Transfer of absolute continuity by a flow generated by a stochastic equation with reflection

Let ϕ_t(x), x ∈ ℝ₊ be a value taken at time t ≥ 0 by a solution of a stochastic equation with normal reflection from a hyperplane starting at initial time from x. We characterize the absolutely continuous (with respect to Lebesgue measure) component and the singular component of a stochastic measure-valued process μ_t = μ ○ ϕ _t ⁻¹ that is the image of a certain absolutely continuous measure μ under random mapping ϕ_t(·). We prove that the restriction of the Hausdorff measure H ^d−1 to the support of the singular component is σ-finite and give sufficient conditions guaranteeing that the singular component is absolutely continuous with respect to H ^d−1. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1663–1673, December, 2006.     

Almost sure asymptotics for the continuous parabolic Anderson model

We consider the parabolic Anderson problem ∂ _t u = κΔu + ξ(x)u on ℝ₊×ℝ ^d with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t→∞. Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000     

Maximal subspaces for solutions of the second order abstract cauchy problem

For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.     

A simple formula for an analogue of conditional wiener integrals and its applications II

Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w _ϕ of Wiener measure and for a partition 0 = t ₀ < t ₁ < ... < t _n < t _n+1 = T of [0, T], let X _n : C[0, T] → ℝ ⁿ⁺¹ and X ⁿ⁺¹: C[0, T] → ℝ ⁿ⁺² be given by X _n (x) = (x(t ₀), x(t ₁), ..., x(t _n )) and X _n+1(x) = (x(t ₀), x(t ₁), ..., x(t _n+1)), respectively. In this paper, using a simple formula for the conditional w _ϕ-integral of functions on C[0, T] with the conditioning function X _n+1, we derive a simple formula for the conditional w _ϕ-integral of the functions with the conditioning function X _n . As applications of the formula with the function X _n , we evaluate the conditional w _ϕ-integral of the functions of the form F _m (x) = ∫₀ ^T (x(t)) ^m for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X _n , we evaluate the conditional w _ϕ-integral of the functions in a Banach algebra which is an analogue of the Cameron and Storvick’s Banach algebra . Finally, we derive the conditional analytic Feynman w _ϕ-integrals of the functions in .      

Correlation structure of intermittency in the parabolic Anderson model

Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤ^d, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant, and ξ = {ξ(x): x∈ℤ ^d } is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate, then the solution u is asymptotically intermittent. In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e ^{s /θ}] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result is that, for fixed x, y∈ℤ ^d and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w _ρ∥⁻² _ℓ2Σ_{z ∈ℤd} w _ρ(x+z)w _ρ(y+z). In this expression, ρ = θ/κ while w _ρ:ℤ^d→ℝ⁺ is given by w _ρ = (v _ρ)^{⊗ d} with v _ρ: ℤ→ℝ⁺ the unique centered ground state (i.e., the solution in ℓ²(ℤ) with minimal l ²-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞). empty It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation coefficient of u(x, t) and u(y, t) converges to δ _{x, y} (resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure. Received: 5 March 1997 / Revised version: 21 September 1998     

Reciprocal polynomials with all zeros on the unit circle

Let f(x)=a _d x ^d +a _d−1 x ^d−1+⋅⋅⋅+a ₀∈ℝ[x] be a reciprocal polynomial of degree d. We prove that if the coefficient vector (a _d ,a _d−1,…,a ₀) or (a _d−1,a _d−2,…,a ₁) is close enough, in the l ¹-distance, to the constant vector (b,b,…,b)∈ℝ ^d+1 or ℝ ^d−1, then all of its zeros have moduli 1.     

