1,143 research outputs found

    Convergence of Non-Symmetric Diffusion Processes on RCD spaces

    Full text link
    We construct non-symmetric diffusion processes associated with Dirichlet forms consisting of uniformly elliptic forms and derivation operators with killing terms on RCD spaces by aid of non-smooth differential structures introduced by Gigli '16. After constructing diffusions, we investigate conservativeness and the weak convergence of the laws of diffusions in terms of a geometric convergence of the underling spaces and convergences of the corresponding coefficients.Comment: 41 pages. To appear in Calc. Var. PDEs. In the second version, the following have been modified: Section 2.3, 2.4, 2.5, 2.6 were added. Assumption 3.3, Proposition 3.4, Remark 3.5, and Example 3.7 were deleted. Example 7.2 was replaced with Corollary 7.2. Theorem 4.4 was modifie

    Convergence of Brownian Motions on Metric Measure Spaces Under Riemannian Curvature-Dimension Conditions

    Full text link
    We show that the pointed measured Gromov convergence of the underlying spaces implies (or under some condition, is equivalent to) the weak convergence of Brownian motions under Riemannian Curvature-Dimension (RCD) conditions.Comment: 39 pages, 3 figures. Final version. To appear in EJP. Theorem 1.4 has been modified with the time interval [\epsilon, \infty

    Convergences and projection Markov property of Markov processes on ultrametric spaces

    Full text link
    Let (S,ρ)(S,\rho) be an ultrametric space with certain conditions and SkS^k be the quotient space of SS with respect to the partition by balls with a fixed radius ϕ(k)\phi(k). We prove that, for a Hunt process XX on SS associated with a Dirichlet form (E,F)(\mathcal E, \mathcal F), a Hunt process XkX^k on SkS^k associated with the averaged Dirichlet form (Ek,Fk)(\mathcal E^k, \mathcal F^k) is Mosco convergent to XX, and under certain additional conditions, XkX^k converges weakly to XX. Moreover, we give a sufficient condition for the Markov property of XX to be preserved under the canonical projection πk\pi^k to SkS^k. In this case, we see that the projected process πkX\pi^k\circ X is identical in law to XkX^k and converges weakly to XX.Comment: 17 page

    On the ergodicity of interacting particle systems under number rigidity

    Get PDF
    In this paper, we provide relations among the following properties:(a) the tail triviality of a probability measure μ on the configuration space ϒ; (b) the finiteness of a suitable L2-transportation-type distance d¯ϒ; (c) the irreducibility of local μ-symmetric Dirichlet forms on ϒ. As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction and arising from determinantal/permanental point processes including sine2, Airy2, Besselα,2 (α ≥ 1), and Ginibre point processes. In particular, the case of the unlabelled Dyson Brownianmotion is covered. For the proof, the number rigidity of point processes in the sense of Ghosh–Peres plays a key role

    An Algorithmic Framework for Computing Validation Performance Bounds by Using Suboptimal Models

    Full text link
    Practical model building processes are often time-consuming because many different models must be trained and validated. In this paper, we introduce a novel algorithm that can be used for computing the lower and the upper bounds of model validation errors without actually training the model itself. A key idea behind our algorithm is using a side information available from a suboptimal model. If a reasonably good suboptimal model is available, our algorithm can compute lower and upper bounds of many useful quantities for making inferences on the unknown target model. We demonstrate the advantage of our algorithm in the context of model selection for regularized learning problems
    corecore