Strichartz Estimates for Schr(?)dinger Equations with Non-degenerate Coefficients

In the present paper,the full range Strichartz estimates for homogeneous Schr(?)dinger equations with non-degenerate and non-smooth coefficients are proved.For inhomogeneous equation,the non-endpoint Strichartz estimates are also obtained.     

A parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces

In this paper we construct a parametrix for the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of (□−λ)u=f and to obtain a uniform Lp estimate for a family of bump functions traveling to infinity.     

球面上热核的渐近展开

具体计算了球面Ｓｎ（１）（ｎ２）上热核的渐近展开式中前五项的系数，而根据已有的公式只能算出前四项．最后给出了展开式一般项的递推公式，发现它与Ｂｅｒｎｏｕｌｉ数有未曾想到的联系．根据不变量理论，我们可以确定任意ｎ维紧致无边Ｒｉｅｍａｎｎ流形上热核的渐近展开式中第五项的系数．     

Strichartz Estimates for Schr(o)dinger Equations with Non-degenerate Coefficients

In the present paper, the full range Strichartz estimates for homogeneous Schr(o)dinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz estimates are also obtained.     

Parametrices and Exact Paralinearization of Semi-Linear Boundary Problems

The subject is parametrices for semi-linear problems, based on parametrices for linear boundary problems and on non-linearities that decompose into solution-dependent linear operators acting on the solutions. Non-linearities of product type are shown to admit this via exact paralinearization. The parametrices give regularity properties under weak conditions; improvements in subdomains result from pseudo-locality of type 1,1-operators. The framework encompasses a broad class of boundary problems in Hölder and L _p -Sobolev spaces (and also Besov and Lizorkin–Triebel spaces). The Besov analyses of homogeneous distributions, tensor products and halfspace extensions have been revised. Examples include the von Karman equation.     

On weak uniqueness and distributional properties of a solution to an SDE with α-stable noise

For an SDE driven by a rotationally invariant $\alpha$-stable noise we prove weak uniqueness of the solution under the balance condition $\alpha +\gamma >1$, where $\gamma$ denotes the Hölder index of the drift coefficient. We prove the existence and continuity of the transition probability density of the corresponding Markov process and give a representation of this density with an explicitly given “principal part”, and a “residual part” which possesses an upper bound. Similar representation is also provided for the derivative of the transition probability density w.r.t. the time variable.     

The parametrix method for parabolic SPDEs

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Itô–Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.     

Second order probabilistic parametrix method for unbiased simulation of stochastic differential equations

In this article, following the paradigm of bias–variance trade-off philosophy, we derive parametrix expansions of order two, based on the Euler–Maruyama scheme with random partitions, for the purpose of constructing an unbiased simulation method for multidimensional stochastic differential equations. These formulas lead to Monte Carlo simulation methods which can be easily parallelized. The second order method proposed here requires further regularity of coefficients in comparison with the first order method but achieves finite moments even when Poisson sampling is used for the partitions, in contrast to Andersson and Kohatsu-Higa (2017). Moreover, using an exponential scaling technique one achieves an unbiased simulation method which resembles a space importance sampling technique which significantly improves the efficiency of the proposed method. A hint of how to derive higher order expansions is also presented.     

Probability density function of SDEs with unbounded and path-dependent drift coefficient

In this paper, we first prove that the existence of a solution of SDEs under the assumptions that the drift coefficient is of linear growth and path-dependent, and diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. We apply Gaussian upper bound for a probability density function of a solution of SDE without drift coefficient and local Novikov condition, in order to use Maruyama–Girsanov transformation. The aim of this paper is to prove the existence with explicit representations (under linear/super-linear growth condition), Gaussian two-sided bound and Hölder continuity (under sub-linear growth condition) of a probability density function of a solution of SDEs with path-dependent drift coefficient. As an application of explicit representation, we provide the rate of convergence for an Euler–Maruyama (type) approximation, and an unbiased simulation scheme.     

Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type

Let (M,g) be a globally symmetric space of noncompact type, of arbitrary rank, and Δ its Laplacian. We introduce a new method to analyze Δ and the resolvent (Δ-σ)^-1; this has origins in quantum N-body scattering, but is independent of the ‘classical’ theory of spherical functions, and is analytically much more robust. We expect that, suitably modified, it will generalize to locally symmetric spaces of arbitrary rank. As an illustration of this method, we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a Riemann surface multiply covering the plane. We also show how this continuation may be deduced using the theory of spherical functions. In summary, this paper establishes a long-suspected connection between the analysis on symmetric spaces and N-body scattering.