On a construction of weak solutions to non-stationary Stokes type equations by minimizing variational functionals and their regularity

summary:In this paper, we prove that the regularity property, in the sense of Gehring-Giaquinta-Modica, holds for weak solutions to non-stationary Stokes type equations. For the construction of solutions, Rothe's scheme is adopted by way of introducing variational functionals and of making use of their minimizers. Local estimates are carried out for time-discrete approximate solutions to achieve the higher integrability. These estimates for gradients do not depend on approximation

A graph discretized approximation of semigroups for diffusion with drift and killing on a complete Riemannian manifold

In the present paper, we prove that the $C_{0}$-semigroup generated by a Schr\"odinger operator with drift on a complete Riemannian manifold is approximated by the discrete semigroups associated with a family of discrete time random walks with killing in a flow on a sequence of proximity graphs, which are constructed by partitions of the manifold. Furthermore, when the manifold is compact, we also obtain a quantitative error estimate of the convergence. Finally, we give examples of the partition of the manifold and the drift term on two typical manifolds: Euclidean spaces and model manifolds

AN EXPLICIT EFFECT OF NON-SYMMETRY OF RANDOM WALKS ON THE TRIANGULAR LATTICE

In the present paper, we study an explicit effect of non-symmetry on asymptotics of the n-step transition probability as n → ∞ for a class of non-symmetric random walks on the triangular lattice. Realizing the triangular lattice into R2 appropriately, we observe that the Euclidean distance in R2 naturally appears in the asymptotics. We characterize this realization from a geometric view point of Kotani-Sunada’s standard realization of crystal lattices. As a corollary of the main theorem, we obtain that the transition semigroup generated by the non-symmetric random walk approximates the heat semigroup generated by the usual Brownian motion on R2

Stochastic quantization associated with the exp⁡(Φ)2\exp(\Phi)_2-quantum field model driven by space-time white noise on the torus in the full L1L^1-regime

The present paper is a continuation of our previous work on the stochastic quantization of the $\exp(\Phi)_2$-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full "$L^{1}$-regime" $\vert\alpha\vert<\sqrt{8\pi}$ of the charge parameter $\alpha$. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.Comment: References are correcte

Stochastic quantization associated with the exp(Φ)2exp(Φ)_2-quantum field model driven by space-time white noise on the torus

We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $exp(Φ)_2$-quantum field model or Høegh-Krohn’s model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation and identify it with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach