Wick-Itô Formula for Gaussian Processes

Abstract

A Wick-Itô formula for Gaussian processes is obtained. This is a change of variables formula, which is to Wick-Itô integrals what the usual Itô formula is to Itô integrals. The conditions are weak enough to allow processes with infinite quadratic variation. They are satisfied by fractional Brownian motion with parameter 1/4 < H < 1.     

Stability of Stochastic Delay Differential Equation with a Small Parameter

Abstract

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution x ^?(·) of the underlying delay equation for ? > 0 small enough. Perturbed Liapunov function method is used in the analysis.     

On the Doubly Singular Equation γ(u)t = Δpu

Abstract

We prove that local weak solutions of a nonlinear parabolic equation with a doubly singular character are locally continuous. One singularity occurs in the time derivative and is due to the presence of a maximal monotone graph; the other comes up in the principal part of the PDE, where the p-Laplace operator is considered. The paper extends to the singular case 1 < p < 2, the results obtained previously by the second author for the degenerate case p > 2; it completes a regularity theory for a type of PDEs that model phase transitions for a material obeying a nonlinear law of diffusion.     

Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion

Abstract

We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H > 1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration and the classical Itô stochastic calculus. The existence result is based on the Yamada–Watanabe theorem.     

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets ${K\subset H}$ of a Hilbert space H by the metric entropy of the set K where the covering numbers ${N(K, \varepsilon)}$ of K by ${\varepsilon}$ -balls of H satisfy the Lorentz condition $$ \int\limits_{0}^{\infty} \left(\log N(K,\varepsilon) \right)^{r/s}\, d\varepsilon^{s} < \infty $$ for some fixed ${0 < r, s \le \infty}$ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 < r < ∞, s = ∞. The integral here is an improper Stieltjes integral. Moreover, we obtain optimal estimates of Gelfand numbers of absolutely convex hulls if the metric entropy satisfies the entropy condition $$\sup_{\varepsilon >0 }\varepsilon \left(\log N(K,\varepsilon) \right)^{1/r}\left(\log(2+\log N(K,\varepsilon))\right)^\beta < \infty$$ for some fixed 0 < r < ∞, ?∞ < β < ∞. Using inequalities between Gelfand and entropy numbers we also get optimal estimates of the metric entropy of the absolutely convex hull cov(K). As an interesting feature of the estimates, a sudden jump of the asymptotic behavior of Gelfand numbers as well as of the metric entropy of absolutely convex hulls occurs for fixed s if the parameter r crosses the point r = 2 and, if r = 2 is fixed, if the parameter β crosses the point β = 1. The results established in Hilbert spaces extend and recover corresponding results of several authors.     

PRESENTATIONS FOR SOME MONOIDS OF PARTIAL TRANSFORMATIONS ON A FINITE CHAIN

ABSTRACT

In this paper we calculate presentations for some natural monoids of transformations on a chain X _n  = {1 < 2 <?s < n}. First we consider 𝒪𝒟 _n [𝒫𝒪𝒟 _n ], the monoid of all full [partial] transformations on X _n that preserve or reverse the order. Two other monoids of partial transformations on X _n we look at are 𝒫𝒪𝒫 _n and 𝒫𝒪? _n –-the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.     

Depth of Initial Ideals of Normal Edge Rings

Let G be a finite graph on the vertex set [d] = {1,…, d} with the edges e ₁,…, e _n and K[t] = K[t ₁,…, t _d ] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ring K[G] which is generated by those monomials t ^e  = t _i t _j such that e = {i, j} is an edge of G. Let K[x] = K[x ₁,…, x _n ] be the polynomial ring in n variables over K, and define the surjective homomorphism π: K[x] → K[G] by setting π(x _i ) = t ^{e _i} for i = 1,…, n. The toric ideal I _G of G is the kernel of π. It will be proved that, given integers f and d with 6 ≤ f ≤ d, there exists a finite connected nonbipartite graph G on [d] together with a reverse lexicographic order <_rev on K[x] and a lexicographic order <_lex on K[x] such that (i) K[G] is normal with Krull-dim K[G] = d, (ii) depth K[x]/in_<rev (I _G ) = f and K[x]/in_<lex (I _G ) is Cohen–Macaulay, where in_<rev (I _G ) (resp., in_<lex (I _G )) is the initial ideal of I _G with respect to <_rev (resp., <_lex) and where depth K[x]/in_<rev (I _G ) is the depth of K[x]/in_<rev (I _G ).     

