Language：English   Publishing type：Research paper (scientific journal)   Publisher：OSAKA JOURNAL OF MATHEMATICS  

In this paper we shall prove the weak convergence of the associated diffusion processes of regular subspaces with monotone characteristic sets for a fixed Dirichlet form. More precisely, given a fixed 1-dimensional diffusion process and a sequence of its regular subspaces, if the characteristic sets of regular subspaces are decreasing or increasing, then their associated diffusion processes are weakly convergent to another diffusion process. This is an extended result of [14].

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

We give a unified method to obtain the conservativeness of a class of Markov processes associated with lower bounded semi-Dirichlet forms on L (2)(X;m), including symmetric diffusion processes, some non-symmetric diffusion processes and jump type Markov processes on X, where X is a locally compact separable metric space and m is a positive Radon measure on X with full topological support. Using the method, we give an example in each section, providing the conservativeness of the processes, that are given by the "increasingness of the volume of some sets(balls)" and "that of the coefficients on the sets" of the Markov processes.

DOI： 10.1007/s11118-016-9596-4

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：OSAKA JOURNAL OF MATHEMATICS  

We give sufficient conditions for Mosco convergences for the following three cases: symmetric locally uniformly elliptic diffusions, symmetric Levy processes, and symmetric jump processes in terms of the L-1(R-d; dx)-local convergence of the (elliptic) coefficients, the characteristic exponents and the jump density functions, respectively. We stress that the global path properties of the corresponding Markov processes such as recurrence/transience, and conservativeness/explosion are not preserved under Mosco convergences and we give several examples where such situations indeed happen.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WALTER DE GRUYTER GMBH  

Recurrence criteria and related capacitary inequalities for a class of symmetric pure jump processes are given under suitable upper volume growth conditions for the underlying metric measure space. Recurrence problems for symmetric jump processes on d-sets are also investigated.

DOI： 10.1515/forum-2013-0190

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：OSAKA JOURNAL OF MATHEMATICS  

Let G be an open set of R-d (d >= 2) and dx denotes the Lebesgue measure on it. We construct a diffusion process with jumps associated with diffusion data (diffusion coefficients {a(ij)(x)}, a drift coefficient {b(i)(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;dx). When G is the whole space, we allow that the diffusion coefficients may degenerate. We also show some Sobolev inequalities for the Dirichlet form and then show the absolute continuity of its resolvent.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER/PLENUM PUBLISHERS  

We give a sufficient condition for a class of jump-type symmetric Dirichlet forms on a"e (d) to be conservative in terms of the jump kernel and the associated measure. Our condition allows the coefficients dominating big jumps to be unbounded. We derive the conservativeness for Dirichlet forms related to symmetric stable processes. We also show that our criterion is sharp by using time changed Dirichlet forms. We finally remark that our approach is applicable to jump-diffusion type symmetric Dirichlet forms on R-d .

DOI： 10.1007/s10959-012-0424-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WYDAWNICTWO UNIWERSYTETU WROCLAWSKIEGO  

Let k(x, y) be a measurable function defined on E x E off the diagonal, where E is a locally compact separable metric space, and let m be a positive Radon measure on E with full support. In 2012, we showed that a quadratic form having k as a Levy kernel becomes a lower bounded semi-Dirichlet form on L-2(E; m) which is non-local and regular. Then there associates a Hunt process corresponding to the semi-Dirichlet form. In the case where E = R-d, we will show that the dual form of the semi-Dirichlet form also produces a Hunt process by taking a killing. As a byproduct, a precise description of the infinitesimal generator of the dual form is also given.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Sufficient conditions for a symmetric jump-diffusion process to be conservative and recurrent are given in terms of the volume of the state space and the jump kernel of the process. A number of examples are presented to illustrate the optimality of these conditions; in particular, the situation is allowed to be that the state space is topologically disconnected but the particles can jump from a connected component to the other components. (C) 2012 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jfa.2012.09.014

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：INST MATHEMATICAL STATISTICS  

Let E be a locally compact separable metric space and in be a positive Radon measure on it. Given a nonnegative function k defined on E x E off the diagonal whose anti-symmetric part is assumed to be less singular than the symmetric part, we construct an associated regular lower bounded semi-Dirichlet form eta on L-2(E; m) producing a Hunt process X-0 on E whose jump behaviours are governed by k. For an arbitrary open subset D C E, we also construct a Hunt process X-D,X-0 on D in an analogous manner. When D is relatively compact, we show that X-D,X-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 In is the Lebesgue measure, a typical example of X is the stable-like process that will be also identified with the solution of a martingale problem up to an eta-polar set of starting points. Approachability to the boundary partial derivative D in finite time of its censored process X-D,X-0 on a bounded open subset D will be examined in terms of the polarity of partial derivative D for the symmetric stable processes with indices that bound the variable exponent alpha(x).

DOI： 10.1214/10-AOP633

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Language：English   Publishing type：Research paper (scientific journal)  

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 suffcient 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. © 2012 Research Institute for Mathematical Sciences, Kyoto University.

DOI： 10.2977/PRIMS/58

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：INST MATHEMATICAL STATISTICS  

Motivated by the recent development in the theory of jump processes, we investigate its conservation property. We will show that a jump process is conservative under certain conditions for the volume-growth of the underlying space and the jump rate of the process. We will also present examples of jump processes which satisfy these conditions.

