On G 2 Continuous Spline Interpolation of Curves in ℝ d

In this paper the problem of G ² continuous interpolation of curves in ^d by polynomial splines of degree n is studied. The interpolation of the data points and two tangent directions at the boundary is considered. The case n = r + 2 = d, where r is the number of interior points interpolated by each segment of the spline curve, is studied in detail. It is shown that the problem is uniquely solvable asymptotically, e., when the data points are sampled regularly and sufficiently dense, and lie on a regular, convex parametric curve in ^d . In this case the optimal approximation order is also determined.     

On Positive Random Objects

The idea of defining the expectation of a random variable as its integral with respect to a probability measure is extended to certain lattice-valued random objects and basic results of integration theory are generalized. Conditional expectation is defined and its properties are developed. Lattice valued martingales are also studied and convergence of sub- and supermartingales and the Optional Sampling Theorem are proved. A martingale proof of the Strong Law of Large Numbers is given. An extension of the lattice is also studied. Studies of some applications, such as on random compact convex sets in R ⁿ and on random positive upper semicontinuous functions, are carried out, where the generalized integral is compared with the classical definition. The results are also extended to the case where the probability measure is replaced by a -finite measure.     

Stochastic two-target pursuit-evasion differential games in the plane

A stochastic version of a two-target homicidal chauffeur, pursuit-evasion differential game (using polar coordinates) is considered. This is used to model a dogfight between a very agile playerQ and a less maneuverable playerP. First, the case where both players have complete observation of the state of the game is considered. A numerical study is conducted, by solving numerically a nonlinear partial differential equation on a torus in ², to investigate the role of the parameters of speed, maneuverability, and performance of the weapon systems, in the encounter. Second, the model is extended to include the case where playerP is jamming playerQ's measurements of , where denotes the bearing ofQ fromP. A numerical study is conducted, by solving numerically a nonlinear partial differential equation on a generalized torus in ³, to investigate the role of the jamming parameter on the outcome of the combat.     

Excursion theory for rotation invariant Markov processes

Summary Let (X _t,P ^x) be a rotation invariant (RI) strong Markov process onR ^d{0} having a skew product representation [|X _t |, ], where ( _t ) is a time homogeneous, RI strong Markov process onS ^d–1, |X _t|, and _t are independent underP ^x andA _t is a continuous additive functional of |X _t|. We characterize the rotation invariant extensions of (X _t,P ^x) toR ^d. Two examples are given: the diffusion case, where especially the Walsh's Brownian motion (Brownian hedgehog) is considered, and the case where (X _t,P ^x) is self-similar.     

Diffusion Approximation in Overloaded Switching Queueing Models

The asymptotic behavior of a queueing process in overloaded state-dependent queueing models (systems and networks) of a switching structure is investigated. A new approach to study fluid and diffusion approximation type theorems (without reflection) in transient and quasi-stationary regimes is suggested. The approach is based on functional limit theorems of averaging principle and diffusion approximation types for so-called Switching processes. Some classes of state-dependent Markov and non-Markov overloaded queueing systems and networks with different types of calls, batch arrival and service, unreliable servers, networks (M _SM,Q /M _SM,Q /1/) ^r switched by a semi-Markov environment and state-dependent polling systems are considered.     

Functional inequalities on arbitrary Riemannian manifolds

For any connected (not necessarily complete) Riemannian manifold, we construct a probability measure of type , where dx is the Riemannian volume measure and V is a function C^∞-smooth outside a closed set of zero volume, satisfying Poincaré–Sobolev type functional inequalities. In particular, V is C^∞-smooth on the whole manifold when the Poincaré and the super-Poincaré inequalities are considered. The Sobolev inequality for infinite measures are also studied.     

