non linear rough heat equations this article is devoted to define and solve an evolution equation of the form dyt yt dt dx t yt wherestands for the laplace operator on a space of the form l p r n and x is a finite dimensional noisy nonlinearity whose typical form is given by xt n i 1 x i fi where each x x 1 x n is a holder function generating a rough path and each fi is a smooth enough function defined on l p r n the generalization of the usual rough path theory allowing to cope with such

random dynamical systems rough paths and rough flows abstract we analyze common lifts of stochastic processes to rough paths rough drivers valued processes and give sufficient conditions for the cocycle property to hold for these lifts we show that random rough differential equations driven by such lifts induce random dynamical systems in particular our results imply that rough differential equations driven by the lift of fractional brownian motion in the sense of friz victoir 26 induce random dynamical systems

a table of elliptic integrals of the second kind by evaluating elliptic integrals in terms of standard r functions instead of legendre s integrals many in one case 144 formulas in previous tables are unified the present table includes only integrals of the first and second kinds having integrands with real singular points the 216 integrals of this type listed in gradshteyn and ryzhik s table constitute a small fraction of the special cases of 13 integrals evaluated here the interval of integration is not required as it is in previous table to begin or end at a singular point of the integrand fortran codes for the standard r functions are included in a supplement

rates of clustering for some gaussian self similar processes the analogue of strassen s functional law of the iterated logarithm in known for many gaussian processes which have suitable scaling properties and here we establish rates at which this convergence takes place we provide a new proof of the best upper bound for the convergence tok by suitably normalized brownian motion and then continue with this method to get similar bounds for the brownian sheet and other self similar gaussian processes the previous method which produced these results for brownian motion in 1 was highly dependent on many special properties unavailable when dealing with other gaussian processes

weak poincare inequalities on domains defined by brownian rough paths we prove weak poincare inequalities on domains which are inverse images of open sets in wiener spaces under continuous functions of brownian rough paths the result is applicable to dirichlet forms on loop groups and connected open subsets of path spaces over compact riemannian manifolds

extensions of the preiss differentiability theorem abstract the preiss differentiability theorem for lipschitz functions on banach spaces is generalized to locally lower upper semi lipschitz functions and several extensions are presented

general rough integration levy rough paths and a levy kintchine type formula we consider rough paths with jumps in particular the analogue of lyons extension theorem and rough integration are established in a jump setting offering a pathwise view on stochastic integration against cadlag processes a class of levy rough paths is introduced and characterized by a sub ellipticity condition on the left invariant diffusion vector fields and a certain integrability property of the carnot caratheodory norm with respect to the levy measure on the group using hunt s framework of lie group valued levy processes examples of levy rough paths include a standard multi dimensional levy process enhanced with a stochastic area as constructed by d williams the pure area poisson process and brownian motion in a magnetic field an explicit formula for the expected signature is given

uniform approximation of convex function in smooth banach spaces abstract this article is devoted to smooth approximation of convex functions on banach spaces with smooth norm we prove that if x is a smooth space and f is a w lower semicontinuous lipschitzian convex function on x then there exist two w lower semicontinuous g teaux differentiable convex function sequences f n n 1 and g n n 1 such that 1 f n f n 1 f and f g n 1 g n 2 f n f and g n f uniformly on x 3 cl x x d f n x x cl x x d g n x x x

levy area of wiener processes in banach spaces the goal of this paper is to construct canonical levy area processes for banach space valued brownian motions via dyadic approximations the significance of the existence of canonical levy area processes is that a stochastic integration theory can be established for such brownian motions in banach spaces existence of flows for stochastic differential equations with infinite dimensional noise then follows via the results of lyons and lyons and qian see e g system control and rough paths 2000 oxford univ press this investigation involves a careful analysis on the choice of tensor norms motivated by the applications to infinite dimensional stochastic differential equations

reduction theorems for elliptic integrands with the square root of two quadratic factors abstract recent methods of making integral tables and symbolic integration programs for elliptic integrals depend critically on reduction theorems for two particular integrals these theorems assume that certain variables have positive real part and the assumption is not always satisfied if the integrand contains the square root of two quadratic polynomials each with conjugate complex zeros a new remedy for this difficulty is the use of duplication theorems leading to the weaker assumption that only sums of two variables must have positive real part numerical examples are given in an appendix related methods are used to simplify part of an algorithm for numerical computation of a symmetric elliptic integral of the third kind

