Conditioned super-Brownian motion

Summary We investigate classes of conditioned super-Brownian motions, namely H-transformsP ^H with non-negative finitely-based space-time harmonic functionsH(t, ). We prove thatH ^H is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching Brownian motions. We identify the limit behaviour of H-transforms with functionsH(t, )=h(t, (1)) depending only on the total mass (1). Using the Palm measures of the super-Brownian motion we describe for an additive spacetime harmonic functionH(t, )=h(t, x) (dx) theH-transformP ^H as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion.     

Local times on curves and uniform invariance principles

Summary Sufficient conditions are given for a family of local times |L _t ^µ | ofd-dimensional Brownian motion to be jointly continuous as a function oft and . Then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of Brownian motion, uniformly over a large family of measures. Applications included some new results for intersection local times for Brownian motions on ² and ².Research partially supported by NSF grant DMS-8822053     

Partial applicability of Moser's method to nonlinear elliptic systems

We consider nonlinear elliptic systems of divergent-type second-order partial differential equations with solutionsu W _p ¹ . It is proved thatDu L _q with someq (p; +) and it is explicitly shown howq depends on the ellipticity modulus of the system. Some conditions on the ellipticity modulus are obtained under which the solutions satisfy the Hölder conditions and the Liouville theorem holds.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 547–557, October, 1995.     

Graphs that admit square 1-factorizations are hamiltonian Cayley graphs

A square 1-factorization of a graph is a 1-factorization in which the union of any two distinct 1-factors is a disjoint union of 4-cycles. We show that a graph admits a square 1-factorization if and only if it is a Cayley graph with group (₂) ⁿ for somen. The rest of the title follows since Cayley graphs of abelian groups are known to be hamiltonian.     

α-self-similar Markov processes

Summary Let ((X(t)), P ^x) be an -self-similar isotropic Markov process on R ^d {0}. A representation of (X(t)), in terms of the radial and angular process which generalizes the skew product representation for Brownian motion is given.     

Occupation Time Fluctuations in Branching Systems

We consider particle systems in locally compact Abelian groups with particles moving according to a process with symmetric stationary independent increments and undergoing one and two levels of critical branching. We obtain long time fluctuation limits for the occupation time process of the one- and two-level systems. We give complete results for the case of finite variance branching, where the fluctuation limits are Gaussian random fields, and partial results for an example of infinite variance branching, where the fluctuation limits are stable random fields. The asymptotics of the occupation time fluctuations are determined by the Green potential operator G of the individual particle motion and its powers G ²,G ³, and by the growth as t of the operator and its powers, where T _t is the semigroup of the motion. The results are illustrated with two examples of motions: the symmetric -stable Lévy process in , and the so called c-hierarchical random walk in the hierarchical group of order N (0<c<N). We show that the two motions have analogous asymptotics of G _t and its powers that depend on an order parameter for their transience/recurrence behavior. This parameter is =d/–1 for the -stable motion, and =log c/log(N/c) for the c-hierarchical random walk. As a consequence of these analogies, the asymptotics of the occupation time fluctuations of the corresponding branching particle systems are also analogous. In the case of the c-hierarchical random walk, however, the growth of G _t and its powers is modulated by oscillations on a logarithmic time scale.     

L q -Estimates of spherical functions and an invariant mean-value property

We find someL ^q -estimates for the spherical functions on Cartan domains. As an application we prove that if the rank of the Cartan domainD is greater than one, then for any 1<-q<, the invariant mean-value property forL ^q -function onD does not imply harmonicity (the converse is known to be true even in the context of general non-compact Riemannian symmetric spacesG/K).     

Etude asymptotique des enlacements du mouvement brownien autour des droites de l'espace

Summary The existence of a joint asymptotic distribution for the windings of a three-dimensional Brownian motion around a finite number of straight lines is obtained. This complements the recent studies, by Pitman- Yor, and the authors, of the joint asymptotic distribution for the windings of planar Brownian motion around a finite number of points.The following principle governs the passage from results in the plane to results in space:Let B be a three-dimensional Brownian motion, and P ₁, ..., P _k, k planes which intersect two by two. Then, the convergences in distribution concerning the planar Brownian motions B ⁱ (1ik), defined respectively as the orthogonal projections of B on P _i (1ik), take place jointly, and the corresponding limit variables are independent.     

Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains

Let be such that each is a signed measure on R ^d belonging to the Kato class K _{d, 1}. A Brownian motion in R ^d with drift is a diffusion process in R ^d whose generator can be informally written as . When each is given by U ⁱ (x)dx for some function U ⁱ , a Brownian motion with drift is a diffusion in R ^d with generator . In Kim and Song (Ill J Math 50(3):635–688, 2006), some properties of Brownian motions with measure-value drifts in bounded smooth domains were discussed. In this paper we prove a scale invariant boundary Harnack principle for the positive harmonic functions of Brownian motions with measure-value drifts in bounded Lipschitz domains. We also show that the Martin boundary and the minimal Martin boundary with respect to Brownian motions with measure-valued drifts coincide with the Euclidean boundary for bounded Lipschitz domains. The results of this paper are also true for diffusions with measure-valued drifts, that is, when is replaced by a uniformly elliptic divergence form operator with C ¹ coefficients or a uniformly elliptic non-divergence form operator with C ¹ coefficients. The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167. The research of P. Kim is supported by Research Settlement Fund for the new faculty of Seoul National University.     

Homomorphisms onC ∞ (E) andC ∞-bounding sets

We prove, for the class of real locally convex spacesE that are continuously and linearly injectable into somec ₀(), that every non-zero homomorphism on the algebraC (E) ofC -functions onE is given by a point evaluation at some point ofE. Furthermore, if every real-valuedC -function on the weak topology of a quasi-complete locally convex spaceE is bounded on a subsetA ofE, thenA is relatively weakly compact.     

