Modularity in orbifold theory for vertex operator superalgebras

This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C_2-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C_2-cofinite and g-rational for any g in G.Comment: 31 page

Markov Selection and WW-strong Feller for 3D Stochastic Primitive Equations

This paper studies some analytical properties of weak solutions of 3D stochastic primitive equations with periodic boundary conditions. The martingale problem associated to this model is shown to have a family of solutions satisfying the Markov property, which is achieved by means of an abstract selection principle. The Markov property is crucial to extend the regularity of the transition semigroup from small times to arbitrary times. Thus, under a regular additive noise, every Markov solution is shown to have a property of continuous dependence on initial conditions, which follows from employing the weak-strong uniqueness principle and the Bismut-Elworthy-Li formula

On the small time asymptotics of 3D stochastic primitive equations

In this paper, we establish a small time large deviation principle for the strong solution of 3D stochastic primitive equations driven by multiplicative noise. Both the small noise and the small, but highly nonlinear, unbounded nonlinear terms should be taken into consideration

Malliavin Matrix of Degenerate SDE and Gradient Estimate

In this paper, we prove that the inverse of Malliavin matrix is p integrable for a kind of degenerate stochastic differential equation under some conditions, which like to Hormander condition, but don't need all the coefficients of the SDE are smooth. Furthermore, we obtain a uniform estimation for Malliavin matrix, a gradient estimate, and prove that the semigroup generated by the SDE is strong Feller. Also some examples are given

Modularity of trace functions in orbifold theory for Z-graded vertex operator superalgebras

We study the trace functions in orbiford theory for Z-graded vertex operator superalgebras and obtain a modular invariance result. More precisely, let V be a C_2-cofinite Z-graded vertex operator superalgebra and G a finite automorphism group of V. Then for any commuting pairs (g,h) in G, the h\sigma-trace functions associated to the simple g-twisted V-modules are holomorphic in the upper half plane where \sigma is the canonical involution on V coming from the superspace structure of V. If V is further g-rational for every g n G, the trace unctions afford a representation for the full modular group SL(2,Z).Comment: 14 page

Ergodicity for a class of semilinear stochastic partial differential equations

In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise

3D tamed Navier-Stokes equations driven by multiplicative L\'{e}vy noise: Existence, uniqueness and large deviations

In this paper, we show the existence and uniqueness of a strong solution to stochastic 3D tamed Navier-Stokes equations driven by multiplicative Levy noise with periodic boundary conditions. Then we establish the large deviation principles of the strong solution on the state space $\mathcal{D}([0,T];\mathbb{H}^1)$, where the weak convergence approach plays a key role.Comment: arXiv admin note: text overlap with arXiv:1801.09565; text overlap with arXiv:1701.00314 by other author

Twisted representations of vertex operator superalgebras

This paper gives an analogue of A_g(V) theory for a vertex operator superalgebra V and an automorphism g of finite order. The relation between the g-twisted V-modules and A_g(V)-modules is established. It is proved that if V is g-rational, then A_g(V) is finite dimensional semisimple associative algebra and there are only finitely many irreducible g-twisted V-modules.Comment: 23 page

Ergodicity of the 2D Navier-Stokes Equations with Degenerate Multiplicative Noise

Consider the two-dimensional, incompressible Navier-Stokes equations on the torus We prove that the semigroup P_t generated by the solutions to stochastic Navier-stokes equations is asymptotically strong Feller. Moreover, we also prove that semigroup P_t is exponentially ergodic in some sens

Origin of Cosmic Ray Electrons and Positrons

With experimental results of AMS on the spectra of cosmic ray (CR) $e^{-}$, $e^{+}$, $e^{-}+e^{+}$ and positron fraction, as well as new measurements of CR $e^{-}+e^{+}$ flux by HESS, one can better understand the CR lepton ($e^{-}$ and $e^{+}$) spectra and the puzzling electron-positron excess above $\sim$10 GeV. In this article, spectra of CR $e^{-}$ and $e^{+}$ are fitted with a physically motivated simple model, and their injection spectra are obtained with a one-dimensional propagation model including the diffusion and energy loss processes. Our results show that the electron-positron excess can be attributed to uniformly distributed sources that continuously inject into the galactic disk electron-positron with a power-law spectrum cutting off near 1 TeV and a triple power-law model is needed to fit the primary CR electron spectrum. The lower energy spectral break can be attributed to propagation effects giving rise to a broken power-law injection spectrum of primary CR electrons with a spectral hardening above $\sim$40 GeV