The integral representation of the exponential product of stochastic semigroups

An integral representation is obtained for the exponential product of stochastic semigroups $$X_s^t \otimes Z_s^t = X_s^t + \mathop \smallint \limits_{s< u< t} X_u^t dV_u X_s^u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} X_{u_2 }^t dV_{u_2 } X_{u_1 }^{u_2 } dV_{u_1 } X_s^{u_1 } + \cdots ,$$ whereV _t is the generating process of the semigroupZ _s ^t and the integrals are understood in the sense of mean-square limits of the Riemann-Stieltjes sums. This representation is different from the traditional representation $$X_s^t \otimes Z_s^t = E + \mathop \smallint \limits_{s< u< t} dW_u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} dW_{u_2 } dW_{u_1 } + \cdots ,$$ in which the integration extends over the processW _t=Y_t+V_t that is the generating process of the exponential productX _s ^t ?Z _s ^t andY _t is the generator of the semigroupX _s ^t .     

High extrema of Gaussian chaos processes

Let ξ ( t)=(ξ ₁(t),…,ξ _d (t)) be a Gaussian stationary vector process. Let \(g:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}\) be a homogeneous function. We study probabilities of large extrema of the Gaussian chaos process g(ξ(t)). Important examples include \(g(\mathbf {\boldsymbol {\xi }}(t))={\prod }_{i=1}^{d}\xi _{i}(t)\) and \(g(\mathbf {\boldsymbol {\xi }}(t))={\sum }_{i=1}^{d}a_{i}{\xi _{i}^{2}}(t)\). We review existing results partially obtained in collaboration with E. Hashorva, D. Korshunov, and A. Zhdanov. We also present the principal methods of our investigations which are the Laplace asymptotic method and other asymptotic methods for probabilities of high excursions of Gaussian vector process’ trajectories.     

On Toeplitz Operators Between Fock Spaces

We study some mapping properties of Toeplitz operators T _μ associated with nonnegative Borel measures μ on the complex space ${\mathbb{C}^n}$ . In particular, we describe the bounded and compact properties of T _μ acting between Fock spaces in terms of the objects t-Berezin transforms, averaging functions, and averaging sequences of μ. We also obtain an asymptotic estimate for the norms of the operators. The results extend and complete a recent work of Z. Hu and X. Lv when both the smallest and the largest Banach–Fock spaces are taken into account.     

Restricted one way analysis of variance using the empirical likelihood ratio test

Empirical likelihood (EL) ratio tests are developed for testing for or against the hypothesis that k-population means μ₁,μ₂,…,μ_k are isotonic with respect to some quasi-order ? on {1,2,…,k}. The null asymptotic distributions are derived and are shown to be of chi-bar squared type. The asymptotic power of the proposed test for testing for equality of these means against the order restriction is derived under contiguous alternatives and a simulation study is carried out to investigate the finite sample behaviors of this test. In addition, an adjusted EL test is used to improve the small size performance of our test and an example is also discussed to illustrate the theoretical results.     

Dynamics of a Two-Prey One-Predator System in Random Environments

In this paper, we propose and investigate a stochastic two-prey one-predator model. Firstly, under some simple assumptions, we show that for each species x _i , i=1,2,3, there is a π _i which is represented by the coefficients of the model. If π _i <1, then x _i goes to extinction (i.e., lim _t→+∞ x _i (t)=0); if π _i >1, then x _i is stable in the mean (i.e., $\lim_{t\rightarrow+\infty}t^{-1} \int_{0}^{t}x_{i}(s)\,\mathrm {d}s=\mbox{a positive constant}$ ). Secondly, we prove that there is a stationary distribution to this model and it has the ergodic property. Thirdly, we establish the sufficient conditions for global asymptotic stability of the positive solution. Finally, we introduce some numerical simulations to illustrate the theoretical results.     

