a solution theory for quasilinear singular spdes we give a construction allowing us to build local renormalized solutions to general quasilinear stochastic pdes within the theory of regularity structures thus greatly generalizing the recent results of 1 5 11 loosely speaking our construction covers quasilinear variants of all classes of equations for which the general construction of 3 4 7 applies including in particular one dimensional systems with kpz type nonlinearities driven by space time white noise in a less singular and more specific case we furthermore show that the counterterms introduced by the renormalization procedure are given by local functionals of the solution the main feature of our construction is that it allows exploitation of a number of existing results developed for the semilinear case so that the number of additional arguments it requires is relatively small

one sided compactness results for aronszajn gagliardo functors interpolating compactness properties of operators is a long standing and important problem in this paper the authors consider the problem in a very general setting of aronszajn gagliardo functors in simplest terms they show that if t a0 b0 is compact and t a1 b1 is bounded then t is compact on some interpolation spaces constructed in the aronszajn gagliardo methods these methods do not include the complex interpolation method but the authors show that if the two couples are formed by banach lattices then the theorem holds in the complex method as well additional results are in the context of interpolation of compactness properties in the context of n tuples of banach spaces the paper includes a number of related interesting results

uniqueness of solutions of the stochastic navier stokes equation with invariant measure given by the enstrophy a stochastic navier stokes equation with space time gaussian white noise is considered having as infinitesimal invariant measure a gaussian measure whose covariance is given in terms of the enstrophy pathwise uniqueness for a e initial velocity is proven for solutions having as invariant measure

mod points on shimura varieties of abelian type we show that the mod p points on a shimura variety of abelian type with hyperspecial level have the form predicted by the conjectures of kottwitz and langlands rapoport along the way we show that the isogeny class of a mod p point contains the reduction of a special point

a stochastic partial differential equation with values in a manifold abstract we investigate a certain stochastic partial differential equation which is defined on the unit interval with periodic boundary condition and takes values in a manifold such equation has particularly two different applications namely it determines the evolution law of an interacting constrained system of continuum distributed over the unit circle while it defines a diffusive motion of loops on a manifold we establish the existence and uniqueness results and then show the smoothness property of the solutions some examples are given in the final section

a non linear parabolic pde with a distributional coefficient and its applications to stochastic analysis abstract we consider a non linear parabolic partial differential equation pde on r d with a distributional coefficient in the non linear term the distribution is an element of a besov space with negative regularity and the non linearity is of quadratic type in the gradient of the unknown under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the pde and investigate properties like continuity with respect to the initial condition and blow up times we prove a global existence and uniqueness result assuming further properties on the non linearity to conclude we consider an application of the pde to stochastic analysis in particular to a class of non linear backward stochastic differential equations with distributional drivers

connected components of closed affine deligne lusztig varieties for rese fgln abstract let f be a p adic field and let e f be a finite unramified extension for the weil restriction res e f g l n of the general linear group g l n we determine the connected components of closed affine deligne lusztig varieties of arbitrary parahoric level

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

pathwise mild solutions for quasilinear stochastic partial differential equations abstract stochastic partial differential equations spdes have become a key modeling tool in applications yet there are many classes of spdes where the existence and regularity theory for solutions is not completely developed here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear cauchy problems including among others cross diffusion systems as a key application our solutions are local in time and are derived via a fixed point argument in suitable function spaces the key idea is to combine in a suitable way the classical theory of deterministic quasilinear parabolic partial differential equations pdes with recent theory of evolution semigroups we also show how to apply our theory to the shigesada kawasaki teramoto skt model furthermore we provide examples of blow up and ill posed operators which can occur after finite time showing that solutions can only be local in time for general quasilinear spdes while they might be global in time for special subclasses of problems

ramification of rough paths the stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended chen s multiplicative property involves the durr connes kreimer coproduct on rooted trees this turns out to be the natural setting for a non geometric theory of rough paths

fundamental elements of an affine weyl group fundamental elements are certain special elements of affine weyl groups introduced by gortz haines kottwitz and reuman they play an important role in the study of affine deligne lusztig varieties in this paper we obtain characterizations of the fundamental elements and their natural generalizations we also derive an inverse to a version of newton hodge decomposition in affine flag varieties as an application we obtain a group theoretic generalization of oort s results on minimal p divisible groups and we show that in certain good reduction of pel shimura datum each newton stratum contains a minimal ekedahl oort stratum this generalizes a result of viehmann and wedhorn

connected components of closed affine deligne lusztig varieties for split reductive algebraic groups we determine the sets of connected components of closed affine deligne lusztig varieties with parahoric level structure