DOI： 10.1214/09-AIHP368

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

For each n let Y-t((n)) be a continuous time symmetric Markov chain with state space n(-1)Z(d). Conditions in terms of the conductances are given for the convergence of the Y-t((n)) to a symmetric Markov process Y-t on R-d. We have weak convergence of {Y-t((n)) : t <= t(0)} for every t(0) and every starting point. The limit process Y has a continuous part and may also have jumps.

DOI： 10.1007/s00440-009-0224-8

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：CROATIAN MATHEMATICAL SOC  

It is known that the Feller property of a semigroup is stable under a bounded perturbation of the infinitesimal generator. Applying this, we derive the Feller property for a class of integro-differential operators including symmetric stable-like processes.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

We obtain upper and lower bounds of the exit times from balls of a jump-type symmetric Markov process. The proofs are delivered separately. The upper bounds are obtained by using the L,vy system corresponding to the process, while the precise expression of the (L (2)-)generator of the Dirichlet form associated with the process is used to obtain the lower bounds.

DOI： 10.1007/s10114-010-6173-4

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：OSAKA JOURNAL OF MATHEMATICS  

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

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TOHOKU UNIVERSITY  

We show that a large class of regular symmetric Dirichlet forms is generated by pseudo differential operators. We calculate the symbols which are closely related to the sentimartingale characteristics (Levy system) of the associated stochastic processes. Using the symbol we obtain estimates for the mean sojourn time of the process for balls. These estimates and a perturbation argument enable us to prove Holder regularity of the resolvent and semigroup; this entails that the semigroup has the Feller property.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：UNIV WASHINGTON, DEPT MATHEMATICS  

We show that any element from the (L2-) maximal domain of a jump-type symmetric Dirichlet form can be approximated by test functions under some conditions. This gives us a direct proof of the fact that the test functions is dense in Bessel potential spaces.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KLUWER ACADEMIC/PLENUM PUBL  

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.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KLUWER ACADEMIC PUBL  

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

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WALTER DE GRUYTER & CO  

An intrinsic necessary and sufficient condition for a measure being conditionally gaugeable was obtained in [Che2] and [T2]. We apply it to symmetric alpha-stable processes and give an interesting example of conditionally gaugeable measure. We show that for a Green-tight measure mu there exist constants theta(-) and theta(+) with theta(-) < 0 < theta(+) such that the operator -1/2(-Delta)(alpha/2) + thetamu is subcritical if and only if theta belongs to the interval (theta(-); theta(+)).

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：GAUTHIER-VILLARS/EDITIONS ELSEVIER  

Given a regular transient Dirichlet space on L-2(X; m) and an associated m-symmetric Hunt process M on X, we show the equivalence of the capacitary isoperimetric inequality mu(K)(kappa) less than or equal to Theta Cap(K) for a Radon measure mu on X and the ultracontractivity (p) over cap (t) (x, y) less than or equal to (H/t)(1/(1-kappa)) for the transition function (p) over cap (t) of the time changed process of M on the support Of mu by the corresponding additive functional. We shall also show how the constants Theta and H control each other. When the Dirichlet space is the Riesz potential space and M is the symmetric stable process on R-n, we show further that the isoperimetric constant can be replaced by the d-bound sup(xis an element ofRn, r>0) mu(B(x, r))r(-d) of the measure mu. (C) 2003 Editions scientifiques et medicales Elsevier SAS. All rights reserved.

DOI： 10.1016/S0021-7824(03)00034-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KLUWER ACADEMIC PUBL  

We derive Sobolev inequalities for Besov spaces B-alpha(p, p) (F), 0 < α < 1, 1 less than or equal to p < ∞, on d-sets F in R-n, d ≤ n, from a metric property of the Bessel capacity on R-n. We first extend Kaimanovitch's result on the equivalence of Sobolev and capacitary inequalites for contractive p-norms in a general setting allowing unbounded Levy kernels. A simple part of the Jonsson-Wallin trace theorem for Besov spaces and some basic properties of Bessel and Besov capacities on R-n 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 M-α over the d-set. The upper bound of the transition function of M-α and metric properties of M-α-polar sets are then exhibited.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KLUWER ACADEMIC PUBL  

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.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER VERLAG  

In this paper, we show the convergence of forms in the sense of Mosco associated with the part form on relatively compact open set of Dirichlet forms with locally uniform ellipticity and the locally uniform boundedness of ground states tinder regular Dirichlet space setting, We also ger the same assertion under Dirichlet space in infinite dimensional setting. As a result of this, we get the weak convergence under some conditions on initial distributions and the growth order of the volume of the balls defined by (modified) pseudo metric used in K. Th. Sturm.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：OSAKA JOURNAL OF MATHEMATICS  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

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.

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Venue：Mathematisches Forschungsinstitut Oberwolfach  

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Venue：Department of Mathematical Sciences, Seoul National University, South Korea  

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Venue：Kansai University  

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Venue：RIMS, Kyoto University  

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Venue：Mathematisches Institut der Universitat Leipzig  

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Venue：関西大学システム理工学部  

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Venue：Będlewo, Poland  

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Venue：Nara Women's University  

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Venue：Saga University  

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Venue：Saga University  

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Venue：Banff International Research Station, Canada  

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Venue：National Taiwan University, Taipei  

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Venue：National Taiwan University, Taipei  

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Venue：Nagoya Univeristy  

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Venue：Senri Life Center, Osaka Japan  

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Venue：Kansai University, Osaka, Japan  

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Venue：Kansai University, Osaka Japan  

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Venue：Kansai University, Osaka, Japan  

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Venue：Bielefeld University, Germany  

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Venue：Kyushu University, Fukuoka, Japan  

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