On the interrelation of almost sure invariance principles for certain stochastic adaptive algorithms and for partial sums of random variables

Since the novel work of Berkes and Philipp⁽³⁾ much effort has been focused on establishing almost sure invariance principles of the form (1) $$\left| {\sum\limits_{i = 1}^{|\_t\_|} {x_1 - X_t } } \right| \ll t^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - \gamma } $$ where {x _i ,i=1,2,3,...} is a sequence of random vectors and {X _t ,t>-0} is a Brownian motion. In this note, we show that if {A _k ,k=1,2,3,...} and {b _k ,k=1,2,3,...} are processes satisfying almost-sure bounds analogous to Eq. (1), (where {X _t ,t≥0} could be a more general Gauss-Markov process) then {h _k ,k=1,2,3...}, the solution of the stochastic approximation or adaptive filtering algorithm (2) $$h_{k + 1} = h_k + \frac{1}{k}(b_k - A_k h_k )for{\text{ }}k{\text{ = 1,2,3}}...$$ also satisfies and almost sure invariance principle of the same type.     

Sharp L 2 boundedness of the oscillatory hyper-Hilbert transform along curves

Consider the oscillatory hyper-Hilbert transform Hn,α,βf（x）=∫0^1 f（x-Г（t））e^it-βt^-1-α dt along the curve P（t） = （tp1, tP2,..., tpn）, where β 〉 α ≥ 0 and 0 〈 p1 〈 p2 〈 ... 〈 Pn. We prove that H n,α,β is bounded on L2 if and only if β ≥ （n ＋ 1）α. Our work extends and improves some known results.     

Self-interacting diffusions: a simulated annealing version

We study asymptotic properties of processes X, living in a Riemannian compact manifold M, solution of the stochastic differential equation (SDE) $$dX_t = dW_t(X_t) - \beta(t)\nabla V\mu_t(X_t)dt$$ with W a Brownian vector field, β(t) = alog(t + 1), $\mu_t = \frac{1}{t} \int_0^t \delta_{X_s}ds$ and $V\mu_t(x) = \frac{1}{t}\int_0^t V(x, X_s)ds$ , V being a smooth function. We show that the asymptotic behavior of μ _t can be described by a non-autonomous differential equation. This class of processes extends simulated annealing processes for which V(x, y) is only a function of x. In particular we study the case $M = {\mathbb{S}}^n$ , the n-dimensional sphere, and V(x, y) = ?cos(d(x, y)), where d(x, y) is the distance on ${\mathbb{S}}^n$ , which corresponds to a process attracted by its past trajectory. In this case, it is proved that μ _t converges almost surely towards a Dirac measure.     

Finite determinacy and Whitney equisingularity of map germs from {\mathbb{C}^n} to {\mathbb{C}^{2n-1}}

We show that a holomorphic map germ ${f : (\mathbb{C}^n,0)\to(\mathbb{C}^{2n-1},0)}$ is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n ≥ 3, we have that μ(D ²(f)) = 2μ(D ²(f)/S ₂)+C(f)?1, where D ²(f) is the lifting of the double point curve in ${(\mathbb{C}^n\times \mathbb{C}^n,0)}$ , μ(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f _t (x)) of f and show that if F is μ-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f _t is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.     

Strong Law of Large Numbers for a Class of Superdiffusions

In this paper we prove that, under certain conditions, a strong law of large numbers holds for a class of superdiffusions X corresponding to the evolution equation ? _t u _t =Lu _t +βu _t ?ψ(u _t ) on a domain of finite Lebesgue measure in ? ^d , where L is the generator of the underlying diffusion and the branching mechanism $\psi(x,\lambda)=\frac{1}{2}\alpha(x)\lambda^{2}+\int_{0}^{\infty}(e^{-\lambda r}-1+\lambda r)n(x, \mathrm{d}r)$ satisfies $\sup_{x\in D}\int_{0}^{\infty}(r\wedge r^{2}) n(x,\mathrm{d}r)<\infty$ .     

