impacts of noise on a class of partial differential equations abstract this paper is concerned with effects of noise on the solutions of partial differential equations we first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic partial differential equations then we prove that noise could induce singularities finite time blow up of solutions finally we show that a stochastic allen cahn equation does not have finite time singularities and the unique solution exists globally

atmospheres of polygons and knotted polygons in this paper we define two statistics a and a the positive and negative atmospheres of a lattice polygon of fixed length n these statistics have the property that a a pn 2 pn where pn is the number of polygons of length n counted modulo translations we use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres using these data we directly estimate the growth constants of polygons in two and three dimensions we find that where the error bars are 67 confidence intervals and the second error bar in the three dimensional estimate of is an estimated systematic error we also compute atmospheres of polygons of fixed knot type k sampled by the bfacf algorithm we discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n

monotonicity of the number of self avoiding walks it is shown that the numberc n of self avoiding walks of lengthn in d is an increasing function ofn

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

large deviations for a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the non gaussian stochastic reaction diffusion equation srde partial t nu varepsilon mathscr l nu varepsilon f x nu varepsilon varepsilon sigma x nu varepsilon ddot w tx as a random perturbation of the deterministic rde partial t nu 0 mathscr l nu 0 f x nu 0 here the space variable takes values on the unit circle s 1 and mathscr l is a strongly elliptic second order operator with constant coefficients the functions f and sigma are sufficiently regular so that there is a unique solution to the above srde for any continuous initial condition we also assume that there are positive constants m and m such that m leq sigma x y leq m for all x in s 1 and all y in mathbb r the perturbation ddot w tx is the formal derivative of a brownian sheet it is known that if the initial condition is continuous then the solution will also be continuous and moreover if the initial condition is assumed to be holder continuous of exponent kappa for some 0 kappa frac 1 2 then the solution will be holder continuous of exponent kappa 2 as a function of t x in this paper we establish the large deviations principle for nu varepsilon in the holder norm of exponent kappa 2 when the initial condition is holder continuous of exponent kappa for any 0 kappa frac 1 2 and when the initial condition is assumed only to be continuous we establish the large deviations principle for nu varepsilon in the supremum norm moreover we prove that these large deviations principles are uniform with respect to the initial condition

on the effect of a noise on the solutions of the focusing supercritical nonlinear schrodinger equation we investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear schrodinger equation we prove that contrary to the deterministic case any initial data gives birth to a solution which develops singularities moreover the singularities appear immediately we use a stochastic generalization of the variance identity and a control argument

pruned enriched rosenbluth method simulations of θ polymers of chain length up to 1 000 000 we present an algorithm for simulating flexible chain polymers it combines the rosenbluth rosenbluth method with recursive enrichment although it can be applied also in more general situations it is most efficient for three dimensional theta polymers on the simple cubic lattice there it allows high statistics simulations of chains of length up to n 10 sup 6 for storage reasons this is feasable only for polymers in a finite volume for free theta polymers in infinite volume we present very high statistics runs with n 10000 these simulations fully agree with previous simulations made by hegger and grassberger j chem phys bold 102 6681 1995 with a similar but less efficient algorithm showing that logarithmic corrections to mean field behavior are much stronger than predicted by field theory but the finite volume simulations show that the density inside a collapsed globule scales with the distance from the theta point as predicted by mean field theory in contrast to claims in the work mentioned above in addition to the simple cubic lattice we also studied two versions of the bond fluctuation model but with much shorter chains finally we show that our method can be applied also to off lattice models and illustrate this with simulations of amore model studied in detail by freire ital et al macromolecules bold 19 452 1986 and later work copyright ital 1997 ital the american physical society less

uniqueness and blow up for a stochastic viscous dyadic model we consider the dyadic model with viscosity and additive gaussian noise as a simplified version of the stochastic navier stokes equations with the purpose of studying uniqueness and emergence of singularities we prove pathwise uniqueness and absence of blow up in the intermediate intensity of the non linearity morally corresponding to the 3d case and blow up for stronger intensity moreover blow up happens with probability one for regular initial data

enumeration of self avoiding walks on the square lattice we describe a new algorithm for the enumeration of self avoiding walks on the square lattice using up to 128 processors on a hp alpha server cluster we have enumerated the number of self avoiding walks on the square lattice to length 71 series for the metric properties of mean square end to end distance mean square radius of gyration and mean square distance of monomers from the end points have been derived to length 59 an analysis of the resulting series yields accurate estimates of the critical exponents and confirming predictions of their exact values likewise we obtain accurate amplitude estimates yielding precise values for certain universal amplitude combinations finally we report on an analysis giving compelling evidence that the leading non analytic correction to scaling exponent 1 3 2

long time existence for the heat equation with a noise term we consider the equationu t u xx u w forx on a finite interval with dirichlet boundary conditions w is spacetime white noise the initial condition is continuous and nonnegative we show existence and uniqueness for all time provided 1 3 2

large deviations for the invariant measure of a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the invariant measure of the non gaussian stochastic partial differential equation spde t v e v e f x v e e x v e ddot w tx here is a strongly elliptic second order operator with constant coefficients h dh xx h and the space variablex takes values on the unit circles 1 the functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic pde and in particular we require that 0 m m wherem andm are some finite positive constants the perturbationw is a brownian sheet it is well known that under some simple assumptions the solutionv 2 is ac k s 1 valued markov process for each 0 1 2 wherec s 1 is the banach space of real valued continuous functions ons 1 which are holder continuous of exponent we prove under some further natural assumptions onf and which imply that the zero element ofc s 1 is a globally exponentially stable critical point of the unperturbed equation t 0 0 f x 0 that e has a unique stationary distributionv k on c s 1 c k s 1 when the perturbation parameter e is small enough some further calculations show that as e tends to zero v k tends tov k 0 the point mass centered on the zero element ofc s 1 the main goal of this paper is to show that in factv k is governed by a large deviations principle ldp our starting point in establishing the ldp forv k is the ldp for the process e which has been shown in an earlier paper our methods of deriving the ldp forv k based on the ldp for e are slightly non standard compared to the corresponding proofs for finite dimensional stochastic differential equations since the state spacec s 1 is inherently infinite dimensional

