constrained rough paths we introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural on the way we develop a notion of rough integration and an ecient and intrinsic theory of rough dierential equations rdes on manifolds the theory of rdes is then used to construct parallel translation along manifold valued rough paths finally this framework is used to show there is a one to one correspondence between rough paths on a d dimensional manifold and rough paths on d dimensional euclidean space this last result is a rough path analogue of cartan s development map and its stochastic version which was developed by eells and elworthy and malliavin

a spatial version of the ito stratonovich correction we consider a class of stochastic pdes of burgers type in spatial dimension 1 driven by space time white noise even though it is well known that these equations are well posed it turns out that if one performs a spatial discretization of the nonlinearity in the wrong way then the sequence of approximate equations does converge to a limit but this limit exhibits an additional correction term this correction term is proportional to the local quadratic cross variation in space of the gradient of the conserved quantity with the solution itself this can be understood as a consequence of the fact that for any fixed time the law of the solution is locally equivalent to wiener measure where space plays the role of time in this sense the correction term is similar to the usual ito stratonovich correction term that arises when one considers different temporal discretizations of stochastic odes

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

the stochastic burgers equation we study burgers equation perturbed by a white noise in space and time we prove the existence of solutions by showing that the cole hopf transformation is meaningful also in the stochastic case the problem is thus reduced to the anaylsis of a linear equation with multiplicative half white noise an explicit solution of the latter is constructed through a generalized feynman kac formula typical properties of the trajectories are then discussed a technical result concerning the regularizing effect of the convolution with the heat kernel is proved for stochastic integrals

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

rough evolution equations we generalize lyons rough paths theory in order to give a pathwise meaning to some nonlinear infinite dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise as an illustration we discuss a class of linear and nonlinear 1d spdes driven by a space time gaussian noise with singular space covariance and brownian time dependence

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

erratum to rough burgers like equations with multiplicative noise unfortunately the proof of proposition 4 8 in hw13 is not correct in fact equation 4 60 wherer is rewritten as a stochastic integral is not correct because the integrand is not adapted to the filtration hence in 4 61 we are not allowed to apply the burkholder davies gundy inequality the argument was corrected in hmw12 lemma 3 6 and corollary 3 7 in a more complicated situation in order to transfer these statement to the situation in hw13 some smaller changes are needed first of all the regularity of the linearised process x should be measured in a holder norm with slightly bigger index than the solution u this can be done without further problems then the definition of the stopping time in 4 53 becomes for k 0 and for an 1 2 we introduce the stopping time

temporal integrators for fluctuating hydrodynamics including the effect of thermal fluctuations in traditional computational fluid dynamics requires developing numerical techniques for solving the stochastic partial differential equations of fluctuating hydrodynamics these langevin equations possess a special fluctuation dissipation structure that needs to be preserved by spatio temporal discretizations in order for the computed solution to reproduce the correct long time behavior in particular numerical solutions should approximate the gibbs boltzmann equilibrium distribution and ideally this will hold even for large time step sizes we describe finite volume spatial discretizations for the fluctuating burgers and fluctuating incompressible navier stokes equations that obey a discrete fluctuation dissipation balance principle just like the continuum equations we develop implicit explicit predictor corrector temporal integrators for the resulting stochastic method of lines discretization these stochastic runge kutta schemes treat diffusion implicitly and advection explicitly are weakly second order accurate for additive noise for small time steps and give a good approximation to the equilibrium distribution even for very strong fluctuations numerical results demonstrate that a midpoint predictor corrector scheme is very robust over a broad range of time step sizes

heat semigroup and singular pdes we provide in this work a semigroup approach to the study of singular pdes in the line of the paracontrolled approach developed recently by gubinelli imkeller and perkowski starting from a heat semigroup we develop a functional calculus and introduce a paraproduct based on the semigroup for which commutator estimates and schauder estimates are proved together with their paracontrolled extensions this machinery allows us to investigate singular pdes in potentially unbounded riemannian manifolds under mild geometric conditions as an illustration we study the generalized parabolic anderson model equation and prove under mild geometric conditions its well posed character in small time on a potentially unbounded 2 dimensional riemannian manifold for an equation driven by a coloured noise and for all times for the linear parabolic anderson model equation in 2 dimensional unbounded manifolds this machinery can be extended to an even more singular setting and deal with sobolev spaces rather than holder spaces

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

weighted stochastic sobolev spaces and bilinear spdes driven by space time white noise in this paper we develop basic elements of malliavin calculus on a weightedl2 this class of generalized wiener functionals is a hilbert space it turns out to be substantially smaller than the space of hida distributions while large enough to accommodate solutions of bilinear stochastic pdes as an example we consider a stochastic advection diffusion equation driven by space time white noise in rd it is known that ford 1 this equation has no solutions inl2 in contrast it is shown in the paper that in an appropriately weightedl2 there is a unique solution to the stochastic advection diffusion equation for anyd 1 in addition we present explicit formulas for the hermite fourier coefficients in the wiener chaos expansion of the solution