Optimal attitude control of a rigid body

Being mainly interested in the control of satellites, we investigate the problem of maneuvering a rigid body from a given initial attitude to a desired final attitude at a specified end time in such a way that a cost functional measuring the overall angular velocity is minimized.This problem is solved by applying a recent technique of Jurdjevic in geometric control theory. Essentially, this technique is just the classical calculus of variations approach to optimal control problems without control constraints, but formulated for control problems on arbitrary manifolds and presented in coordinate-free language. We model the state evolution as a differential equation on the nonlinear state spaceG=SO(3), thereby completely circumventing the inevitable difficulties (singularities and ambiguities) associated with the use of parameters such as Euler angles or quaternions. The angular velocities _k about the body's principal axes are used as (unbounded) control variables. Applying Pontryagin's Maximum Principle, we lift any optimal trajectorytg^*(t) to a trajectory onT ^*G which is then revealed as an integral curve of a certain time-invariant Hamiltonian vector field. Next, the calculus of Poisson brackets is applied to derive a system of differential equations for the optimal angular velocitiest _k ^* (t); once these are known the controlling torques which need to be applied are determined by Euler's equations.In special cases an analytical solution in closed form can be obtained. In general, the unknown initial values _k ^* (t₀) can be found by a shooting procedure which is numerically much less delicate than the straightforward transformation of the optimization problem into a two-point boundary-value problem. In fact, our approach completely avoids the explicit introduction of costate (or adjoint) variables and yields a differential equation for the control variables rather than one for the adjoint variables. This has the consequence that only variables with a clear physical significance (namely angular velocities) are involved for which gooda priori estimates of the initial values are available.     

Density results for uniform families

A familyF of subsets is calledk-dense if there exists ak-element setA such that all 2 ^k of its subsets can be obtained in the formAF for someFF. Best possible bounds are obtained for the maximum number of sets in notk-densek-partite families. This is a consequence of a new rank formula for inclusion matrices.     

Combinatorial Properties of Brownian Motion on the Compact Classical Groups

We consider the probability distribution on a classical group G which naturally generalizes the normal distribution (the heat kernel), defined in terms of Brownian motions on G. As Brownian motion can be defined in terms of the Laplacian on G, much of this work involves the computation of the Laplacian. These results are then used to study the behavior of the normal distribution on U(n( as . In addition, we show how the results on computing the Laplacian on the classical groups can be used to compute expectations with respect to Haar measure on those groups.     

Proportional navigation and the game of two cars

A stochastic version of Isaacs's game of two cars is considered. The motion of the players is confined to the pursuer's effective operation zoneD _P, and the cost function of the game is the probability of the event: {Before the evader enters his safe zone, the evader enters the pursuer's killing zoneK _P, at somet, 0tT, or the evader stays at the domainD _P–K _P, for allt[0,t ₀],t ₀>T}. By numerically solving a nonlinear parabolic boundary-value problem on a generalized torus in ³, it is shown that, for a range of values of some parameter, a proportional navigation guidance law is an optimal feedback pursuit strategy.     

Closability and resolvent of Dirichlet forms perturbed by jumps

We give a necessary and sufficient condition for the closability of a form _J =+J, where is a Dirichlet form on someL ² (E, m) andJ is a symmetric measure onE×E. We then exhibit the resolvent family of such a form.     

Natural extensions of holomorphic motions

We consider an arbitrary real analytic family X_z, , over the closed unit disc , of real analytic plane Jordan curves X_z. Ifj _e ^iθ ,e ^iθ ∋ ∂D, is an arbitrary real-analytic family of orientation-reversing homeomorphisms of fixingX _e ^iθ pointwise, we show that there is a unique holomorphic motion of extending the given motion of Jordan curves and consistent with the given family of involutions. If these generalized reflections are defined using the barycentric extension construction of Douady-Earle-Nag, then the resulting extension method for holomorphic motions of X is natural, that is Moebius-invariant and continuous with respect to variation of the given motion of X₀.     

On detachment conditions in the problem on the motion of a rigid body on a rough plane

The classical mechanical problem about the motion of a heavy rigid body on a horizontal plane is considered within the framework of theory of systems with unilateral constraints. Under general assumptions about the character of friction, we examine the question on the possibility of detachment of the body from the plane under the action of reaction of the plane and forces of inertia. For systems with rolling, we find new scenarios of the appearing of motions with jumps and impacts. The results obtained are applied to the study of stationary motions of a disk. We have showed the following.

Constructing cospectral graphs

Some new constructions for families of cospectral graphs are derived, and some old ones are considerably generalized. One of our new constructions is sufficiently powerful to produce an estimated 72% of the 51039 graphs on 9 vertices which do not have unique spectrum. In fact, the number of graphs of ordern without unique spectrum is believed to be at leastn ³ g _–1 for some>0, whereg _n is the number of graphs of ordern andn 7.     

Functional calculus and its application to operator algebras

In this paper we will prove that any unital isometric representation ofH () with finitely connected domain has property (A ₁(r)) for somer1, which generalizes the same conclusion in [2] and [8].     

Time asymptotics for some degenerate models of evolution of systems with an infinite number of particles

The paper is devoted to the problem of convergence to the equilibrium state in the motion of infinite systems of classical particles. Two models of the motion are considered: free motion of point particles in Euclidean spaceR ^v , 1, and motion of solid rods on the line R¹. The paper contains new results obtained by the authors and also a survey of previous results in this direction.K. Boldrigini took part in the work on the paper.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 14, pp. 147–254, 1979