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M. Arisawa

Publications and source records attributed to M. Arisawa.

12 recordsLinked to original sources
arXiv2011

Extended groups of semigroups and backward problems of heat equations

In this paper, we are concerned with backward solvabilities of heat equations, in an abstract framework. We show that semigroups $T_t$ in Banach spaces $X$, generated by heat operators, are extendable to groups in an extended space $E$, which is obtained by considering a sequence of wider Banach spaces containing $X$, i.e. $X$$/subset$$X_t$$/subset$$X_s$... $(t 0$. Another is the backward uniqueness of the semigroup $T_t$. For example, we prove the holomorphic semigroup satisfies the above conditions, and thus is extendable to a group in a larger functional space $E$. We also studied structual properties of the extended space $E$.

math.AP↗
arXiv2011

Quasi-periodic and almost periodic homogenizations of integro-differential equations with Levy operators

In this paper, we studied quasi-periodic and almost periodic homogenizations of integro-differential equations with Levy operators, which contain alpha stable Levy densities. This is the intermediate stage between the periodic homogenization and the stochastic homogenization of the non-local problem, and is the straight forward generalization of the case of partial differential equations.

math.AP↗
arXiv2010

An Interior Gradient Estimate for a class of Second Order Partial Differential Inequalities

A uniform gradient for functions u which satisfy a system of N second-order partial differential inequalities is given in this paper. Some structure conditions are given for the coefficients of the matrices of second-order terms and of first-order terms. This result can be applied to study the gradient estimate of a class of second-order degenerate elliptic partial differential equations.

math.AP↗
arXiv2010

Homogenization of a class of integro-differential equations with L{é}vy operators

The periodic homogenization problem of integro-differential equations of the alpha stable L{é}vy operators is studied in this paper. Thanking to the symmetry of the L{é}vy density, we can use the method of the formal asymptotic expansion, to connect the problem to the ergodic cell problem. A rigorous proof is given by the perturbed test function's method.

math.AP↗
arXiv2010

Multiscale Homogenizations for first-order Hamilton-Jacobi-Bellman Equations

The quasi-periodic homogenization for some classes of first-order Hamilton-Jaconi-Bellman equation is studied in this paper. The cell problem of the quasi-periodic homogenization satisfies the non-resonance condition, under which the corresponding deterministic system is ergodic. The almost periodic homogenization for the same classes of equations is also solved, as a limit of a sequence of quasi-periodic homogenizations. Here, the almost periodicity is in the sense of H. Bohr. This result has been cited by some authors, for example: by H. Ishii, "Almost periodic homogenization of Hamilton-Jacobi equations", in Int. Conf. on Diff. Eqs., vol.1, Berlin, 1999, World Scientific, River Edge, NJ 2000, pp. 600-605; and by P.-L. Lions, and P.E. Souganidis, "Correctors for the Homogenizations of Hamilton-Jacobi Equations in the stationary ergodic setting", Comm. Pure Appl. Math. LVI, (2003), pp. 1501-1524.

math.AP↗
arXiv2010

Homogenizations of integro-differential equations with L{é}vy operators with asymmetric and degenerate densities

We consider periodic homogenization problems for the L{é}vy operators with asymmetric L{é}vy densities. The formal asymptotic expansion used for the $\a$-stable (symmetric) L{é}vy operators ($\a\in (0,2)$) is not applicable directly to such asymmetric cases. We rescale the asymmetric densities, extract the most singular part of the measures, which average out the microscopic dependences in the homogenization procedures. We give two conditions (A) and (B), which characterize such a class of asymmetric densities, that the above "rescaled" homogenization is available.

math.AP↗
arXiv2010

A localization of the L{é}vy operators arising in mathematical finances

The comparison principle and the existence of the solution of the integro-differential equation with L{é}vy operators, in the framework of the viscosity solution, are shown in this paper. For the one dimensional case, a detailed estimate of the H{ö}lder continuity of solutions is presented, by localizing the singularity of the L{é}vy measure.

math.AP↗
arXiv2010

Strong maximum principle for radiative tranfer type operators

The strong maximum principle ((SMP) in short) for subsolutions of the radiative transfer type equations is shown in this paper. We treat a general class of integro-differential equations, defined in the product space of the space variable "x" and the velocity variable "v". The equations consist of two terms : a nonlocal integral operator in v variable, and a first-order partial-differential operator in x variable. The nonlocal term represents the jump process in v direction, and the term of the first-order partial differential operator describes the drift in x direction. In particular, the drift in x is generated by the velocity variable v. Based on the idea of the propagation of maxima, we give a general sufficient condition ((A) in the paper) so that the (SMP) holds for the present class of nonlocal equations. The framework of the viscosity solution is used to formulate the problem, and the related existence and uniqueness of solutions are also given.

math.AP↗