Kazuhiro Kuwae

Stochastic calculus for symmetric Markov processes

Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an Ito formula for Dirichlet processes is obtained.

Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms

Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

On Generalized Measure Contraction Property and Energy Functionals over Lipschitz Maps

We construct Sobolev spaces and energy functionals over maps between metric spaces under the strong measure contraction property of Bishop–Gromov type, which is a generalized notion of Ricci curvature bounded below. We also present the notion of generalized measure contraction property, which gives a characterization of energies by approximating energies of Sturm type over Lipschitz maps.

STOCHASTIC CALCULUS OVER SYMMETRIC MARKOV PROCESSES WITHOUT TIME REVERSAL

We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakaos divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain class of functions locally in the domain of Dirichlet form and a generalized Ito formula.

On doubly Feller property

Let X be a Feller process that has strong Feller property. In this p aper, we investigate the Feller as well as strong Feller properties o f the semigroups generated by multiplicative functionals of X in open sets. Special attention is given to the Feynman-Kac and Girsanov transforms of X. Three examples of local Kato class measure that are not of Kato class are given in the last sectio n so that Feller and strong Feller properties hold for corresponding Feynman-Kac semigroup of X in open sets.

Reflected Dirichlet Forms and the Uniqueness of Silverstein's Extension

Reflected Dirichlet space for quasi-regular Dirichlet forms is presented in this paper. We show the closedness of the (active) reflected Dirichlet forms without using the first definition of reflected Dirichlet space by Silverstein and the characterization by Chen. As an application of the closedness, the closability of distorted forms are discussed. We also show the maximality of (active) reflected Dirichlet space in the class of Silversteins extensions and consider the uniqueness problem. Only the techniques of the transfer method and the change of underlying measures are used.

Variational convergence over metric spaces

We introduce a natural definition of L p -convergence of maps, p > 1, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the L p -convergence, we establish a theory of variational convergences. We prove that the Poincare inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are CAT(0)-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature.

Conservativeness of diffusion processes with drift

We show the conservativeness of the Girsanov transformed diffusion process by drift b ∈ L p (R d → R d ) with p > 4/(2 - √2δ(|b| 2 )/λ) or p > 4d/(d+2), or p = 2 if |b| 2 is of the Hardy class with sufficiently small coefficient of energy δ(|b| 2 ) 0 is the lower bound of the symmetric measurable matrix-valued function a(x):= (a i,j (x)) i,j appearing in the given Dirichlet form. In particular, our result improves the conservativeness of the transformed process by b ∈ L d (R d → R d ) when d ≥ 3.

Stochastic calculus over symmetric Markov processes with time reversal

We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakaos divergence-like continuous additive functional of zero energy and the sto chastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain class of functions locally in the domain of Dirichlet form and a generalized ltd for mula.

Resolvent Flows for Convex Functionals and p-Harmonic Maps

Abstract We prove the unique existence of the (non-linear) resolvent associated to a coercive proper lower semicontinuous function satisfying a weak notion of p-uniform λ-convexity on a complete metric space, and establish the existence of the minimizer of such functions as the large time limit of the resolvents, which generalizing pioneering work by Jost for convex functionals on complete CAT(0)-spaces. The results can be applied to Lp-Wasserstein space over complete p-uniformly convex spaces. As an application, we solve an initial boundary value problem for p-harmonic maps into CAT(0)-spaces in terms of Cheeger type p-Sobolev spaces.