measures

Some aspects of multi-parameter potential theory are developed: we give a Choquet-type integral representation for measures which are supermedian for a countable family of submarkovian resolvents of commuting kernels on a Radon measurable space. For the subclass of polysupermedian measures we prove a Riesz-type decomposition, and we show that there is a unique integral representation by minimal polysupermedian measures. The setting covers a variety of very different examples like random fields, measures on product spaces which are supermedian for resolvents on the factor spaces, and completely supermedian measures.     

Self-similarity and spectral asymptotics for the continuum random tree

We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application, we determine the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form.     

On reflecting diffusion processes and Skorokhod decompositions

Summary LetG be ad-dimensional bounded Euclidean domain, H¹ (G) the set off in L²(G) such that f (defined in the distribution sense) is in L²(G). Reflecting diffusion processes associated with the Dirichlet spaces (H¹(G), ) on L²(G, dx) are considered in this paper, where A=(a^ij is a symmetric, bounded, uniformly ellipticd×d matrix-valued function such thata ^ij H¹(G) for eachi,j, and H¹(G) is a positive bounded function onG which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with (H¹(G), ) having starting points inG under a mild condition which is satisfied when G has finite (d–1)-dimensional lower Minkowski content.     

Martin–Kuramochi boundary and reflecting symmetric diffusion

In this paper, we will give sufficient conditions for the existence of the reflecting diffusion process on a locally compact space. In constructing reflecting diffusion process, we consider the corresponding Martin–Kuramochi boundary as the reflecting barrier and introduce the notion of strong (ℰ, u)-Caccioppoli set. Our method covers reflecting diffusion processes with diffusion coefficient degenerating on the boundary. Received: 23 June 1997 / Revised version: 28 September 1991/ Published online: 14 June 2000     

Reflected Symmetric α-Stable Processes and Regional Fractional Laplacian

In this paper we investigate the reflected symmetric α-stable processes and their generators. We show that the generators are regional fractional Laplacians on the closed region. In the case of 1≤α<2 their existence requires that on the boundary. Among other things we obtain the integration by parts formula of the regional fractional Laplacian and the semi-martingale decomposition of the reflected symmetric α-stable processes.