Large deviations and concentration properties for ∇ϕ interface models

We consider the massless field with zero boundary conditions outside D _N ≡D∩ (ℤ ^d /N) (N∈ℤ⁺), D a suitable subset of ℝ ^d , i.e. the continuous spin Gibbs measure ℙ _N on ℝ ^ℤd/N with Hamiltonian given by H(ϕ) = ∑ _{x,y:|x−y|=1} V(ϕ(x) −ϕ(y)) and ϕ(x) = 0 for x∈D _N ^C . The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base D _N . Due to the choice of scaling of the base, we scale the height with the same factor by setting ξ _N = ϕ/N. We study various concentration and relaxation properties of the family of random surfaces {ξ _N } and of the induced family of gradient fields ∇ ^N ξ _N as the discretization step 1/N tends to zero (N→∞). In particular, we prove a large deviation principle for {ξ _N } and show that the corresponding rate function is given by ∫ _D σ(∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ℙ _N under the volume constraint, i.e. the constraint that (1/N ^d ) ∑ _x∈DN ξ _N (x) stays in a neighborhood of a fixed volume v > 0, and the hard–wall constraint, i.e. ξ _N (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξ _N of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇ ^N ξ _N (·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ? _p estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer–Sj?strand [31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields. Received: 3 March 1999 / Revised version: 9 August 1999 / Published online: 30 March 2000     

On a Function Related to Multinomial Coefficients (I)

Let F _p,t (n) denote the number of the coefficients of (x ₁+1x ₂+...+x _t ) ^j , 0 ≤j≤n− 1, which are not divisible by the prime p. Define G _p,t (n) = F _p,t /n ^θ and β(p,t) = lim infF _p,t )(n)/n ^θ, where θ = (log)/(log p). In this paper, we mainly prove that G _p,t can be extended to a continuous function on ℝ⁺, and the function G _p,t is nowhere monotonic. Both the set of differential points of the function G _p,t and the set of non-differential points of the function G _p,t are dense in ℝ⁺. Received February 18, 2000, Accepted December 7, 2000     

Gibbsianness of fermion random point fields

We consider fermion (or determinantal) random point fields on Euclidean space ℝ^d. Given a bounded, translation invariant, and positive definite integral operator J on L²(ℝ^d), we introduce a determinantal interaction for a system of particles moving on ℝ^d as follows: the n points located at x₁,· · ·,x_n ∈ ℝ^d have the potential energy given by where j(x−y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d≥2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I+J)⁻¹ is a Gibbs measure admitted to the specification.     

On the behavior of solutions of a semilinear parabolic or elliptic equation satisfying a nonlinear boundary condition in a cylindrical domain

One considers a semilinear parabolic equation u _t = Lu − a(x)f(u) or an elliptic equation u _tt + Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ₊ with the nonlinear boundary condition , where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝⁿ; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems for t → ∞. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007.     

Large antipodal families

A family {A _i | i ∈ I} of sets in ℝ ^d is antipodal if for any distinct i, j ∈ I and any p ∈ A _i , q ∈ A _j , there is a linear functional ϕ:ℝ ^d → ℝ such that ϕ(p) ≠ ϕ(q) and ϕ(p) ≤ ϕ(r) ≤ ϕ(q) for all r ∈ ∪ _i∈I A _i . We study the existence of antipodal families of large finite or infinite sets in ℝ³. The research was supported by the Hungarian-South African Intergovernmental Scientific and Technological Cooperation Programme, NKTH Grant no. ZA-21/2006 and South African National Research Foundation Grant no. UID 61853, as well as Hungarian National Foundation for Scientific Research Grants no. NK 67867, no. T47102, and no. K72537.     

Three-Dimensional Polyhedra Can Be Described by Three Polynomial Inequalities

Bosse et al. conjectured that for every natural number d≥2 and every d-dimensional polytope P in ℝ ^d , there exist d polynomials p ₁(x),…,p _d (x) satisfying P={x∈ℝ ^d :p ₁(x)≥0,…,p _d (x)≥0}. We show that every three-dimensional polyhedron can be described by three polynomial inequalities, which confirms the conjecture for the case d=3 but also provides an analogous statement for the case of unbounded polyhedra. The proof of our result is constructive. Work supported by the German Research Foundation within the Research Unit 468 “Methods from Discrete Mathematics for the Synthesis and Control of Chemical Processes”.     