Hierarchical pinning models, quadratic maps and quenched disorder

We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by Derrida et al. (J Stat Phys 66:1189–1213, 1992), which can be re-interpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {R _n } _{n=1,2, ...}, which in absence of disorder (i.e., when the initial condition is degenerate) reduces to a particular case of the well-known logistic map. The large-n limit of the sequence of random variables 2^?n log R _n , a non-random quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter α ? (0, 1), related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R ₀ is larger than a certain threshold value, and it is zero otherwise. It was conjectured in Derrida et al. (J Stat Phys 66:1189–1213, 1992) that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/2 < α < 1 (respectively, α < 1/2 or α = 1/2), in the sense that an arbitrarily small amount of randomness in the initial condition modifies the critical point with respect to that of the pure (i.e., non-disordered) model if α ≥ 1/2, but not if α <  1/2. Our main result is a proof of these conjectures for the case α ≠ 1/2. We emphasize that for α >  1/2 we find the correct scaling form (for weak disorder) of the critical point shift.     

Decay parameter and related properties of 2-type branching processes

We consider the decay parameter, invariant measures/vectors and quasi-stationary dis- tributions for 2-type Markov branching processes. Investigating such properties is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for 2-type Markov branching q-matrix are firstly investigated in detail. The exact value of the decay parameter λC of such model is given for the communicating class C = Z+2 \ 0. It is shown that this λC can be directly ...     

Global Continuity of the Integrated Density of States for Random Landau Hamiltonians

Abstract

We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent 0 < q < 1. The random Anderson-type potential is constructed with a nonnegative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly supported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179--209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is Hölder continuous as a function of the nonzero magnetic field strength.     

A least squares estimator for discretely observed Ornstein–Uhlenbeck processes driven by symmetric α-stable motions

We study the problem of parameter estimation for Ornstein–Uhlenbeck processes driven by symmetric α-stable motions, based on discrete observations. A least squares estimator is obtained by minimizing a contrast function based on the integral form of the process. Let h be the length of time interval between two consecutive observations. For both the case of fixed h and that of h → 0, consistencies and asymptotic distributions of the estimator are derived. Moreover, for both of the cases of h, the estimator has a higher order of convergence for the Ornstein–Uhlenbeck process driven by non-Gaussian α-stable motions (0 < α < 2) than for the process driven by the classical Gaussian case (α = 2).     

Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type

We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ_* > 0 such that for λ < λ_*, the problem has no positive solution; for λ = λ_*, it has at least one positive solution; and for λ > λ_*, it has at least two positive solutions.     

On Factorizations of Baer-Local Formations

Abstract

It is proved that if the product ? of formations 𝔐 and ? ≠ ? is either a Baer-local formation or an ω-local formation and if ? ? 𝔉 for some one-generated ω-local formation 𝔉, then 𝔐 is also an ω-local formation.     

Factoriality of Certain Threefolds Complete Intersection in P 5 with Ordinary Double Points

Abstract

Let V ? P ⁵ be a reduced and irreducible threefold of degree s, complete intersection on a smooth hypersurface of degree t, with s > t ² ? t. In this paper, we prove that if the singular locus of V consists of δ < 3s/8t ordinary double points, then any projective surface contained in V is a complete intersection on V. In particular, V is Q-factorial.     

Random Measures and Applications

Abstract

A mapping Z(·) from a δ-ring ?₀(?) into the vector space of random variables L ^p (P) is a vector-valued measure if it is σ-additive in the metric of its range. It is a vector measure if the range is a Banach space and a random measure if also its values are independent on disjoint sets. An important reason for this study is to construct integrals relative to such Zs, which typically do not have finite variation. For this, it is essential to find a controlling (σ-finite) measure for Z that is not available if 0 <p < 1, and here the random measure is taken to be p-stable and utilize properties of infinitely divisible distributions. In the case of p = 2, Z(·) induces a bimeasure, and if p > 2 is an integer it induces a polymeasure, either of which need not be (signed) measures on product spaces. Important applications lead to all these possibilities. In all those cases, a detailed analysis of vector-valued set functions is presented, with special focus for the cases of 0 <p < 1 and p = 2 where probability and Bochner's L ^{2, 2} boundedness plays a key role. Specialization if Z is stationary, harmonizable, and/or isotropic are discussed using the group structure of ? ⁿ , n ≥ 1, extending it for an lca group G. If Z is Banach valued or a quasi-martingale measure, methods of obtaining integrals are outlined in the last section, and open problems motivated by applications are pointed out at various places.     

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u _t  ? Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ?u/?n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = ?βu ^p , β ≥ 0, and g(x, u) = u ^q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T _b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T _b  ≤ T < τ. On the other hand, for the case f(x, u) = ?βu ^p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.     

Minimizing the L ∞ Norm of the Gradient with an Energy Constraint

ABSTRACT

The variational problem in L ^∞ considered is to minimize F(u) = ‖Du‖ _{L ^∞(Ω)} subject to ∈ t _Ω |Du|² dx ≤ E for given E > 0. It is proven that a constrained minimizer exists and satisfies an Aronsson-Euler equation in the viscosity sense which depends on a parameter Λ_∞ ≥ 0. This parameter splits Ω into two parts. In one part the minimizer satisfies the infinity laplace equation and in the remaining part the minimizer is the solution of the elasto-plastic torsion problem with constraint ‖Du‖ _{L ^∞}  ≤ Λ_∞.