Precise Asymptotics for Lévy Processes

Let {X(t), t ≥ 0} be a Lévy process with EX(1) = 0 and EX ²(1) < ∞. In this paper, we shall give two precise asymptotic theorems for {X(t), t ≥ 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables. This work is supported by the National Natural Science Foundation (Grant No. 10671188) and Special Foundation of USTC     

On a class of threshold AR(k) processes

We consider the model $$Z_t = \sum\limits_{i = 1}^k {\phi (i,j)Z_{t - i} } + a_t (j)when\left[ {Z_{t - 1} ,Z_{t - 2,...,} Z_{t - k} } \right]^\prime \in R(j),$$ where {R(j);1?j? ?}is a partition of ? ^k , and for each 1?j??,{a _t (j);t? 0} are i.i.d. zero-mean random variables, having a strictly positive density. Sufficient conditions are obtained for this process to be transient. In addition, for a particular class of such models, necessary and sufficient conditions for ergodicity are obtained. Least-squares estimators of the parameters are obtained and are, under mild regularity conditions, shown to be strongly consistent and asymptotically normal.     

L 1 decay properties for a semilinear parabolic system

This article is concerned with the decay property in theL ¹ norm ast»∞ of the nonnegative solutions of the initial value problem in ? ⁿ $\left\{ {\begin{array}{*{20}c} {u_t = \Delta u + \mu |\nabla \upsilon |^q } \\ {\upsilon _t = \Delta \upsilon + \upsilon |\nabla \upsilon |^p } \\ \end{array} } \right.$ for different values of the parametersp, q≥1 and when μ, ν<0. If $pq > \frac{{\inf \left( {p,q} \right)}}{{n + 1}} + \left( {n + 2} \right)/\left( {n + 1} \right)$ then lim _t→∞∥u(t)+v(t)∥₁>0 and when $pq< \frac{{\inf \left( {p,q} \right)}}{{n + 1}} + \left( {n + 2} \right)/\left( {n + 1} \right)$ then lim _t→∞∥u(t)+v(t)∥₁>0.     

A generalized Cole-Hopf transformation for a two-dimensional burgers equation with a variable coefficient

The direct method is applied to the two dimensional Burgers equation with a variable coefficient (u _t + uu _x ? u _xx ) _x + s(t)u _yy = 0 is transformed into the Riccati equation $H' - \tfrac{1} {2}H^2 + \left( {\tfrac{\rho } {2} - 1} \right)H = 0$ via the ansatz $u\left( {x,y,t} \right) = \tfrac{1} {{\sqrt t }}H(\rho ) + \tfrac{y} {{2\sqrt t }}\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , provided that s(t) = t ^?3/2. Further, a generalized Cole-Hopf transformations $u\left( {x,y,t} \right) = \tfrac{y} {{2\sqrt t }} - \tfrac{2} {{\sqrt t }}\tfrac{{U_\rho (\rho ,r)}} {{U(\rho ,r)}}$ , $\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , r(t) = log t is derived to linearize (u _t + uu _x ? u _xx ) _x + t ^?3/2 u _yy to the parabolic equation $U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho$ .     

The mean value of the zeta-function on <Emphasis Type="Italic">σ</Emphasis>=1

Let R(T): = ò₁^T|z(1+it)|² dt - z(2)T+plogTR(T):= \int_{1}^{T}|\zeta (1+it)|^{2}\,\mathrm{d}t - \zeta (2)T+\pi\log T. We derive a precise explicit expression for R(t) which is used to derive asymptotic formulas for ò₁^TR(t) dt\int_{1}^{T}R(t)\,\mathrm{d}t and ò₁^TR²(t) dt\int_{1}^{T}R^{2}(t)\,\mathrm{d}t. These results improve on earlier upper bounds of Balasubramanian, Ramachandra and the author for the integrals in question.     

