Essential self-adjointness of Dirichlet operators on a path space with Gibbs measures via an SPDE approach

The main objective of this paper is to prove the essential self-adjointness of Dirichlet operators in L²(μ) where μ is a Gibbs measure on an infinite volume path space C(R,Rd). This operator can be regarded as a perturbation of the Ornstein-Uhlenbeck operator by a nonlinearity and corresponds to a parabolic stochastic partial differential equation (= SPDE, in abbreviation) on R. In view of quantum field theory, the solution of this SPDE is called a P₁(?)-time evolution.     

Fractional Brownian motion and packing dimension

LetX(t) (tR ^N ) be a fractional Brownian motion of index inR ^d . For any compact setER ^N , we compute the packing dimension ofX(E).Partially supported by an NSF grant.     

An upper bound for the cardinality of ans-distance subset in real euclidean space,II

It is shown that ifX is ans-distance subset inR ^d , then |X|≦( _s ^d+s ). Supported in part by NSF grant MCS7903128 A01. Supported in part by NSF grant MCS.     

The occupation measure of super-Brownian motion conditioned to nonextinction

We study super-Brownian motion inR ^d starting from a nontrivial finite measure and conditioned to nonextinction as defined by Evans. If (Y _t ) _t0 denotes this process, we provide a new approach to the immortal particle representation of (Y _t ) _t0 . We then show that the measureZ onR ^d defined byZ(B)= _o ¹ Y _t (B) dt is almost surely finite on compact sets whend5 and almost surely infinite on every ball whend4.     

Quadratic forms which have only large zeros

We give lower bounds for zeros of quadratic forms. For example, givenn2d>0 there are infinitely many nonsingular quadratic formsF with integral coefficients inn variables which vanish on ad-dimensional rational subspace such that anyn linearly independent integral zerosx ₁,...,x _n satisfy |x ₁|...|x _n |F(ⁿ²/^2d)–d, whereF is the maximum modulus of the coefficients ofF. This complements a recent result of the authors in the opposite direction and shows that it is best possible.Written whileSchmidt had a Senior Scientist Award from the Alexander von Humboldt-Stiftung in the Federal Republic of Germany.     

On the existence of solutions of stochastic differential equations with singular drifts

Summary In this paper we prove the existence of solutions for a stochastic differential equation inR ^d, when the drift and the diffusion term are allowed to depend on a specific way on the local time of thedth coordinate of the process to be constructed. The methods of our construction are of purely probabilistic nature.     

Efficient construction of a small hitting set for combinatorial rectangles in high dimension

We describe a deterministic algorithm which, on input integersd, m and real number (0,1), produces a subset S of [m] ^d ={1,2,3,...,m} ^d that hits every combinatorial rectangle in [m] ^d of volume at least , i.e., every subset of [m] ^d the formR ₁×R ₂×...×R _d of size at least m ^d . The cardinality of S is polynomial inm(logd)/, and the time to construct it is polynomial inmd/. The construction of such sets has applications in derandomization methods based on small sample spaces for general multivalued random variables.A preliminary version of this paper appeared in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993.Research partially done while visiting the International Computer Science Institute. Research supported in part by a grant from the Israel-USA Binational Science Foundation.A large portion of this research was done while still at the International Computer Science Institute in Berkeley, California. Research supported in part by National Science Foundation operating grants CCR-9304722 and NCR-9416101, and United States-Israel Binational Science Foundation grant No. 92-00226.Supported in part by NSF under grants CCR-8911388 and CCR-9215293 and by AFOSR grants AFOSR-89-0512 AFOSR-90-0008, and by DIMACS, which is supported by NSF grant STC-91-19999 and by the New Jersey Commission on Science and Technology. Research partially done while visiting the International Computer Science Institute.Partially supported by NSF NYI Grant No. CCR-9457799. Most of this research was done while the author was at MIT, partially supported by an NSF Postdoctoral Fellowship. Research partially done while visiting the International Computer Science Institute.     

Existence,uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces

Summary A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroupR _t are obtained. We show thatR _t is a compactC ₀-semigroup in all Sobolev spacesW ^n,p which are built on its invariant measure . Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations inL ^p() spaces and spacesW ^1,p . As a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to . It is shown also that the density of this measure with respect to is inL ^p() for allp1.This work was done during the first author's stay at UNSW supported by ARC Grant 150.346 and the second author's stay at ód University supported by KBN Grant 2.1020.91.01     

Triangulating point sets in space

A setP ofn points inR ^d is called simplicial if it has dimensiond and contains exactlyd + 1 extreme points. We show that whenP containsn interior points, there is always one point, called a splitter, that partitionsP intod + 1 simplices, none of which contain more thandn/(d + 1) points. A splitter can be found inO(d ⁴ +nd ²) time. Using this result, we give anO(nd ⁴ log_1+1/d n) algorithm for triangulating simplicial point sets that are in general position. InR ³ we give anO(n logn +k) algorithm for triangulating arbitrary point sets, wherek is the number of simplices produced. We exhibit sets of 2n + 1 points inR ³ for which the number of simplices produced may vary between (n – 1)² + 1 and 2n – 2. We also exhibit point sets for which every triangulation contains a quadratic number of simplices.Research supported by the Natural Science and Engineering Research Council grant A3013 and the F.C.A.R. grant EQ1678.     

