Toshihiro Uemura

Conservation property of symmetric jump processes

Many papers concerning the conservation property for various Markov processes are known so far. In particular, the conservativeness for symmetric diffusion processes is known as giving the volume growth conditions with respect to the ‘intrinsic metric’ relative to the processes. On the other hand, there are few papers for which the conservativeness property is derived in the case of jump processes. In this talk, we will focus on symmetric jump processes and give a condition on the jump (Levy) kernel to derive the conservation property.

A law of large numbers for random sets

Abstract We show a law of large numbers for random sets taking values in a class of subsets of a Banach space, which is at least larger than the class of compact subsets of the space, if the Banach space is of infinite dimension. To see this we first define the Hausdorff distance on the family of all the non-empty subsets of a Banach space and give some properties of it. Motivated by the fact that a study of fuzzy sets is to study of set-valued analysis.

On Some Path Properties of Symmetric Stable-Like Processes for One Dimension

We show that a symmetric stable-type form becomes a Dirichlet form in the wide sense under a quite mild assumption and give a necessary and sufficiently condition that the domain contains the family of all uniformly Lipschitz continuous functions with compact support. Moreover we give some path properties of the corresponding Markov processes (we call the processes symmetric stable-like processes) in one dimension such as exceptionality of points and recurrence of the processes. We then note that the recurrence of the processes depend on the behavior of the index functions at the infinity.

On Sobolev and Capacitary Inequalities for Contractive Besov Spaces over d-sets

We derive Sobolev inequalities for Besov spaces Bαp,p(F), 0<α<1, 1≤p<∞ on d-sets F in Rn, d≤n, from a metric property of the Bessel capacity on Rn. We first extend Kaimanovitchs result on the equivalence of Sobolev and capacitary inequalites for contractive p-norms in a general setting allowing unbounded Lévy kernels. A simple part of the Jonsson–Wallin trace theorem for Besov spaces and some basic properties of Bessel and Besov capacities on Rn are then utilized in getting the desired inequalities. When p=2, the Besov space being considered is a non-local regular Dirichlet space and gives rise to a jump type symmetric Markov process Mα over the d-set. The upper bound of the transition function of Mα and metric properties of Mα-polar sets are then exhibited.

On Symmetric Stable-Like Processes: Some Path Properties and Generators

We derive some path properties of symmetric stable-like processes constructed via Dirichlet form theory and then sufficient conditions in order that the generators of the forms contain a nice functions space, are given.

Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms

Let E be a locally compact separable metric space and m be a positive Radon measure on it. Given a nonnegative function k de- fined on E×E off the diagonal whose anti-symmetric part is assumed to be less singular than the symmetric part, we construct an associ- ated regular lower bounded semi-Dirichlet formon L 2 (E;m) pro- ducing a Hunt process X 0 on E whose jump behaviours are governed by k. For an arbitrary open subset DE, we also construct a Hunt process X D,0 on D in an analogous manner. When D is relatively compact, we show that X D,0 is censored in the sense that it admits no killing inside D and killed only when the path approaches to the boundary. When E is a d-dimensional Euclidean space and m is the Lebesgue measure, a typical example of X 0 is the stable-like process that will be also identified with the solution of a martingale prob- lem up to an �-polar set of starting points. Approachability to the boundary @D in finite time of its censored process X D,0 on a bounded open subset D will be examined in terms of the polarity of @D for the symmetric stable processes with indices that bound the variable exponent �(x).

On the Structure of the Domain of a Symmetric Jump-type Dirichlet Form

We characterize the structure of the domain of a pure jump-type Dirichlet form which is given by a Beurling{Deny formula. In particular, we obtain sucient conditions in terms of the jumping kernel guaranteeing that the test functions are a core for the Dirichlet form and that the form is a Silverstein extension. As an application we show that for recurrent Dirichlet forms the extended Dirichlet space can be interpreted in a natural way as a homogeneous Dirichlet space. For reected Dirichlet spaces this leads to a simple purely analytic proof that the active reected Dirichlet space (in the sense of Chen, Fukushima and Kuwae) coincides with the extended active reected Dirichlet space.

A remark on non-local operators with variable order

We reveal a relationship between the non-local operator L with variable order having n as a L´ evy-type kernel and the symmetric quadratic form defined by t he kernel n. The relationship is obtained through the carr ´ e du champ operator relative to L.

ON MULTIDIMENSIONAL DIFFUSION PROCESSES WITH JUMPS

Let G be an open set of d (d 2) and d x denotes the Lebesgue measure on it. We construct a diffusion process with jumps associated with diffusion data (diffusion coefficients {ai j (x)}, a drift coefficient {bi (x)} and a killing function c(x)) and a Levy kernel k(x, y) in terms of a lower bounded semi-Dirichlet form on L 2 (G d x). When G is the whole space, we allow that the diffusion coefficients m ay degenerate. We also show some Sobolev inequalities for the Dirichlet form and then show the absolute continuity of its resolvent.

On Spectral Synthesis for Contractive p-Norms and Besov Spaces

We prove that spectral synthesis is possible for a general function space Fp with a contractive p-norm, namely, any quasi-continuous function in Fp vanishing q.e. outside an open set G can be approximated in this norm by continuous functions in Fp with compact support in G. The result is applied to contractive Besov spaces over d-sets in RN and censored stable processes over N-sets.