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 a modelled rough heat equation we use the formalism of hairer s regularity structures theory to study a heat equation model with non linear perturbation driven by a space time fractional noise different regimes are observed depending on the global pathwise roughness of the noise to this end and following the procedure exhibited in hairer invent math 198 2 269 504 2014 the equation is first lifted into some abstract regularity structure and therein solved through a fixed point argument then we construct a consistent model above the fractional noise by relying on a smooth fourier type approximation of the process

point process model of 1 f noise vs a sum of lorentzians we present a simple point process model of 1 f noise covering different values of the exponent the signal of the model consists of pulses or events the interpulse interevent interarrival recurrence or waiting times of the signal are described by the general langevin equation with the multiplicative noise and stochastically diffuse in some interval resulting in a power law distribution our model is free from the requirement of a wide distribution of relaxation times and from the power law forms of the pulses it contains only one relaxation rate and yields 1 f spectra in a wide range of frequencies we obtain explicit expressions for the power spectra and present numerical illustrations of the model further we analyze the relation of the point process model of 1 f noise with the bernamont surdin mcwhorter model representing the signals as a sum of the uncorrelated components we show that the point process model is complementary to the model based on the sum of signals with a wide range distribution of the relaxation times in contrast to the gaussian distribution of the signal intensity of the sum of the uncorrelated components the point process exhibits asymptotically a power law distribution of the signal intensity the developed multiplicative point process model of 1 f noise may be used for modeling and analysis of stochastic processes in different systems with the power law distribution of the intensity of pulsing signals

scaling limits of wick ordered kpz equation consider the kpz equation x d where w t x is a space time white noise this paper investigates the question of whether for some exponents and z k u k z t kx converges in some sense as k to infty and if so what are the values of these exponents the non linear term in the kpz equation is interpreted as a wick product and the equation is solved in a suitable space of stochastic distributions the main tools for establishing the scaling properties of the solution are those of white noise analysis in particular the wiener chaos expansion a notion of convergence in law in the sense of wiener chaos is formulated and convergence in this sense of k u k z t kx as k is established for various values of and z depending on the dimension d

a priori estimates for rough pdes with application to rough conservation laws we introduce a general framework to study pdes driven by rough paths we develop new a priori estimates based on a rough gronwall lemma argument for weak solutions to rough pdes this will allow us to follow standard pde strategies to obtain existence and uniqueness results in particular our approach does not rely on any sort of transformation formula flow transformation feynman kac representation formula etc and is therefore rather flexible as an application we study conservation laws driven by rough paths establishing well posedness for the corresponding kinetic formulation

a long range memory stochastic model of the return in financial markets abstract we present a nonlinear stochastic differential equation sde which mimics the probability density function pdf of the return and the power spectrum of the absolute return in financial markets absolute return as a measure of market volatility is considered in the proposed model as a long range memory stochastic variable the sde is obtained from the analogy with an earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework the proposed stochastic model generates time series of the return with two power law statistics i e the pdf and the power spectral density reproducing the empirical data for the one minute trading return in the nyse

singular perturbations to semilinear stochastic heat equations we consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof the interesting feature of these perturbations is that as the small parameter e tends to zero their solutions converge to the wrong limit i e they do not converge to the solution obtained by simply setting e 0 a similar effect is also observed for some formally small stochastic perturbations of a deterministic semilinear parabolic pde our proofs are based on a detailed analysis of the spatially rough component of the equations combined with a judicious use of gaussian concentration inequalities

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

a white noise approach to a class of non linear stochastic heat equations abstract cauchy problems for a class of non linear stochastic evolution equations are studied they are formulated as integral equations for generalized random fields by methods of white noise analysis s transformation characterization theorem etc these problems are reduced to fixed point problems in appropriatly constructed banach spaces this technique provides a systematic treatment of existence and uniqueness questions for a variety of equations the method is applied to non linear heat equations non linear volterra equations and non linear ordinary differential equations which may all be anticipating for each case examples are given such as the stochastic burgers equation and stochastic reaction diffusion equations

the jain monrad criterion for rough paths and applications to random fourier series and non markovian hormander theory we discuss stochastic calculus for large classes of gaussian processes based on rough path analysis our key condition is a covariance measure structure combined with a classical criterion due to jain and monrad ann probab 11 1983 46 57 this condition is verified in many examples even in absence of explicit expressions for the covariance or volterra kernels of special interest are random fourier series with covariance given as fourier series itself and we formulate conditions directly in terms of the fourier coefficients we also establish convergence and rates of convergence in rough path metrics of approximations to such random fourier series an application to spde is given our criterion also leads to an embedding result for cameron martin paths and complementary young regularity cyr of the cameron martin space and gaussian sample paths cyr is known to imply malliavin regularity and also ito like probabilistic estimates for stochastic integrals resp stochastic differential equations despite their rough pathwise construction at last we give an application in the context of non markovian hormander theory

maximal monotone model with history term abstract models involving maximal monotone terms and history delay term are considered in a mathematical and numerical point of view mechanical example of elasto plastic material with creep is presented

explanation of power law behavior of autoregressive conditional duration processes based on the random multiplicative process autoregressive conditional duration acd processes which have the potential to be applied to power law distributions of complex systems found in natural science life science and social science are analyzed both numerically and theoretically an acd 1 process exhibits the singular second order moment which suggests that its probability density function pdf has a power law tail it is verified that the pdf of the acd 1 has a power law tail with an arbitrary exponent depending on a model parameter on the basis of theory of the random multiplicative process a relation between the model parameter and the power law exponent is theoretically derived it is confirmed that the relation is valid from numerical simulations an application of the acd 1 to intervals between two successive transactions in a foreign currency market is shown

study of a maximal monotone model with a delay term we present a model involving a friction term and a delay term by using results of existence and uniqueness for maximal monotone differential inclusions we give theoretical results for this model an implicit euler numerical scheme and results related to order of convergence are also provided from both a theoretical and a numerical point of view