On reflecting diffusion processes and Skorokhod decompositions

Summary LetG be ad-dimensional bounded Euclidean domain, H¹ (G) the set off in L²(G) such that f (defined in the distribution sense) is in L²(G). Reflecting diffusion processes associated with the Dirichlet spaces (H¹(G), ) on L²(G, dx) are considered in this paper, where A=(a^ij is a symmetric, bounded, uniformly ellipticd×d matrix-valued function such thata ^ij H¹(G) for eachi,j, and H¹(G) is a positive bounded function onG which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with (H¹(G), ) having starting points inG under a mild condition which is satisfied when G has finite (d–1)-dimensional lower Minkowski content.     

A ratio ergodic theorem for increasing additive functionals

Summary Let B be a 1-dimensional Brownian motion. In this paper ratios of the form A ⁺ (t)/A ^- (t), where A ⁺ is the (0, )-occupation time functional of B and A ^-is a local time integral of an infinite (but locally finite) measure m with support in (-, 0), are studied. Conditions on m are given which ensure that such a ratio will be unbounded a.s. (or go to zero a.s.) as t».The work was supported in part by a grant from the National Science Foundation.     

A semi-discrete convergent scheme for a quasilinear hyperbolic equation

We establish here the convergence (thereby proving the existence) of a semi-discrete scheme for the quasilinear hyperbolic equation where x ∈ Rⁿ, t ∈ [0,T], and ? ∈ L^∞ (Rⁿ). It is well known that the above problem does not necessarily have global classical solutions and the usual concepts of weak solution. do not lead to a unique solution The existence of a unique solution to the above problem in a suitable sense was established in [3], where a parabolic problem obtained by introducing the term ??Δu was studied and then the behavior as ? → 0 was discussed. A difference scheme approach to a problem of the above type where ?_i does not depend on x and t and Ψ does not depend on u was also studied in [2]. The aim of this paper is to present a proof for the case when ? depends on x, Ψ depends on u, and the technical complications in this case are nontrivial. The discussions in this paper my be considered as continuation of the ideas in the above papers.     

Diffusion approximation forGI/G/1 controlled queues

A queueing model is considered in which a controller can increase the service rate. There is a holding cost represented by functionh and the service cost proportional to the increased rate with coefficientl. The objective is to minimize the total expected discounted cost.Whenh andl are small and the system operates in heavy traffic, the control problem can be approximated by a singular stochastic control problem for the Brownian motion, namely, the so-called reflected follower problem. The optimal policy in this problem is characterized by a single numberz ^* so that the optimal process is a reflected diffusion in [0,z ^*]. To obtainz ^* one needs to solve a free boundary problem for the second order ordinary differential equation. For the original problem the policy which increases to the maximum the service rate when the normalized queue-length exceedsz ^* is approximately optimal.     

Time-Space Harmonic Polynomials for Continuous-Time Processes and an Extension

A time-space harmonic polynomial for a stochastic process M=(M _t) is a polynomial P in two variables such that P(t, M _t) is a martingale. In this paper, we investigate conditions for the existence of such polynomials of each degree in the second, space, argument. We also describe various properties a sequence of time-space harmonic polynomials may possess and the interaction of these properties with distributional properties of the underlying process. Thus, continuous-time conterparts to the results of Goswami and Sengupta,⁽²⁾ where the analoguous problem in discrete time was considered, are derived. A few additional properties are also considered. The resulting properties of the process include independent increments, stationary independent increments and semi-stability. Finally, a generalization to a measure proposed by Hochberg⁽³⁾ on path space is obtained.     

A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

Summary A system ofN particles inR ^d with mean field interaction and diffusion is considered. Assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by Brownian sheets (microscopic equation), where all coefficients depend on the position of the particles and on the empirical mass distribution process. This empirical mass distribution process satisfies a quasilinear stochastic partial differential equation (SPDE). This SPDE (mezoscopic equation) is solved for general measure valued initial conditions by extending the empirical mass distribution process from point measure valued initial conditions with total mass conservation. Starting with measures with densities inL ₂(R ^d ,dr), wheredr is the Lebesgue measure, the solution will have densities inL ₂(R ^d ,dr) and strong uniqueness (in the Itô sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDE's.This research was supported by NSF grant DMS92-11438 and ONR grant N00014-91J-1386     

