Narrow Results

We establish sharp (or ’refined’) comparison theorems for the Klein–Gordon equation. We show that the condition $V_a\le V_b$, which leads to $E_a\le E_b$, can be replaced by the weaker assumption $U_a\le U_b$ which still implies the spectral ordering $E_a\le E_b$. In the simplest case, for $d=1$, $U_i(x)={\int }_0^x{V}_i(t)dt$, $i=a$ or $b$ and for $d>1$, $U_i(r)={\int }_0^r{V}_i(t)t^{d-1}dt$, $i=a$ or $b$. We also consider sharp comparison theorems in the presence of a scalar potential $S$ (a ‘variable mass’) in addition to the vector term $V$ (the time component of a four-vector). The theorems are illustrated by a variety of explicit detailed examples.

By recasting the Klein–Gordon equation as an eigen-equation in the coupling parameter $v>0,$ the basic Klein–Gordon comparison theorem may be written $f_1\le f_2\Rightarrow G_1(E)\le G_2(E)$, where $f_1$ and $f_2$ are the monotone nondecreasing shapes of two central potentials $V_1(r)=v_1{f}_1(r)$ and $V_2(r)=v_2{f}_2(r)$ on $[0,\infty )$. Meanwhile, $v_1=G_1(E)$ and $v_2=G_2(E)$ are the corresponding coupling parameters that are functions of the energy $E\in (-m,m)$. We weaken the sufficient condition for the ground-state spectral ordering by proving (for example in $d=1$ dimension) that if ${\int }_0^x[f_2(t)-f_1(t)]{\phi }_i(t)dt\ge 0$, the couplings remain ordered $v_1\le v_2$, where $i=1\mathrm{\text{or}}2,$ and $\{{\phi }_1,{\phi }_2\}$ are the ground-states corresponding, respectively to the couplings $\{v_1,v_2\}$ for a given $E\in (-m,m).$. This result is extended to spherically symmetric radial potentials in $d>1$ dimensions.

In this paper, we study stochastic partial differential equations with two reflecting smooth walls h¹ and h², driven by space-time white noise with non-constant diffusion coefficients. The existence and uniqueness of the solutions are established.

Let Mⁿ be an n-dimensional Riemannian manifold with boundary ∂M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k ∈ ℝ, we give a sharp estimate of the upper bound of ρ(x) = d(x, ∂M), in terms of the mean curvature bound of the boundary. When ∂M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kähler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kähler manifold and also estimate the first eigenvalue of the real Laplacian.

In this paper we consider stochastic equations in Banach spaces. Our first result is a comparison theorem. As an application we prove an existence theorem in the case when the drift coefficient is nonsmooth. The present studies extend some results both for deterministic and stochastic equations in infinite dimensional case.

We prove an existence and uniqueness result for quasilinear Stochastic PDEs with obstacle (in short OSPDE) under a weaker integrability condition on the coefficient and the barrier.

By means of Itô’s formula and a comparison theorem for integral equations, we study the blow up in finite time of semilinear stochastic differential equations of the form

This paper is devoted to comparison of strong solutions of stochastic equations with respect to optional semimartingales. Optional semimartingales have right and left limits but are not necessarily continuous and therefore defined on “unusual” probability spaces. Integration theory with respect to optional semimartingales is well-developed. However, not much attention is given to stochastic integral equations of optional semimartingales. A pathwise comparison result for strong solutions of a very general class of optional stochastic equations with non-lipshitz coefficients is given. Moreover, simple applications to mathematical finance is presented.

In this paper, we consider two classes of backward stochastic differential equations (BSDEs). First, under a Lipschitz-type condition on the generator of the equation, which can also be unbounded, we give sufficient conditions for the existence of a unique solution pair. The method of proof is that of Picard iterations and the resulting conditions are new. We also prove a comparison theorem. Second, under the linear growth and continuity assumptions on the possibly unbounded generator, we prove the existence of the solution pair. This class of equations is more general than the existing ones.

We prove the existence and uniqueness of solution to obstacle problem for quasilinear stochastic partial differential equations with Neumann boundary condition. Our method is based on the analytical techniques coming from parabolic potential theory. The solution is expressed as a pair $(u,\nu )$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ is a random regular measure satisfying minimal Skohorod condition.

We establish well-posedness for a class of systems of SDEs with non-Lipschitz coefficients in the diffusion and jump terms and with two sources of interdependence: a monotone function of all the components in the drift of each SDE and the correlation between the driving Brownian motions and jump random measures. Pathwise uniqueness is derived by employing some standard techniques. Then, we use a comparison theorem along with our uniqueness result to construct non-negative, $L^1$-integrable càdlàg solutions as monotone limits of solutions to approximating SDEs, allowing for time-inhomogeneous drift terms to be included. Our approach allows also for a comparison property to be established for the solutions to the systems we investigate. The applicability of certain systems in financial modeling is also discussed.

In this paper, we consider the reflected backward stochastic differential equations driven by $G$-Brownian motion (reflected $G$-BSDEs) with time-varying Lipschitz coefficients. We obtain the uniqueness result by establishing a priori estimates. For the existence, the solution can be approximated by a family of reflected $G$-BSDEs with Lipschitz conditions and by penalized $G$-BSDEs with time-varying coefficients. The latter approximation is useful to get the comparison theorem. Finally, we study the reflected $G$-BSDEs with infinite time horizon.

This paper studies the binary classification problem associated with a family of Lipschitz convex loss functions called large-margin unified machines (LUMs), which offers a natural bridge between distribution-based likelihood approaches and margin-based approaches. LUMs can overcome the so-called data piling issue of support vector machine in the high-dimension and low-sample size setting, while their theoretical analysis from the perspective of learning theory is still lacking. In this paper, we establish some new comparison theorems for all LUM loss functions which play a key role in the error analysis of large-margin learning algorithms. Based on the obtained comparison theorems, we further derive learning rates for regularized LUMs schemes associated with varying Gaussian kernels, which maybe of independent interest.

Briand et al. gave a counterexample showing that given g, Jensen's inequality for g-expectation usually does not hold in general. This paper proves that Jensen's inequality for g-expectation holds in general if and only if the generator g(t, z) is super-homogeneous in z. In particular, g is not necessarily convex in z.

In this paper, a discrete n-species Lotka–Volterra type food-chain system with time delays and feedback controls is proposed. By applying the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system.

In this paper we improve the comparison theorem on magnetic Jacobi fields for Kähler magnetic fields under an assumption that sectional curvatures are bounded from above or from below by some constants.

In this paper we study balls and spheres formed by trajectories for Kähler magnetic fields on a Kähler manifold and give estimates of their volume elements under some assumptions on sectional curvatures.