Heavy tails of a Lévy process and its maximum over a random time interval

Let {Xt,t0} be a Lévy process with Lévy measure ν on(∞,∞),and let τ be a nonnegative random variable independent of {Xt,t0}.We are interested in the tail probabilities of X τ and X(τ) = sup0≤t≤τXt.For various cases,under the assumption that either the Lévy measure ν or the random variable τ has a heavy right tail we prove that both Pr(X τ > x) and Pr(X(τ) > x) are asymptotic to Eτν((x,∞)) + Pr(τ > x/(0 ∨ EX 1)) as x →∞,where Pr(τ > x/0) = 0 by convention.     

The ergodicity of stochastic generalized porous media equations with lévy jump

In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of(pt)t≥0 to equilibrium uniform on any bounded subset in H.     

THE ERGODICITY OF STOCHASTIC GENERALIZED POROUS MEDIA EQUATIONS WITH LVY JUMP

In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of(pt)t≥0 to equilibrium uniform on any bounded subset in H.     

A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N-1Va(xi-xj) where Va(x) = a-2V (x/a). It is well known that the Gross-Pitaevskii (GP) equation is a nonlinear Schrdinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices {k ut, k ≥ 1} solves the GP hierarchy. Denote by ψN,t the solution to the N-particle Schrdinger equation. Under the assumption that a = N-ε for 0 ε 3/4, we prove that as N →∞ the limit points of the k-particle density matrices of ψN,t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V (x) dx.     

Monotone positive solution for three-point boundary value problem

In this paper, the existence of monotone positive solution for the following secondorder three-point boundary value problem is studied：
x″（t）＋f（t,x（t））=0,0〈t〈1,
x′（0）=0,x（1）＋δx′（η）=0,
where η ∈ （0, 1）, δ∈ [0, ∞）, f ∈ C（[0, 1] × [0, ∞）, [0, ∞））. Under certain growth conditions on the nonlinear term f and by using a fixed point theorem of cone expansion and compression of functional type due to Avery, Anderson and Krueger, sufficient conditions for the existence of monotone positive solution are obtained and the bounds of solution are given. At last, an example is given to illustrate the result of the paper.     

Exponential convergence in probability for empirical means of Lévy processes

Let （Xt）t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato＇s class, we show that the empirical mean 1/t ∫ f（Xs）ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Lévy processes.     

LARGE DEVIATIONS FOR SYMMETRIC DIFFUSION PROCESSES

Let a(x)=(a_(ij)(x)) be a uniformly continuous, symmetric and matrix-valued function satisfying uniformly elliptic condition, p(t, x, y) be the transition density function of the diffusion process associated with the Diriehlet space (, H_0~1 (R~d)), where(u, v)=1/2 integral from n=R~d sum from i=j to d(u(x)/x_i v(x)/x_ja_(ij)(x)dx).Then by using the sharpened Arouson's estimates established by D. W. Stroock, it is shown that2t ln p(t, x, y)=-d~2(x, y).Moreover, it is proved that P_y~6 has large deviation property with rate functionI(ω)=1/2 integral from n=0 to 1<(t), α~(-1)(ω(t)),(t)>dtas s→0 and y→x, where P_y~6 denotes the diffusion measure family associated with the Dirichlet form (ε, H_0~1(R~d)).     

Infinitely Many Solutions for a Class of Quasi-linear Elliptic Problem

In this paper,we study the following quasi-linear elliptic equation:■where Ω?R^N is a bounded domain,λ>0 is a parameter.The function ψ(|t|)t is the subcritical term,and φ(|t|)t is the critical Orlicz-Sobolev growth term with respect to φ.Under appropriate conditions on φ,ψ and φ,we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation,for λ∈(0,λ₀),where λ₀> 0 is a fixed constant.     