Diffusion processes on fractal fields: heat kernel estimates and large deviations

A fractal field is a collection of fractals with, in general, different Hausdorff dimensions, embedded in ². We will construct diffusion processes on such fields which behave as Brownian motion in ² outside the fractals and as the appropriate fractal diffusion within each fractal component of the field. We will discuss the properties of the diffusion process in the case where the fractal components tile ². By working in a suitable shortest path metric we will establish heat kernel bounds and large deviation estimates which determine the trajectories followed by the diffusion over short times.     

On recovering density in a plane domain from incomplete spectral data

The inverse spectral problem of recovering density in a bounded domain is considered. The first N eigenvalues and traces on the boundary of the normal derivatives of the eigenfunctions of the Dirichlet problem are considered as the input data. It is shown that the error of the density recovery does not exceed cln^– N, where c and are certain positive constants.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 218–224.     

Persistence of critical multitype particle and measure branching processes

Summary We consider a class of systems of particles ofk types inR ^d undergoing spatial diffusion and critical multitype branching, where the diffusions, the particle lifetimes and the branching laws depend on the types. We prove persistence criteria for such systems and for their corresponding high density limits known as multitype Dawson-Watanabe processes. The main tool is a representation of the Palm distributions for a general class of inhomogeneous critical branching particle systems, constructed by means of a backward tree.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

Volume,diameter and the minimal mass of a stationary 1-cycle

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold Mⁿ is bounded from above by (n + 2)!d/4, where d is the diameter of a manifold Mⁿ. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold Mⁿ is bounded from above by where where is the filling radius of a manifold, and where is its volume.     

Generalized elements of best approximation

It is proved that generalized elements of best approximation (GEBA's) are approximated by minimizing nets in the(X^**, X^*) topology. Criteria for GEBA's are established. GEBA's are considered in ^* where is a 1-normalizing subspace of X^*.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 513–523, April, 1976.     

Some current questions on solving linear elliptic problems

Summary Common properties of elliptic problems and their solutions are compared with those of difference equations, with emphasis on the classical Dirichlet problem. Resistance network analogies are used to construct discrete harmonic functions; a new 9 POINT NET approximation is defined on elementary domains whose solutions haveO(h ⁴) max-norm error. The convection-diffusion equation is studied with especial emphasis on new fast iterative methods for solving its difference approximations. A new analysis of the structure of the matrix representation of difference approximations reveals special features which can naturally be associated with diffusion and with convective processes.Dedicated to R. S. Varga on the occasion of his sixtieth birthday     

The rate function of hypoelliptic diffusions

Let be a hypoelliptic diffusion operator on a compact manifold M. Given an a priori smooth reference measure λ on M, we can then rewrite L as the sum of a λ-symmetric part L⁰ and a first-order drift part Y. The paper investigates the effect of the drift Y on the Donsker-Varadhan rate function corresponding to the large deviations of the empirical measure of the diffusion. When Y is in the linear span of the first and second-order Lie brackets of the X_i's, we derive an affine bound relating the rate functions associated with L and L⁰. As soon as one point exists where Y is not in the linear span of the first and second-order Lie brackets of the X_i's, we show that such an affine bound is impossible. © 1994 John Wiley & Sons, Inc.     

Random Walks Associated with Non-Divergence Form Elliptic Equations

This paper is concerned with the study of the diffusion process associated with a nondivergence form elliptic operator in d dimensions, d2. The authors introduce a new technique for studying the diffusion, based on the observation that the probability of escape from a d–1 dimensional hyperplane can be explicitly calculated. They use the method to estimate the probability of escape from d–1 dimensional manifolds which are C ^1, , and also d–1 dimensional Lipschitz manifolds. To implement their method the authors study various random walks induced by the diffusion process, and compare them to the corresponding walks induced by Brownian motion.