Doubling measures,monotonicity, and quasiconformality

We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure. L.V.K. was supported by an NSF Young Investigator award under grant DMS 0601926. J.-M.W. was supported by the NSF grant DMS 0400810.     

Uniqueness and stability in determining the heat radiative coefficient,the initial temperature and a boundary coefficient in a parabolic equation

We consider an inverse parabolic problem. We prove that the heat radiative coefficient, the initial temperature and a boundary coefficient can be simultaneously determined from the final overdetermination, provided that the heat radiative coefficient is a priori known in a small subdomain. Moreover we establish a stability estimate for this inverse problem.     

Localization and completion of nilpotent groups of automorphisms

We study nilpotent subgroups of automorphism groups in the category of groups and the homotopy category of spaces. We establish localization and completion theorems for nilpotent groups of automorphisms of nilpotent groups. We then apply these algebraic theorems to prove analogous results for certain groups of self-homotopy equivalences of spaces.     

The martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree

Consider a closed subgroup of the automorphism group of a homogeneous treeT, and assume that acts transitively on the vertex set. Suppose that is a probability measure on which has continuous density with respect to Haar measure and whose support is compact open and generates as a closed semigroup. It is shown that the Martin boundary of with respect to the random walk with law coincides with the space of ends ofT. This extends known results for free groups and applies, for example, to the affine group over a non archimedean local field.     

Logarithmic order and type of indeterminate moment problems II

We show that there exist indeterminate Stieltjes moment problems with prescribed common logarithmic order and type.     

Ergodic behavior of diffusions with random jumps from the boundary

We consider a diffusion process on D⊂R^d$D\subset {\mathbb{R}}^d$, which upon hitting ∂D$\partial D$, is redistributed in D$D$ according to a probability measure depending continuously on its exit point. We prove that the distribution of the process converges exponentially fast to its unique invariant distribution and characterize the exponent as the spectral gap for a differential operator that serves as the generator of the process on a suitable function space.     

Periodic-like solutions of variable coefficient and Wick-type stochastic NLS equations

Variable coefficient and Wick-type stochastic nonlinear Schrödinger (NLS) equations are investigated. By using white noise analysis, Hermite transform and extended F-expansion method, we obtain a number of Wick versions of periodic-like wave solutions and periodic wave solutions expressed by various Jacobi elliptic functions for Wick-type stochastic and variable coefficient NLS equations, respectively. In the limit cases, the soliton-like wave solutions are showed as well. Since Wick versions of functions are usually difficult to evaluate, we get some nonWick versions of the solutions for Wick-type stochastic NLS equations in special cases.     

Inner coefficient of quasiconformality of a pair of dihedral wedges

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 6, pp. 12–16, November–December, 1988.     

Invariant tensorfields and the canonical connection of a 3-web

In the following we shall give a new approach to web geometry. Instead of using the reduced adapted coframe bundles over the web manifolds we define invariant tensorfields corresponding to the 3-web structure and express the covariant derivation of the Chern connection with the help of these tensorfields, working only on the tangent bundle of the web manifold. The invariant tensorfields of a 3-web define a more general, so-called {H, J}-structure on the manifold which can be considered as an infinitesimal, non-integrable version of a web structure. We introduce a canonical connection of a {H, J}-structure which reduces to the Chern connection in the case of a 3-web. Using the tensorial expression of the covariant derivation of the Chern connection we give direct proof for the torsion and curvature identities. Finally, we apply our formulae to an algebraic characterization of the parallel translation with respect to the Chern connection.     

On a modification of the classical risk process

In this paper, we present the classical risk process with two-step premium function. This means that the gross risk premium rate changes if the insurer’s surplus reaches a certain threshold level. The formula for the infinite-time ruin probability is obtained. The asymptotic behaviour of the ruin probability in the case where the claim size distribution has a light tail is considered as well.     

Local automorphisms of the Hilbert ball

Every holomorphic mapping which takes a piece of the boundary of the unit ball in complex Hilbert space into the boundary of the unit ball and whose differential at some point of this boundary is onto is the restriction of an automorphism of the ball. We also show that it is enough to assume that the mapping is only Gâteaux-holomorphic.

     

On the equivalence of the static and dynamic points of view for diffusions in a random environment

We study the equivalence of the static and dynamic points of view for diffusions in a random environment in dimension one. First we prove that the static and dynamic distributions are equivalent if and only if either the speed in the law of large numbers does not vanish, or b/a$b/a$ is a.s. the gradient of a stationary function, where a$a$ and b$b$ are the covariance coefficient resp. the local drift attached to the diffusion. We moreover show that the equivalence of the static and dynamic points of view is characterized by the existence of so-called “almost linear coordinates”.     

Solvability of strongly nonlinear boundary value problems for second order differential inclusions

In this paper we study a problem for a second order differential inclusion with Dirichlet, Neumann and mixed boundary conditions. The equation is driven by a nonlinear, not necessarily homogeneous, differential operator satisfying certain conditions and containing, as a particular case, the p$p$-Laplacian operator. We prove the existence of solutions both for the case in which the multivalued nonlinearity has convex values and for the case in which it has not convex values. The presence of a maximal monotone operator in the equation make the results applicable to gradient systems with non-smooth, time invariant, convex potential and differential variational inequalities.     

Cooperative hedging with a higher interest rate for borrowing

The paper studies the cooperative hedging problem of contingent claims in an incomplete financial market. Firstly we give the characterization of the optimal cooperative hedging strategy for the Black-Scholes model and the Volatility Jump model explicitly, then we consider the problem of cooperative hedging for the multi-agent case in a market with a higher borrowing interest rate. By the results of concave and linear backward stochastic differential equations, we give the optimal cooperative hedging strategy in our model.     

On the local behavior of mappings with unbounded quasiconformality coefficient

We study space mappings more general than the mappings with bounded distortion in the sense of Reshetnyak. We consider questions related to the local behavior of mappings differentiable almost everywhere, possessing Properties N, N ^?1, ACP, and ACP ^?1, and such that quasiconformality coefficient satisfies a certain restriction on growth. We show that the value of a mapping satisfying these requirements on an arbitrary neighborhood of an essential singularity can be greater in absolute value than the logarithm of the inverse radius of the ball raised to an arbitrary positive power.     

Local asymptotic laws for the Brownian convex hull

Summary We prove a general theorem for the precise rate at which the convex hull of Brownian motion gets created. The latter result relates large deviation theory to P. Lévy's geometric proof of Strassen's law of the iterated logarithm. This also answers a question of S. Evans. Moreover, we give a partial solution to a question of J. Hammersley and P. Lévy regarding the slowness of the growth of the hull process. Several examples, some classical and some new, are given to illustrate the theorems. Finally, we present applications to the convex hull of random walks ind dimensions.     

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains

We study a system of 3D Navier-Stokes equations in a two-layer parallelepiped-like domain with an interface coupling of the velocities and mixed (free/periodic) boundary condition on the external boundary. The system under consideration can be viewed as a simplified model describing some features of the mesoscale interaction of the ocean and atmosphere. In case when our domain is thin (of order ε), we prove the global existence of the strong solutions corresponding to a large set of initial data and forcing terms (roughly, of order ε^−2/3). We also give some results concerning the large time dynamics of the solutions. In particular, we prove a spatial regularity of the global weak attractor.