the discrete wavelet transform for a symmetric anti symmetric multiwavelet family on the interval the chui lian multiwavelet family of approximation order three is extended to serve as a basis for an orthogonal discrete multiwavelet transform on the interval this requires in addition to the original symmetric and antisymmetric pairs of scaling functions and wavelets four new functions of each type at each edge of the interval i e one more than required just to preserve the approximation order the criteria are given that such edge functions must satisfy in order to allow multiresolution analysis of data sequences without special constraints on endpoint behavior or use of periodic boundary conditions and a specific choice is examined in applications to both noiseless and noisy data these are accomplished through the development of interval basis extensions for both an earlier multiwavelet projection based prefilter postfilter combination and a multiwavelet vector thresholding algorithm

large scale behavior of three dimensional continuous phase coexistence models we study a class of three dimensional continuous phase coexistence models and show that under different symmetry assumptions on the potential the large scale behavior of such models near a bifurcation point is described by the dynamical urn x wiley 00103640 media cpa21738 cpa21738 math 0001 models for p 2 3 4 this result is specific to space dimension 3 and does not hold in dimension 2

approximate counting the lovasz local lemma and inference in graphical models in this article we introduce a new approach to approximate counting in bounded degree systems with higher order constraints our main result is an algorithm to approximately count the number of solutions to a cnf formula when the width is logarithmic in the maximum degree this closes an exponential gap between the known upper and lower bounds moreover our algorithm extends straightforwardly to approximate sampling which shows that under lovasz local lemma like conditions it is not only possible to find a satisfying assignment it is also possible to generate one approximately uniformly at random from the set of all satisfying assignments our approach is a significant departure from earlier techniques in approximate counting and is based on a framework to bootstrap an oracle for computing marginal probabilities on individual variables finally we give an application of our results to show that it is algorithmically possible to sample from the posterior distribution in an interesting class of graphical models

singular spdes in domains with boundaries we study spaces of modelled distributions with singular behaviour near the boundary of a domain that in the context of the theory of regularity structures allow one to give robust solution theories for singular stochastic pdes with boundary conditions the calculus of modelled distributions established in hairer invent math 198 2 269 504 2014 https doi org 10 1007 s00222 014 0505 4 is extended to this setting we formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the dirichlet and neumann heat kernels these results are then used to provide solution theories for the kpz equation with dirichlet and neumann boundary conditions and for the 2d generalised parabolic anderson model with dirichlet boundary conditions in the case of the kpz equation with neumann boundary conditions we show that depending on the class of mollifiers one considers a boundary renormalisation takes place in other words there are situations in which a certain boundary condition is applied to an approximation to the kpz equation but the limiting process is the hopf cole solution to the kpz equation with a different boundary condition

algebraic renormalisation of regularity structures we give a systematic description of a canonical renormalisation procedure of stochastic pdes containing nonlinearities involving generalised functions this theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms this subgroup is sufficiently large to be able to implement a version of the bphz renormalisation prescription in this context this is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations our construction is based on bialgebras of decorated coloured forests in cointeraction more precisely we have two hopf algebras in cointeraction coacting jointly on a vector space which represents the generalised functions of the theory two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic birkhoff factorisation that arises naturally in perturbative quantum field theory

unique ergodicity for a class of stochastic hyperbolic equations with additive space time white noise in this paper we consider a certain class of second order nonlinear pdes with damping and space time white noise forcing posed on the d dimensional torus this class includes the wave equation for d 1 and the beam equation for d le 3 we show that the gibbs measure is the unique invariant measure for this system since the flow does not satisfy the strong feller property we introduce a new technique for showing unique ergodicity this approach may be also useful in situations in which finite time blowup is possible

exponential loss of memory for the 2 dimensional allen cahn equation with small noise we prove an asymptotic coupling theorem for the 2 dimensional allen cahn equation perturbed by a small space time white noise we show that with overwhelming probability two profiles that start close to the minimisers of the potential of the deterministic system contract exponentially fast in a suitable topology in the 1 dimensional case a similar result was shown in martinelli et al commun math phys 120 1 25 69 1988 j stat phys 55 3 4 477 504 1989 it is well known that in two or more dimensions solutions of this equation are distribution valued and the equation has to be interpreted in a renormalised sense formally this renormalisation corresponds to moving the minima of the potential infinitely far apart and making them infinitely deep we show that despite this renormalisation solutions behave like perturbations of the deterministic system without renormalisation they spend large stretches of time close to the minimisers of the un renormalised potential and the exponential contraction rate of different profiles is given by the second derivative of the potential in these points as an application we prove an eyring kramers law for the transition times between the stable solutions of the deterministic system for fixed initial conditions

optimal local well posedness for the periodic derivative nonlinear schrodinger equation we prove local well posedness for the periodic derivative nonlinear schrodinger equation which is l 2 critical in fourier lebesgue spaces which scale like h s mathbb t for s 0 our result is optimal in the sense that it covers the full subcritical regime in particular we close the existing gap in the subcritical theory by improving the result of grunrock and herr siam j math anal 39 6 1890 1920 2008 which established local well posedness in fourier lebesgue spaces which scale like h s mathbb t for s frac 1 4 we achieve this result by a delicate analysis of the structure of the solution and the construction of an adapted nonlinear submanifold of a suitable function space together these allow us to construct the unique solution to the given subcritical data this constructive procedure is inspired by the theory of para controlled distributions developed by gubinelli et al forum math pi 3 75 2015 and catellier and chouk ann probab 46 5 2621 2679 2018 in the context of stochastic pde our proof and results however are purely deterministic

