Brownian Chen series and Atiyah-Singer theorem

The purpose of this work is to give a new and short proof of the Atiyah-Singer local index theorem for the Dirac operator on the spin bundle. This proof is obtained by using heat semigroups approximations based on the truncation of Brownian Chen series.     

A representation theorem for smooth Brownian martingales

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a ‘frozen path’. The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete-time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.     

A weak limit theorem for generalized multifractional Brownian motion

We provide a result on an approximation to the generalized multifractional Brownian motion in the space of continuous functions on [0, 1]. The construction of this approximation is based on the Poisson process.     

Comparison theorem for Brownian multidimensional BSDEs via jump processes

In this Note, we provide an original proof of the comparison theorem for multidimensional Brownian BSDEs in the case where at each line k the generator depends on the matrix variable Z only through its row k.     

Weak hyperbolicity and many-dimensional analogues of Picard's theorem

Omsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 1, pp. 153–160, January–February, 1991.     

Higher analogues of Stickelberger's theorem

Let l be an odd prime number, F denote any totally real number field and E/F be an Abelian CM extension of F of conductor f. In this paper we prove that for every n odd and almost all prime numbers l we have where S_n(E/F,l) is the Stickelberger ideal (Ann. of Math. 135 (1992) 325–360; J. Coates, p-adic L-functions and Iwasawa's theory, in: Algebraic Number Fields by A. Fröhlich, Academic Press, London, 1977). In addition if we assume the Quillen–Lichtenbaum conjecture then To cite this article: G. Banaszak, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Multidimensional analogues of Bohr's theorem on power series

Generalizing the classical result of Bohr, we show that if an -variable power series converges in -circular bounded complete domain and its sum has modulus less than 1, then the sum of the maximum of the modulii of the terms is less than 1 in the homothetic domain , where . This constant is near to the best one for the domain

     

6-Valent analogues of Eberhard’s theorem

It is shown that for every sequence of non-negative integers (p _n|1≦n≠3) satisfying the equation {ie19-1} (respectively, =0) there exists a 6-valent, planar (toroidal, respectively) multi-graph that has preciselyp _n n gonal faces for alln, 1≦n≠3. This extends Eberhard’s theorem that deals, in a similar fashion, with 3-valent, 3-connected planar graphs; the equation involved follows from the famous Euler’s equation.     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

A central limit theorem for Gibbs measures relative to Brownian motion

We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position to use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.     

Applications of the continuous-time ballot theorem to Brownian motion and related processes

Motivated by questions related to a fragmentation process which has been studied by Aldous, Pitman, and Bertoin, we use the continuous-time ballot theorem to establish some results regarding the lengths of the excursions of Brownian motion and related processes. We show that the distribution of the lengths of the excursions below the maximum for Brownian motion conditioned to first hit λ>0 at time t is not affected by conditioning the Brownian motion to stay below a line segment from (0,c) to (t,λ). We extend a result of Bertoin by showing that the length of the first excursion below the maximum for a negative Brownian excursion plus drift is a size-biased pick from all of the excursion lengths, and we describe the law of a negative Brownian excursion plus drift after this first excursion. We then use the same methods to prove similar results for the excursions of more general Markov processes.     

Some analogues of eberhard’s theorem on convex polytopes

Letp _k denote the number ofk-gonal faces of a simple 3-polytope. Euler’s relation leads to an equation between thep _k ’s which does not involvep ₆. Eberhard proved in 1891 that every sequence of non-negative integers (p ₃,p ₄,…) satisfying this equation corresponds to a polytope for suitable values ofp ₆. In the present paper it is established that ifp ₃=p ₄=0 then every valuep ₆≧8 is suitable. Research supported in part by the National Science Foundation under grant GP-7536     

Schilder theorem for the Brownian motion on the diffeomorphism group of the circle

We prove a large deviation principle for flows associated to stochastic differential equations with non-Lipschitz coefficients. As an application we establish a Schilder Theorem for the Brownian motion on the group of diffeomorphisms of the circle.     

An explicit Schilder‐type theorem for super‐Brownian motions

Like ordinary Brownian motion, super‐Brownian motion, a central object in the theory of superprocesses, is a universal object arising in a variety of settings. Schilder‐type theorems and Cramér‐type theorems are two of the major topics for large‐deviation theory. A Schilder‐type (which is also a Cramér‐type) sample large deviation for super‐Brownian motions with a good rate function represented by a variation formula was established in 1993 and 1994; since then there have been very valuable contributions for giving an affirmative answer to the question of whether this sample large deviation holds with an explicit good rate function. In this paper, thanks to previous results on this issue and the Brownian snake, we establish such a large deviation for nonzero finite initial measures. © 2010 Wiley Periodicals, Inc.