Karhunen–Loève expansion for a generalization of Wiener bridge

We derive a Karhunen–Loève expansion of the Gauss process \( {B}_t-g(t){\int}_0^1{g}^{\hbox{'}}(u)\mathrm{d}{B}_u,t\in \left[0,1\right] \), where (B_t)_t?∈?[0,?1] is a standardWiener process, and g?:?[0,?1]?→?? is a twice continuously differentiable function with g(0) = 0 and \( {\int}_0^1{\left(g\hbox{'}(u)\right)}^2\mathrm{d}u=1 \). This process is an important limit process in the theory of goodness-of-fit tests. We formulate two particular cases with the functions \( g(t)=\left(\sqrt{2}/\pi \right)\sin \left(\pi t\right),t\in \left[0,1\right] \), and g(t)?=?t, t?∈?[0,?1]. The latter corresponds to the Wiener bridge over [0, 1] from 0 to 0.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

Circulation of lifted processes on trivial principal bundles

A Brownian motion {x _t } _t?0 on a compact Riemannian manifold M with a drift vector field X can be lifted to a diffusion process $\left\{ {\tilde x_t } \right\}_{t \ge 0} $ on M × T^k corresponding to an ?^k valued smooth differential one-form A on M. The circulations (rotation numbers) of the lifted process $\left\{ {\tilde x_t } \right\}_{t \ge 0} $ around the k circles of T^k are studied. By choosing a certain ?^k -valued differential one-form A, these circulations give the hidden circulation of {x _t } _t?0 in M and the rotation numbers of {x _t } _t?0 around some closed curves in M which generalize the first homology group H₁(M,?) of M.     

Analyticity of the Planar Limit of a Matrix Model

Using Chebyshev polynomials combined with some mild combinatorics, we provide an alternative approach to the analytical and formal planar limits of a random matrix model with a 1-cut potential V. For potentials ${V(x)=x^{2}/2-\sum_{n\ge1}a_{n}x^{n}/n}$ , as a power series in all a _n , the formal Taylor expansion of the analytic planar limit is exactly the formal planar limit. In the case V is analytic in infinitely many variables {a _n } _n ≥ 1 (on the appropriate spaces), the planar limit is also an analytic function in infinitely many variables and we give quantitative versions of where this is defined. Particularly useful in enumerative combinatorics are the gradings of ${V,V_{t}(x)=x^{2}/2-\sum_{n\ge1}a_{n}t^{n/2}x^{n}/n}$ and ${V_{t}(x)=x^{2}/2-\sum_{n\ge3}a_{n}t^{n/2 -1}x^{n}/n}$ . The associated planar limits F(t) as functions of t count planar diagram sorted by the number of edges respectively faces. We point out a method of computing the asymptotic of the coefficients of F(t) using the combination of the wzb method and the resolution of singularities. This is illustrated in several computations revolving around the important extreme potential ${V_{t}(x)=x^{2}/2+\log(1-\sqrt{t}x)}$ and its variants. This particular example gives a quantitative and sharp answer to a conjecture of ’t Hooft’s, which states that if the potential is analytic, the planar limit is also analytic.     

Some remarks on spectral representation for symmetric stable processes

For the symmetric α-stable stochastic process X={X_t∶t∈T} with reproducing kernel space H(X) ? L_α constructed in § 1 we define the following parameters: $\alpha _0 = \sup {\mathbf{ }}\{ \beta \in (0.2]:{\mathbf{ }}\mathcal{H}\mathcal{X}$ embeds isometrically into some L^β}, containsl _β ⁿ 's uniformly}. In §2 we show that for α₀ > α the stochastic process X admits the representation $$X_t = \smallint Y_t (w){\mathbf{ }}Z_\alpha (dw),{\mathbf{ }}t \in T,$$ where {Y_t∶t∈T} itself is a symmetric stable process and Z_α is a symmetric α-stable independently scattered random measure. We show also how some properties of the stochastic process {X_t∶t∈T} depend on the corresponding properties of the process {Y_t∶t∈T}.