Infima of superharmonic functions

Let Ω be a Greenian domain in ℝ ^d , d≥2, or—more generally—let Ω be a connected -Brelot space satisfying axiom D, and let u be a numerical function on Ω, , which is locally bounded from below. A short proof yields the following result: The function u is the infimum of its superharmonic majorants if and only if each set {x:u(x)>t}, t∈ℝ, differs from an analytic set only by a polar set and , whenever V is a relatively compact open set in Ω and x∈V.     

On the stability of solutions to quadratic programming problems

We consider the parametric programming problem (Q _p ) of minimizing the quadratic function f(x,p):=x ^T Ax+b ^T x subject to the constraint Cx≤d, where x∈ℝ ⁿ , A∈ℝ ^n×n , b∈ℝ ⁿ , C∈ℝ ^m×n , d∈ℝ ^m , and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q _p ) are denoted by M(p) and M ^loc (p), respectively. It is proved that if the point-to-set mapping M ^loc (·) is lower semicontinuous at p then M ^loc (p) is a nonempty set which consists of at most ? _m,n points, where ? _m,n = is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones. Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000     

Pseudodifferential Operators on Local Hardy Spaces

In this work, we study the continuity of pseudodifferential operators on local Hardy spaces h ^p (ℝ ⁿ ) and generalize the results due to Goldberg and Taylor by showing that operators with symbols in S _1,δ ⁰(ℝ ⁿ ), 0≤δ<1, and in some subclasses of S _1,1⁰(ℝ ⁿ ) are bounded on h ^p (ℝ ⁿ ) (0<p≤1). As an application, we study the local solvability of the planar vector field L=∂ _t +ib(x,t)∂ _x , b(x,t)≥0, in spaces of mixed norm involving Hardy spaces. Work supported in part by CNPq, FINEP, and FAPESP.     

Formula for a solution ofu t +H(u,Du) =g

We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equationu _t +H(u,Du) =g in ℝ ⁿ x ℝ₊ withu(x, 0) =u ₀(x). The HamiltonianH(s,p) is assumed to be convex and positively homogeneous of degree one inp for eachs in ℝ. IfH is non increasing ins, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions.     

The Markov chain associated to a Pick function

To each function ϕ˜(ω) mapping the upper complex half plane ?⁺ into itself such that the coefficient of ω in the Nevanlinna integral representation is one, we associate the kernel p(y, dx) of a Markov chain on ℝ by

A local limit theorem for random walks conditioned to stay positive

We consider a real random walk S_n=X₁+...+X_n attracted (without centering) to the normal law: this means that for a suitable norming sequence a_n we have the weak convergence S_n/a_n⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S₁>0,...,S_n>0) and let S_n⁺ denote the random variable S_n conditioned on : it is known that S_n⁺/a_n ↠ ϕ⁺(x) dx, where ϕ⁺(x):=x exp (−x²/2)1_(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X₁ has an absolutely continuous law: in this case the uniform convergence of the density of S_n⁺/a_n towards ϕ⁺(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.     

Multiple points of trajectories of multiparameter fractional Brownian motion

Consider 0<α<1 and the Gaussian process Y(t) on ℝ ^N with covariance E(Y(s)Y(t))=|t|^2α+|s|^2α−|t−s|^2α, where |t| is the Euclidean norm of t. Consider independent copies X ¹,…,X ^d of Y and␣the process X(t)=(X ¹(t),…,X ^d (t)) valued in ℝ ^d . When kN≤␣(k−1)αd, we show that the trajectories of X do not have k-multiple points. If N<αd and kN>(k−1)αd, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^{k N /α−( k −1) d} (loglog(1/ɛ)) ^k . If N=αd, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^d (log(1/ɛ) logloglog 1/ɛ) ^k . (This includes the case k=1.) Received: 20 May 1997 / Revised version: 15 May 1998