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

on compactness results of lions peetre type for bilinear operators abstract let a a 0 a 1 b b 0 b 1 be banach couples let e be a banach space and let t be a bilinear operator such that t a b e m j a a j b b j for a a 0 a 1 b b 0 b 1 j 0 1 if t a j b j e compactly for j 0 or 1 we show that t may be uniquely extended to a compact bilinear operator from the complex interpolation spaces generated by a and b to e furthermore the corresponding result for the real method is given and we also study the case when e is replaced by a couple e 0 e 1 of banach function spaces on the same measure space

bilinear interpolation theorems and applications abstract interpolation of bilinear operators is studied the celebrated littlewood mixed norm inequality is used to prove interpolation theorems for bilinear operators defined on couples of c 0 weighted sequence spaces generated by parameters of quasi concave functions it is shown that these results can be lifted to a wider class of abstract methods of interpolation the abstract results are used to prove bilinear interpolation theorems to the setting of calderon lozanovskii spaces as an application the boundedness of the bilinear hilbert transform between orlicz spaces including zygmund classes is proved

geometric and homological properties of affine deligne lusztig varieties this paper studies ane deligne lusztig varieties x w b in the ane ag variety of a quasi split tamely ramied group we describe the geometric structure of x w b for a minimal length element w in the conjugacy class of an extended ane weyl group we then provide a reduction method that relates the structure of x w b for arbitrary elements w in the extended ane weyl group to those associated with minimal length elements based on this reduction we establish a connection between the dimension of ane deligne lusztig varieties and the degree of the class polynomial of ane

stochastic pde limit of the six vertex model we study the stochastic six vertex model and prove that under weak asymmetry scaling i e when the parameter delta rightarrow 1 so as to zoom into the ferroelectric disordered phase critical point its height function fluctuations converge to the solution to the kardar parisi zhang kpz equation we also prove that the one dimensional family of stochastic gibbs states for the symmetric six vertex model converge under the same scaling to the stationary solution to the stochastic burgers equation our proofs rely upon the markov self duality of our model the starting point is an exact microscopic hopf cole transform for the stochastic six vertex model which follows from the model s known one particle markov self duality given this transform the crucial step is to establish self averaging for specific quadratic function of the transformed height function we use the model s two particle self duality to produce explicit expressions as bethe ansatz contour integrals for conditional expectations from which we extract time decorrelation and hence self averaging in time the crux of our markov duality method is that the entire convergence result reduces to precise estimates on the one particle and two particle transition probabilities previous to our work markov dualities had only been used to prove convergence of particle systems to linear gaussian spdes e g the stochastic heat equation with additive noise

two dimensional navier stokes equations driven by a space time white noise abstract we study the two dimensional navier stokes equations with periodic boundary conditions perturbed by a space time white noise it is shown that although the solution is not expected to be smooth the nonlinear term can be defined without changing the equation we first construct a stationary martingale solution then we prove that for almost every initial data with respect to a measure supported by negative spaces there exists a unique global solution in the strong probabilistic sense

a note on the notion of geometric rough paths we use simple sub riemannian techniques to prove that every weak geometric p rough path a geometric p rough path in the sense of 20 is the limit in sup norm of a sequence of canonically lifted smooth paths uniformly bounded in p variation thus clarifying the two different definitions of a geometric p rough path our proofs are sufficiently general to include the case of holder and modulus type regularity this allows us to extend a few classical results on holder spaces and p variation spaces to the non commutative setting necessary for the theory of rough paths as an application we give a precise description of the support of enhanced fractional brownian motion and prove a conjecture by ledoux et al

connected components of closed affine deligne lusztig varieties we determine the set of connected components of closed affine deligne lusztig varieties for special maximal compact subgroups of split connected reductive groups we show that there is a transitive group action on this set thus such an affine deligne lusztig variety has isolated points if and only if its dimension is 0 we also obtain a description of the set of these varieties that are zero dimensional