Brownian motion with darning (BMD in abbreviation) is introduced and studied in [5] and [6, Chapter 7]. Roughly speaking, BMD travels across the “darning area" at infinite speed, while it behaves like a regular BM outside of this area. In this paper we show that starting from a single point in its state space, BMD is the weak limit of a family of continuous-time simple random walks on square lattices with diminishing mesh sizes. From any vertex in their state spaces, the approximating random walks jump to its nearest neighbors with equal probability after an exponential holding time.

In this paper, we will show that if the sectional curvature of a Hadamard manifold $M$ is pinched by two negative constants, then $M$-valued jump-diffusion process $\{X_t;0\leq t\lt e\}$ satisfying suitable conditions on the Lévy measure is irreducible, transient and conservative. In order to show such properties of paths, we need the upper and lower estimates of the radial part of the jump-diffusion process.

In this paper, we consider the semilinear elliptic problem $-\Delta u=a(x)|u|^{p-2}u+\lambda b(x)|u|^{q-2}u$ in a bounded domain $\Omega$ with Neumann boundary condition. We show the existence infinitely many solutions by applying critical point theory with a suitable decomposition of the Sobolev space $W^{1,2}(\Omega)$. Also we prove the $C^{\alpha}$ regularity of the solutions.

We give some properties of the dynamical number of base-points of birational self-maps of the complex projective plane.

In particular we give a formula to determine the dynamical number of base-points of non base-wandering Jonquières twists.

In this note we prove the anisotropic version of the Shannon inequality. This can be conveniently realised in the setting of Folland and Stein's homogeneous groups. We give two proofs: one giving the best constant, and another one using the Kubo-Ogawa-Suguro inequality.

In this paper, we study the anabelian geometry of hyperbolic polycurves of dimension $2$ over sub-$p$-adic fields. In $1$-dimensional case, Mochizuki proved the Hom version of the Grothendieck conjecture for hyperbolic curves over sub-$p$-adic fields and the pro-$p$ version of this conjecture. In $2$-dimensional case, a naive analogue of this conjecture does not hold for hyperbolic polycurves over general sub-$p$-adic fields.Moreover, the Isom version of the pro-$p$ Grothendieck conjecture does not hold in general. We explain these two phenomena and prove the Hom version of the Grothendieck conjecture for hyperbolic polycurves of dimension $2$ under the assumption that the Grothendieck section conjecture holds for some hyperbolic curves.

We generalize Murai's conjecture on an upper bound for the number of irreducible $p$-Brauer characters in the principal block to an arbitrary block. We prove that the new conjecture has an affirmative answer for tame blocks and blocks with cyclic defect groups. In addition we confirm Murai's conjecture for symmetric and alternating groups.

This paper studies fine Selmer groups of elliptic curves in abelian $p$-adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic $\mathbb{Z}_p$-extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified.