arXiv subjects
Publications and source records attributed to Kohei Suzuki.
Suppose that metric measure spaces X_n=(X_n, d_n, m_n) satisfy RCD(K,infty) conditions with m_n(X_n)=1. Then the measured Gromov convergence (introduced by Gigili-Mondino-Savare '13) of X_n is equivalent to the weak convergence of the laws of Brownian motions on X_n with initial distributions m_n.
Let $X$ be a compact metric space and $\mathcal M_X$ be the set of isometry classes of compact metric spaces $Y$ such that the Lipschitz distance $d_L(X,Y)$ is finite. We show that $(\mathcal M_X, d_L)$ is not separable when $X$ is a closed interval, or an infinite union of shrinking closed intervals.
Let $(S,ρ)$ be an ultrametric space with certain conditions and $S^k$ be the quotient space of $S$ with respect to the partition by balls with a fixed radius $ϕ(k)$. We prove that, for a Hunt process $X$ on $S$ associated with a Dirichlet form $(\mathcal E, \mathcal F)$, a Hunt process $X^k$ on $S^k$ associated with the averaged Dirichlet form $(\mathcal E^k, \mathcal F^k)$ is Mosco convergent to $X$, and under certain additional conditions, $X^k$ converges weakly to $X$. Moreover, we give a sufficient condition for the Markov property of $X$ to be preserved under the canonical projection $π^k$ to $S^k$. In this case, we see that the projected process $π^k\circ X$ is identical in law to $X^k$ and converges weakly to $X$.
We give sufficient conditions for Mosco convergences for the following three cases: symmetric locally uniformly elliptic diffusions, symmetric Lévy processes, and symmetric jump processes in terms of the $L^1(\mathbb R;dx)$-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.
We introduce a new distance, a Lipschitz-Prokhorov distance $d_{LP}$, on the set $\mathcal {PM}$ of isomorphism classes of pairs $(X, P)$ where $X$ is a compact metric space and $P$ is the law of a continuous stochastic process on $X$. We show that $(\mathcal {PM}, d_{LP})$ is a complete metric space. For Markov processes on Riemannian manifolds, we study relative compactness and convergence.