Independent collections of translates of boxes and a conjecture due to Grünbaum

A collection ofn setsA ₁, ...,A _n is said to beindependent provided every set of the formX ₁ ... X _n is nonempty, where eachX _i is eitherA _i orA _i ^c . We give a simple characterization for when translates of a given box form an independent set inR ^d. We use this to show that the largest number of such boxes forming an independent set inR ^d is given by 3d/2 ford2. This settles in the negative a conjecture of Grünbaum (1975), which states that the maximum size of an independent collection of sets homothetic to a fixed convex setC inR ^d isd+1. It also shows that the bound of 2d of Rényiet al. (1951) for the maximum number of boxes (not necessarily translates of a given one) with sides parallel to the coordinate axes in an independent collection inR ^d can be improved for these special collections.Daniel Q. Naiman was supported by National Science Foundation Grant No. DMS-9103126. Henry P. Wynn was supported by the Science and Engineering Research Council, UK.     

Abstract,weighted, and multidimensional Adamjan-Arov-Krein theorems,and the singular numbers of Sarason commutants

The classical Adamjan-Arov-Krein (A-A-K) theorem relating the singular numbers of Hankel operators to best approximations of their symbols by rational functions is given an abstract version. This provides results for Hankel operators acting in weightedH ²(T; ), as well as inH ²(T ^d ), and an A-A-K type extension of Sarason's interpolation theorem. In particular, it is shown that all compact Hankel operators inH ²(T ^d ) are zero.Author partially supported by NSF grant DMS89-11717.     

Problems for elliptic equations with operator-boundary conditions

We consider, in a Hilbert space H, a problem on [0,1] for a second order elliptic operator-differential equation with operator-boundary conditions. We also consider second order elliptic differential equations with operatorboundary conditions in cylindrical domains in the case when operator-boundary conditions contain integral terms over the whole domain. In this case, the proof of the density of the domain of definition of operators in a space is difficult. When boundary conditions are local, this fact is a simple corollary of the density ofC ₀ () inL _p ().     

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR ⁿ andr(X, d ₂) the rendezvous number ofX, whered ₂ denotes the Euclidean distance inR ⁿ . (The rendezvous numberr(X, d ₂) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x ₁,x ₂,...,x _n inX, there exists somex inX such that .) Then there exists some regular Borel probability measure μ₀ onX such that the value of ∫ _X d ₂(x, y)dμ₀ (y) is independent of the choicex inX, if and only ifr(X, d ₂) = sup_μ ∫ _X ∫ _X d ₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.     

Can you cover your shadows?

A subsetX of thed-dimensional Euclidean space ^d can cover its shadows inR ^d , if every orthogonal projection ofX onto a (d–1)-dimensional linear subspace of ^d is contained in some congruent copy ofX. Whereas every two-dimensional convex discC R ^d has this property, no (d–1)-polytope does, provided thatd>-4.     

Weingarten surfaces and nonlinear partial differential equations

The sine-Gordon equation has been known for a long time as the equation satisfied by the angle between the two asymptotic lines on a surface inR ³ with constant Gauss curvature –1. In this paper, we consider the following question: Does any other soliton equation have a similar geometric interpretation? A method for finding all the equations that have such an interpretation using Weingarten surfaces inR ³ is given. It is proved that the sine-Gordon equation is the only partial differential equation describing a class of Weingarten surfaces inR ³ and having a geometricso(3)-scattering system. Moreover, it is shown that the elliptic Liouville equation and the elliptic sinh-Gordon equation are the only partial differential equations describing classes of Weingarten surfaces inR ³ and having geometricso(3,C)-scattering systems.     

Stochastic differential equations for multi-dimensional domain with reflecting boundary

Summary In this paper we prove that there exists a unique solution of the Skorohod equation for a domain inR ^d with a reflecting boundary condition. We remove the admissibility condition of the domain which is assumed in the work [4] of Lions and Sznitman. We first consider a deterministic case and then discuss a stochastic case.     

Gitterpunkte unter algebraischen Kurven

Lattice points below Algebraic Curves. A generalization of the classical circle problem is treated. An asymptotic formula for the number of lattice points in a region whose boundary is an algebraic curve is obtained. This gives a mean value formula for the number of representations of the positive integers in the formn=g(x,y), whereg is a polynomial with coefficients 0 and leading terma _d0x^d*a_0dy^d. The caseg(x,y)=p₁(x)+p₂(y) was considered inKuba andNowak [4], andKuba [5]. The discrete Hardy-Littlewood method is used along with Rouché's theorem.

     

A characterization ofh-Brownian motion by its exit distributions

Summary LetX ^h be anh-Brownian motion in the unit ballDR ^d withh harmonic, such that the representing measure ofh is not singular with respect to the surface measure on D. IfY is a continuous strong Markov process inD with the same killing distributions asX ^h , thenY is a time change ofX ^h . Similar results hold in simply connected domains inC provided with either the Martin or the Euclidean boundary.     

An upper bound for the cardinality of ans-distance subset in real euclidean space

IfX is ans-distance subset inR ^d , then |X|<( _s ^d+s )+( _s-1 ^d+s-1 . Supported in part by NSF grant MCS—7903128 (OSURF 711977).     

Carleman Estimates and Controllability of Linear Stochastic Heat Equations

   Abstract. This work is concerned with Carleman inequalities and controllability properties for the following stochastic linear heat equation (with Dirichlet boundary conditions in the bounded domain D ⊂ R ^d and multiplicative noise):