Limiting behavior of a class of Hermitian Yang-Mills metrics,I

This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills(HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with K?hler metrics ωε when ε → 0. Here, ωε are flat and have areas ε and ε-1 on the two elliptic curves, respectively.A family of Hermitian metrics on the vector bundle are explicitly constructed and with respect to them, the HYM metrics are normalized. We then compare the family of normalized HYM metrics with the family of constructed Hermitian metrics by doing estimates. We get the higher order estimates as long as the C~0-estimate is provided. We also get the estimate of the lower bound of the C~0-norm. If the desired estimate of the upper bound of the C~0-norm can be obtained, then it would be shown that these two families of metrics are close to arbitrary order in ε in any Cknorms.     

BEST MONOTONE L_■-APPROXIMATIONS IN SEVERAL VARIABLES

Given a continuous function f defined on the unit cube of R~n and a convexfunction _t,_t(0)-0,_t(x)>0,for x>0,we prove that the set ofbest L~(t)-approximations by monotone functions has exactly one elementft,which is also a continuous function.Moreover if the family of convexfunctions {_t}t>0 converges uniformly on compact sets to a function _0,then the best approximation f_t→f_0 uniformly,as t→0,where fo is thebest approximation of f within the Orlicz space L~(0) The best approxima-tions{f_t}are obtained as well as minimizing integrals or the Luxemburgnorm     

A note on homoclinic orbits for second order Hamiltonian system

Some existence and multiplicity of homoelinic orbits for second order Hamiltonian system x-a(t)x f(t,x)=0 are given by means of variational methods, where the function -1/2a（t）|s|^2∫^t0f（t,s）ds is asymptotically quadratic in s at infinity and subquadratic in s at zero, and the function a (t) mainly satisfies the growth condition limt→∞∫^t 1 t a（t）dt= ∞,VI∈R^1.A resonance case as well as a noncompact case is discussed too.     

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

Nonradial Entire Large Solutions of Semilinear Elliptic Equations

We consider the problem of whether the equation △u = p（x）f（u） on RN, N ≥ 3, has a positive solution for which lim ｜x｜→∞（x） = ∞ where f is locally Lipschitz continuous, positive, and nondecreasing on （0,oo） and satisfies ∫1∞[F （t）]^- 1/2dt = ∞ where F（t） = ∫0^tf（s）ds. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies ∫0∞ r min｜x｜=r P （x） dr = ∞. Conversely, we show that a necessary condition for the solution to exist is that p satisfies ∫0∞r1＋ε min ｜x｜=rp（x）dr =∞ for all ε〉0.     

Oscillatory Behavior of Third-order Nonlinear Differential Equations with a Sublinear Neutral Term

The authors present some new criteria for oscillation and asymptotic behavior of solutions of thirdorder nonlinear differential equations with a sublinear neutral term of the form ■where z(t) = x(t)+∫_a~b p(t, ξ)x~γ(τ(t, ξ)) dξ, 0 < γ ≤ 1. Under the conditions∫_t0~∞ r-1/α(t)dt = ∞ or∫_t0~∞ r-1/α(t)dt <∞. The results obtained here extend, improve and complement to some known results in the literature. Examples are provided to illustrate the theorems.     

A Tree-valued Markov Process Associated with an Admissible Family of Branching Mechanisms

Let E?R be an interval. By studying an admissible family of branching mechanisms{ψt,t ∈E} introduced in Li [Ann. Probab., 42, 41-79(2014)], we construct a decreasing Levy-CRT-valued process {Tt, t ∈ E} by pruning Lévy trees accordingly such that for each t ∈E, Tt is a ψt-Lévy tree. We also obtain an analogous process {Tt*,t ∈E} by pruning a critical Levy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of {Tt,t ∈E} at the ascension time A := inf{t ∈E;Tt is finite} can be represented by{Tt*,t∈E}.The results generalize those studied in Abraham and Delmas [Ann. Probab., 40, 1167-1211(2012)].     

QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS ON THE PROJECTIVE PLANE

Consider the differential system \[\frac{{dx}}{{dt}} = P(x,y),\frac{{dy}}{{dt}} = Q(x,y)\](1) where P(x, y), Q(x,y) are defined on the square S: [0, a]X[0, a], continuous and have continuous first partial derivative there, and satisfy the following relations P(0, y)=P(a,a-y), Q(0, y) = -Q(a, a-y), P(x, 0) = —P(a-x,a), Q(x,0)=Q(a-x, a), \[(x,y) \in [0,a] \times [0,a]\], A projective plane will be viewed as the square S in the (x,y) -plane, in which the points (0, y), (a, a -y) or (x, 0), (a—x, a) on opposite sides of the square are identified. Thus, under condition (2), (1) is a differential system defined on the pro?tective plane. On the projective plane, in addition to closed curves in the usual sense (we call it 0-closed curve), there are also closed curves consisting of several arcs in the square, such closed curves are illustrated in Fig. 1 (in which we use arrows and numbers to show that a closed curve can be constructed according to this direction and order). Hereafter, we will call the closed curve JT on the projective plane an closed curve, if .T consists of n arcs which do not meet each other in S, and each arc intersects the sides of 8 at its two end points only. If n is even (odd), then we also call P an even- closed curve (odd-closed curve) on the projective plane. Lemma 1. An even-closed curve on the projective plane divides the projective plane into two parts, but an odd-closed curve does not. So we can define in a certain sense the interior and exterior of an even-closed curve, while for an odd-closed curve, we can not define its interior and exterior. We call L left-right orientead family of directed arcs in S, if the origin of every arc in L is at the left hand side of its end. Similarly, we can define right-left, upper- lower and lower-upper oriented families.? Lemma 2. Let \[\Gamma \] be a closed orbit of system (1) on the projective plane consisting of only oriented ares of the same kind, then \[\Gamma \] contains two arcs atmost. On the projective plane, we can define limit cycle of the differential system (1) as in [1]. In particular, we can define stable and unstable cycles as well as semi-stable cycle for an even-closed orbit, but if an odd-closed orbit is a limit cycle, it must be a stable or unstable limit cycle. Let us extend system (1) to the square S*: [—a, a] x [—a, a] by defining first in [-a, a] X [0, a]: \[\begin{gathered} {P_1}(x,y) = \left\{ {\begin{array}{*{20}{c}} {p(x,y),(x,y) \in [0,a] \times [0,a]} \\ {p(a + x,a - y),(x,y) \in [ - a,0] \times [0,a];} \end{array}} \right. \hfill \ {Q_1}(x,y) = \left\{ {\begin{array}{*{20}{c}} {Q(x,y),(x,y) \in [0,a] \times [0,a]} \\ { - Q(a + x,a - y),(x,y) \in [ - a,0] \times [0,a];} \end{array}} \right. \hfill \ \hfill \\ \end{gathered} \] and then in s*: \[\begin{gathered} {P_2}(x,y) = \left\{ {\begin{array}{*{20}{c}} {{p_1}(x,y),(x,y) \in [ - a,a] \times [0,a]} \\ { - {p_1}( - x, - y),(x,y) \in [ - a,a] \times [ - a,0];} \end{array}} \right. \hfill \ {Q_1}(x,y) = \left\{ {\begin{array}{*{20}{c}} {{Q_1}(x,y),(x,y) \in [ - a,a] \times [0,a]} \\ { - {Q_2}( - x, - y),(x,y) \in [ - a,a] \times [ - a,0];} \end{array}} \right. \hfill \\ \end{gathered} \] It is easily seen that \[\frac{{dx}}{{dt}} = {P_2}(x,y),\frac{{dy}}{{dt}} = {Q_2}(x,y)\](6) is a C1 differential system on the torus formed by identifying opposite sides of S*. A closed orbit of (1) on the projective plane must correspond to some closed orbits of (6) onthe torus. We can prove now the following theorems. Theorem 1. Let \[m = {m_1} \cup {m_2}\] and \[n = {n_1} \cup {n_2}\] be two 2-closed curves in the projective plane, and n is in the interior of m. Suppose the domain Q bounded by m and n contains no stationary points., and trajectories of (1) crossing m all run from exterior to interior, while trajectories crossing n all run from interior to exterior. Then Q contains at least two 2-closed orbits F and L, where F is outer-stable, L is inner- stable limit cycle. Here r may coincide with L, if this takes place, then F=L is a stable limit cycle. Theorem 1.1. Let m be a 2-closed curve in the projective plane which consists of arcs joining opposite sides of the square 8. The interior of m contains no