flow of diffeomorphisms induced by a geometric multiplicative functional in this paper it is shown that the unique multiplicative functional solution to a differential equation driven by a geometric multiplicative functional consitutes a flow of local diffeomorphisms in the case where the driving geometric multiplicative functional is generated by a brownian motion the result in particular presents an answer to an open problem proposed in ikeda and watanabe 4

laplace s method for the laws of heat processes on loop spaces abstract in this paper we will consider laplace s method for a class of heat processes on loop spaces we will obtain the first term of the asymptotics under assumptions that the function under consideration attains its minimum at a unique point and that the hessian at the point is non degenerate this kind of process was first introduced by p malliavin in 1990 for the loop group case and then gradually generalized by various authors our tool is the rough path theory of t lyons this technique was pioneered by s aida for finite dimensional processes in his unpublished paper

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

stochastic analysis rough path analysis and fractional brownian motions in this paper we show by using dyadic approximations the existence of a geometric rough path associated with a fractional brownian motion with hurst parameter greater than 1 4 using the integral representation of fractional brownian motions we furthermore obtain a skohorod integral representation of the geometric rough path we constructed by the results in ly1 a stochastic integration theory may be established for fractional brownian motions and strong solutions and a wong zakai type limit theorem for stochastic differential equations driven by fractional brownian motions can be deduced accordingly the method can actually be applied to a larger class of gaussian processes with covariance functions satisfying a simple decay condition

a table of elliptic integrals one quadratic factor integration in terms of real quantities is accomplished for 33 integrands that are rational except for the square root of a cubic or quartic polynomial with exactly one pair of conjugate complex zeros formulas are provided by which 45 more integrals of the same type can be expressed in terms of real quantities with the help of earlier papers neither limit of integration is assumed to be a singular point of the integrand all the integrals are reduced to r functions for which fortran programs are available most of the integrals are not listed in other tables

acoustic waves in long range random media we consider waves propagating through multiscale media much is known about waves propagating through a medium that satisfies a scale separation assumption with random fluctuations on a microscale here we go beyond this situation and consider waves propagating through a medium defined in terms of a long range process such a medium can for instance be modeled in terms of a one dimensional fractional brownian motion with variations on a continuum of scales fractal medium models are used to model for example the heterogeneous earth and the turbulent atmosphere we set forth a framework using the theory of rough paths in which propagation problems of this nature can be analyzed in the case with anticipative medium fluctuations with a hurst exponent h 1 2 we show how the wave interacts with the medium fluctuations in this case and that the interaction is qualitatively different from the situation where the medium satisfies a separation of scales assumption in the long range case considered here the

two characterizations of super reflexive banach spaces by the behaviour of differences of convex functions abstract a function defined on a banach space x is called convex if it can be represented as a difference of two continuous convex functions in this work we study the relationship between some geometrical properties of a banach space x and the behaviour of the class of all convex functions defined on it more precisely we provide two new characterizations of super reflexivity in terms convex functions

stochastic area for brownian motion on the sierpinski gasket we construct a levy stochastic area for brownian motion on the sierpinski gasket the standard approach via ito integrals fails because this diffusion has sample paths which are far rougher than those of semimartingales we thus provide an example demonstrating the restrictions of the semimartingale framework as a consequence of the existence of the area one has a stochastic calculus and can solve stochastic differential equations driven by brownian motion on the sierpinski gasket

generic frechet differentiability of convex functions dominated by a lower semicontinuous convex function abstract in this paper an extended real valued proper lower semicontinuous convex function f on a banach space is said to have the frechet differentiability property fdp if every proper lower semicontinuous convex function g with g f is frechet differentiable on a dense g subset of int dom g the interior of the effective domain of g we show that f has the fdp if and only if the w closed convex hull of the image of the subdifferential map of f has the radon nikod m property this is a generalization of the main theorem in a paper by lixin and shuzhong to appear according to this result it also gives several new criteria of asplund spaces

stochastic calculus for fractional brownian motion with hurst exponent h a rough path method by analytic extension the d dimensional fractional brownian motion fbm for short b t b 1 t b d t t r with hurst exponent 0 1 is a d dimensional centered self similar gaussian process with covariance e b i s b i t 1 2 i j lsl 2 ltl 2 t s 2 the long standing problem of defining a stochastic integration with respect to fbm and the related problem of solving stochastic differential equations driven by fbm has been addressed successfully by several different methods although in each case with a restriction on the range of either d or the case corresponds to the usual stochastic integration with respect to brownian motion while most computations become singular when gets under various threshhold values due to the growing irregularity of the trajectories as 0 we provide here a new method valid for any d and for 1 4 by constructing an approximation e t e 0 of fbm which allows to define iterated integrals and then applying the geometric rough path theory the approximation relies on the definition of an analytic process z on the cut plane z c r of which fbm appears to be a boundary value and allows to understand very precisely the well known see 5 but as yet a little mysterious divergence of levy s area for 1 4