biorthogonal multiresolution analysis on a triangle and applications we present in this paper new constructions of biorthogonal multiresolution analysis on the triangle we use direct method based on the tensor product to construct dual scaling spaces on next we construct the associated wavelet spaces and we prove that the associated wavelets have compact support and preserve the original regularity finally we describe some regular results which are very useful to establish the norm equivalences as applications we prove that the wavelet bases constructed in this paper are adapted for the study of the sobolev spaces h 0 s and h s s n and are easy to implement wavelets have unfolded their full computation efficiently in numerical and applied analysis the properties of wavelet bases provide a rigorous analysis for dynamical systems we construct in this work biorthogonal wavelet bases on a triangle these bases are adapted to the study of the sobolev spaces the bases allow many concrete numerical examples as numerical simulation for elliptic problems or image processing

the infinitesimal generator of the stochastic burgers equation we develop a martingale approach for a class of singular stochastic pdes of burgers type including fractional and multi component burgers equations by constructing a domain for their infinitesimal generators it was known that the domain must have trivial intersection with the usual cylinder test functions and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions using the new domain we are able to prove existence and uniqueness for the kolmogorov backward equation and the martingale problem we also extend the uniqueness result for energy solutions of the stochastic burgers equation of gubinelli and perkowski j am math soc 31 2 427 471 2018 to a wider class of equations as applications of our approach we prove that the stochastic burgers equation on the torus is exponentially l 2 ergodic and that the stochastic burgers equation on the real line is ergodic

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

the dynamic phi 4_3 φ 3 4 model comes down from infinity we prove an a priori bound for the dynamic phi 4 3 model on the torus which is independent of the initial condition in particular this bound rules out the possibility of finite time blow up of the solution it also gives a uniform control over solutions at large times and thus allows one to construct invariant measures via the krylov bogoliubov method it thereby provides a new dynamic construction of the euclidean phi 4 3 field theory on finite volume our method is based on the local in time solution theory developed recently by gubinelli imkeller perkowski and catellier chouk the argument relies entirely on deterministic pde arguments such as embeddings of besov spaces and interpolation which are combined to derive energy inequalities

on the well posedness of the stochastic allen cahn equation in two dimensions white noise driven nonlinear stochastic partial differential equations spdes of parabolic type are frequently used to model physical systems in space dimensions d 1 2 3 whereas existence and uniqueness of weak solutions to these equations are well established in one dimension the situation is different for d 2 despite their popularity in the applied sciences higher dimensional versions of these spde models are generally assumed to be ill posed by the mathematics community we study this discrepancy on the specific example of the two dimensional allen cahn equation driven by additive white noise since it is unclear how to define the notion of a weak solution to this equation we regularize the noise and introduce a family of approximations based on heuristic arguments and numerical experiments we conjecture that these approximations exhibit divergent behavior in the continuum limit the results strongly suggest that shrinking the mesh size in simulations of the two dimensional white noise driven allen cahn equation does not lead to the recovery of a physically meaningful limit

new constructive aspects of the lovasz local lemma the lovasz local lemma lll is a powerful tool that gives sufficient conditions for avoiding all of a given set of bad events with positive probability a series of results have provided algorithms to efficiently construct structures whose existence is non constructively guaranteed by the lll culminating in the recent breakthrough of moser and tardos 2010 for the full asymmetric lll we show that the output distribution of the moser tardos algorithm well approximates the conditional lll distribution the distribution obtained by conditioning on all bad events being avoided we show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results we also show that when a lll application provides a small amount of slack the number of resamplings of the moser tardos algorithm is nearly linear in the number of underlying independent variables not events and can thus be used to give efficient constructions in cases where the underlying proof applies the lll to super polynomially many events even in cases where finding a bad event that holds is computationally hard we show that applying the algorithm to avoid a polynomial sized core subset of bad events leads to a desired outcome with high probability this is shown via a simple union bound over the probabilities of non core events in the conditional lll distribution and automatically leads to simple and efficient monte carlo and in most cases rnc algorithms we demonstrate this idea on several applications we give the first constant factor approximation algorithm for the santa claus problem by making a lll based proof of feige constructive we provide monte carlo algorithms for acyclic edge coloring nonrepetitive graph colorings and ramsey type graphs in all these applications the algorithm falls directly out of the non constructive lll based proof our algorithms are very simple often provide better bounds than previous algorithms and are in several cases the first efficient algorithms known as a second type of application we show that the properties of the conditional lll distribution can be used in cases beyond the critical dependency threshold of the lll avoiding all bad events is impossible in these cases as the first even nonconstructive result of this kind we show that by sampling a selected smaller core from the lll distribution we can avoid a fraction of bad events that is higher than the expectation max k sat is an illustrative example of this

strong solutions to the stochastic quantization equations we prove the existence and uniqueness of a strong solution of the stochastic quantization equation in dimension 2 for almost all initial data with respect to the invariant measure the method is based on a fixed point result in suitable besov spaces