stationary points, and trajectories crossing m all run into the interior of m, then in the interior of m there is at least a closed orbit of (1) which is an outer stable 2-limit cycle or a stabe 1-limit cycle. Theorem 1.2. Let \[\Gamma \] be a 2-closed orbit of (1) in the projective plane which consists of arcs joining opposite sides of the square S. The interior of \[\Gamma \] contains no stationary points, then in the interior of \[\Gamma \] there is a 1-closed orbit. Theorem 3. Let Q be a domain in the projective plane, and B(x, y) be a single- valued continuous function in G which has continuous first partial derivatives. Suppose \[\frac{\partial }{{\partial x}}(BP) + \frac{\partial }{{\partial y}}(BQ)\] does not change its sign in G, and the set \[\frac{\partial }{{\partial x}}(BP) + \frac{\partial }{{\partial y}}(BQ) = 0\]contains no 2-dimensionaL domain, then the system (1) has no even-closed orbit whose interior is in G. In particular, if G is the whole projective plane, then the system (1) has no closed orbit at all. Theorem 4. Suppose F(x, y) = C is a family of curves, where F(x, y) is a single valued continuous function and has continuous first partial derivatives in the projective plane. \[P\frac{{\partial F}}{{\partial x}} + Q\frac{{\partial F}}{{\partial y}} does not change its sign in a domain G, and the subset of G in which \[P\frac{{\partial F}}{{\partial x}} + Q\frac{{\partial F}}{{\partial y}} = 0\] contains no closed orbits of (1), then system (1) II. Examples The systems in the following examples are all systems of differential equations (defined in the projective plane, and the projective plane is formed by the square [0, pi] x [0, pi]. A, B, C, D are constants other than zero. \[Ex.1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{dx}}{{dt}} = A\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2x + B\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} y,\frac{{dy}}{{dt}} = C\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 4y\] Using theorem 4 and theorem 1, we can prove that (8) has a 2-olosed orbit and an 1-closed orbit, which are stable and unstable limit cycles respectively. \[Ex.2{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{dx}}{{dt}} = B\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} y,\frac{{dy}}{{dt}} = C\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x + \frac{D}{2}\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2y\] system (9) has always a 1-closed orbit, if \[\left| {\frac{D}{{2C}}} \right| > 1\], then we can prove that (9) has no 2-closed orbit. \[Ex.3{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{dx}}{{dt}} = \frac{A}{2}\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2x,\frac{{dy}}{{dt}} = C\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x + \frac{D}{2}\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2y\] where D>-2C>0 This system has ho closed orbit, although the projective plane is a one-sided surface.     

Existence of T-solution for degenerated problem via Minty’s lemma

We study the existence result of solutions for the nonlinear degenerated elliptic problem of the form, -div（a（x, u,△↓u）） = F in Ω, where Ω is a bounded domain of R^N, N≥2, a ：Ω×R×R^N→R^N is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but they verify only the large monotonicity. The second term F belongs to W^-1,p′（Ω, w^＊）. The existence result is proved by using the L^1-version of Minty＇s lemma.     

UNIQUENESS, ERGODICITY AND UNIDIMENSIONALITY OF INVARIANT MEASURES UNDER A MARKOV OPERATOR

Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by
for any f = （f1,f2,… ,fN）, where （pij） is a N x N transition probability matrix and {wij } is an family of continuous transformations on X. The authors study the uniqueness, ergodicity and unidimensionality of T＊-invariant measures where T＊ is the adjoint operator of T.     

On the global existence and uniqueness of solutions to Prandtl’s system

In this paper, we consider the Prandtl system for the non-stationary boundary layer in the vicinity of a point where the outer flow has zero velocity. It is assumed that U（t, x, y） = x^mU1（t, x）, where 0 〈 x 〈 L and m 〉 1. We establish the global existence of the weak solution to this problem. Moreover the uniqueness of the weak solution is proved.