approximation metric entropy and small ball estimates for gaussian measures a precise link proved by kuelbs and li relates the small ball behavior of a gaussian measure on a banach space e with the metric entropy behavior of k the unit ball of the reproducing kernel hilbert space of in e we remove the main regularity assumption imposed on the unknown function in the link this enables the application of tools and results from functional analysis to small ball problems and leads to small ball estimates of general algebraic type as well as to new estimates for concrete gaussian processes moreover we show that the small ball behavior of a gaussian process is also tightly connected with the speed of approximation by finite rank processes

generic frechet differentiability of convex functions on non asplund spaces abstract let f be a continuous convex function on a banach space e this paper shows that every proper convex function g on e with g f is generically frechet differentiable if and only if the image of the subdifferential map f of f has the radon nikod m property and in this case it is equivalent to showing that the image of f is separable on each separable subspace of e

rough paths analysis of general banach space valued wiener processes abstract in this article we carry out a rough paths analysis for banach space valued wiener processes we show that most of the features of the classical wiener process pertain to its rough path analog to be more precise the enhanced process has the same scaling properties and it satisfies a fernique type theorem a support theorem and a large deviation principle in the same holder topologies as the classical wiener process does moreover the canonical rough paths of finite dimensional approximating wiener processes converge to the enhanced wiener process finally a new criterion for the existence of the enhanced wiener process is provided which is based on compact embeddings this criterion is particularly handy when analyzing kunita flows by means of rough paths analysis which is the topic of a forthcoming article

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

asymptotic expansions for the laplace approximations for ito functionals of brownian rough paths abstract in this paper we establish asymptotic expansions for the laplace approximations for ito functionals of brownian rough paths under the condition that the phase function has finitely many non degenerate minima our main tool is the banach space valued rough path theory of t lyons we use a large deviation principle and the stochastic taylor expansion with respect to the topology of the space of geometric rough paths this is a continuation of a series of papers by inahama y inahama laplace s method for the laws of heat processes on loop spaces j funct anal 232 2006 148 194 and by inahama and kawabi y inahama h kawabi large deviations for heat kernel measures on loop spaces via rough paths j london math soc 73 3 2006 797 816 y inahama h kawabi on asymptotics of certain banach space valued ito functionals of brownian rough paths in proceedings of the abel symposium 2005 stochastic analysis and applications a symposium in honor of kiyosi ito springer berlin in press available at http www abelprisen no no abelprisen deltagere 2005 html

a smooth variational principle with applications to hamilton jacobi equations in infinite dimensions abstract we prove that if x is a banach space which admits a smooth lipschitzian bump function then for every lower semicontinuous bounded below function there exists a lipschitzian smooth function g on x such that f g attains its strong minimum on x thus extending a result of borwein and preiss we then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of hamilton jacobi equations in infinite dimensional banach spaces a without assuming the radon nikodym property

the levy area process for the free brownian motion abstract in this paper we construct a levy area process for the free brownian motion and in this way a typical geometric rough path in the sense of t lyons 1998 rev mat iberoamer 14 215 310 lying above the free brownian path thus the general results of lyons on differential equations driven by rough paths may be applied to the free brownian motion case

a table of elliptic integrals of the third kind as many as 72 elliptic integrals of the third kind in previous tables are unified by evaluation in terms of r functions instead of legendre s integrals the present table includes only integrals having integrands with real singular points in addition to 31 integrals of the third kind most of them unavailable elsewhere 10 integrals of the first and second kinds from an earlier table are listed again in new notation in contrast to conventional tables the interval of integration is not required to begin or end at a singular point of the integrand fortran codes for the standard r functions r sub c and r sub j revised to include their cauchy principal values are listed in a supplement

smooth approximation of convex functions in banach spaces abstract this paper shows that every extended real valued lower semi continuous proper respectively lipschitzian convex function defined on an asplund space can be represented as the point wise limit respectively uniform limit on every bounded set of a sequence of lipschitzian convex functions which are locally affine hence c at all points of a dense open subset and shows an analogous for w lower semi continuous proper respectively lipschitzian convex functions defined on dual spaces whose pre duals have the radon nikodym property