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

exponential relaxation to equilibrium for a one dimensional focusing non linear schrodinger equation with noise we construct generalized grand canonical and canonical gibbs measures for a hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear schrodinger equation with a focusing nonlinearity of order p 6 key properties of these gibbs measures in particular absence of phase transitions and regularity properties of field samples are established we then study a time evolution of this system given by the hamiltonian evolution perturbed by a stochastic noise term that mimics effects of coupling the system to a heat bath at some fixed temperature the noise is of ornstein uhlenbeck type for the fourier modes of the field with the strength of the noise decaying to zero as the frequency of the mode tends to we prove exponential approach of the state of the system to a grand canonical gibbs measure at a temperature and chemical potential determined by the stochastic noise term

rough burgers like equations with multiplicative noise we construct solutions to vector valued burgers type equations perturbed by a multiplicative space time white noise in one space dimension due to the roughness of the driving noise solutions are not regular enough to be amenable to classical methods we use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution subject to the choice of the correct reference rough path we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise

rough stochastic pdes in this article we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic pdes of burgers type that exhibit too high spatial roughness for classical analytical methods to apply in fact the class of spdes that we consider is genuinely ill posed in the sense that different approximations to the nonlinearity may converge to different limits using rough path theory a pathwise notion of solution to these spdes is formulated and we show that this yields a well posed problem that is stable under a large class of perturbations including the approximation of the rough driving noise by a mollified version and the addition of hyperviscosity we also show that under certain structural assumptions on the coefficients the spdes under consideration generate a reversible markov semigroup with respect to a diffusion measure that can be given explicitly

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

meshfree methods and boundary conditions summary we perform here some meshfree methods to inhomogeneous laplace equations we prove the efficiency of those methods compared with classical ones for one or two dimensional case for numerics and for one dimensional for theoretical results copyright 2006 john wiley sons ltd

random walks that find perfect objects and the lovasz local lemma we give an algorithmic local lemma by establishing a sufficient condition for the uniform random walk on a directed graph to reach a sink quickly our work is inspired by moser s entropic method proof of the lovasz local lemma lll for satisfiability and completely bypasses the probabilistic method formulation of the lll in particular our method works when the underlying state space is entirely unstructured similarly to moser s argument the key point is that the inevitability of reaching a sink is established by bounding the entropy of the walk as a function of time

solution of cartesian and curvilinear quantum equations via multiwavelets on the interval abstract it is shown how orthogonal compact support multiwavelets may be used for the solution of quantum mechanical eigenvalue problems subject to specific boundary conditions special scaling functions and wavelets with convenient limiting behaviors at the edges of an interval are constructed in analogy to earlier work on single wavelet families all of the integrals required for hamiltonian matrix elements involving both regular and edge functions are calculated efficiently through use of recursion and quadrature methods it is demonstrated through accurate eigenvalue determination that both cartesian and curvilinear degrees of freedom are readily accommodated with such a basis using as examples the particle in a box and the hydrogen atom in spherical polar coordinates

compactly supported wavelets and boundary conditions abstract wavelet bases subject to various homogeneous boundary conditions on an interval are constructed they are obtained from daubechies compactly supported wavelets which are restricted to the interval following an idea by y meyer and then suitably modified the sobolev spaces arising from homogeneous boundary problems are characterized in terms of wavelet coefficients in these bases

a two space dimensional semilinear heat equation perturbed by gaussian white noise a two space dimensional heat equation perturbed by a white noise in a bounded volume is considered the equation is perturbed by a non linearity of the type f au where means wick re ordering with respect to the free solution a are small parameters u denotes a solution f is the fourier transform of a complex measure with compact support

deterministic algorithms for the lovasz local lemma the lovasz local lemma lll p erdos and l lovasz problems and results on 3 chromatic hypergraphs and some related questions in infinite and finite sets vol ii a hajnal r rado and v t sos eds north holland amsterdam 1975 pp 609 627 is a powerful result in probability theory that informally states the following the probability that none of a set of bad events happens is positive if the probability of each event is small compared to the number of events that depend on it the lll is often used for nonconstructive existence proofs of combinatorial structures a prominent application is to k cnf formulas where the lll implies that if every clause in a formula shares variables with at most d leq 2 k e 1 other clauses then such a formula has a satisfying assignment recently a randomized algorithm to efficiently construct a satisfying assignment in this setting was given by moser a constructive proof of the lovasz local lemma in stoc 09 proceedings of the 41st annual acm sym

uniform sampling through the lovasz local lemma we propose a new algorithmic framework called partial rejection sampling to draw samples exactly from a product distribution conditioned on none of a number of bad events occurring our framework builds new connections between the variable framework of the lovasz local lemma and some classical sampling algorithms such as the cycle popping algorithm for rooted spanning trees among other applications we discover new algorithms to sample satisfying assignments of k cnf formulas with bounded variable occurrences