the number of smallest knots on the cubic lattice it has been shown that the smallest knots on the cubic lattice are all trefoils of length 24 in this paper we show that the number of such unrooted knots on the cubic lattice is 3496

some non existence results for a class of stochastic partial differential equations abstract consider the following stochastic partial differential equation t u t x l u t x u t x f t x t 0 and x r d the operator l is the generator of a strictly stable process and f is the random forcing term which is assumed to be gaussian under some additional conditions most notably on and the initial condition we show non existence of global random field solutions our results complement those of p chow in 5 and 6 among others

self avoiding walks neighbour avoiding walks and trails on semiregular lattices we study self avoiding and neighbour avoiding walks and lattice trails on two semiregular lattices the lattice and the lattice for the lattice we find the exact connective constant for both self avoiding walks neighbour avoiding walks and trails for the lattice we generate long series which permit the accurate estimation of the connective constant for self avoiding walks and trails

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

the pivot algorithm and polygons results on the fcc lattice the authors consider the pivot algorithm for polygons on the face centred cubic lattice and prove ergodicity a numerical study of the algorithm is carried out in particular the authors look at the acceptance fraction of the elementary moves and find that it goes only slowly to zero as the length of the polygons is increased the numerical properties of polygons on the fcc lattice are also considered they calculate the exponent nu by considering the mean square radius of gyration of the polygons and compare the result to the expected value from field theory they also consider the mean span of the polygons and discuss corrections to scaling and their influence on the numerical calculation of critical exponents

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

new monte carlo method for the self avoiding walk we introduce a new monte carlo algorithm for the self avoiding walk saw and show that it is particularly efficient in the critical region long chains we also introduce new and more efficient statistical techniques we employ these methods to extract numerical estimates for the critical parameters of the saw on the square lattice we find 2 63820 0 00004 0 00030 1 352 0 006 0 025 v 0 7590 0 0062 0 0042 where the first error bar represents systematic error due to corrections to scaling subjective 95 confidence limits and the second bar represents statistical error classical 95 confidence limits these results are based on saws of average length 166 using 340 hours cpu time on a cdc cyber 170 730 we compare our results to previous work and indicate some directions for future research

self avoiding walk enumeration via the lace expansion we introduce a new method for the enumeration of self avoiding walks based on the lace expansion we also introduce an algorithmic improvement called the two step method for self avoiding walk enumeration problems we obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3 we enumerate 32 step self avoiding polygons in d 3 26 step self avoiding polygons in d 4 30 step self avoiding walks in d 3 and 24 step self avoiding walks and polygons in all dimensions d 4 we analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8 we also provide major extensions of 1 d expansions for the connective constant and for two critical amplitudes

a phase transition for a stochastic pde related to the contact process we consider the one dimensional heat equation with a semilinear term and with a nonlinear white noise term r durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions we show that our equation possesses a phase transition to be more precise we assume that the initial function is nonnegative with bounded total mass if a certain parameter in the equation is small enough then the solution dies out to 0 in finite time with probability 1 if this parameter is large enough then the solution has a positive probability of never dying out to 0 this result answers a question of durett

the bfacf algorithm and knotted polygons the bfacf algorithm applied to polygons involves sampling on a markov chain whose state space is the set of all polygons in three dimensions for the simple cubic lattice the authors prove that the ergodic classes of this markov chain are the knot classes of the polygons

the approximate decomposition of exponential order of slow fast motions in multifrequency systems abstract this paper deals with multifrequency slow fast systems it is shown that under a suitable change of coordinates the system can be reduced to a simple form such that slow motions are described by autonomous equations except for exponential error of perturbations hence the fast and slow motions are decoupled the newton rapid iteration is used in addition for a perturbation only the smallness condition is needed

time space white noise eliminates global solutions in reaction diffusion equations abstract we prove that perturbing the reaction diffusion equation u t u x x u p p 1 with time space white noise produces that solutions explodes with probability one for every initial datum opposite to the deterministic model where a positive stationary solution exists

explosive solutions of stochastic reaction diffusion equations in mean lp norm abstract the paper is concerned with the problem of non existence of global solutions for a class of stochastic reaction diffusion equations of ito type under some sufficient conditions on the initial state the nonlinear term and the multiplicative noise it is proven that in a bounded domain d r d there exist positive solutions whose mean l p norm will blow up in finite time for p 1 while if d r d the previous result holds in any compact subset of r d two examples are given to illustrate some application of the theorems

ergodicity of the bfacf algorithm in three dimensions let omega be a walk in z3 let the endpoints of omega be fixed at x and gamma in z3 suppose that d x gamma maxi xi x x i gamma mod where x i3 x is the ith coordinate of the vertex x in z then the bfacf algorithm is ergodic for walks joining x and gamma provided that d x gamma 1

near invariant tori on exponentially long time for poisson systems this paper deals with the near invariant tori for poisson systems it is shown that the orbits with the initial points near the diophantine torus approach some quasi periodic orbits over an extremely long time in particular the results hold for the classical hamiltonian system and in this case the drift of the motions is smaller than the one in the past works

new lower bounds on the self avoiding walk connective constant we give an elementary new method for obtaining rigorous lower bounds on the connective constant for self avoiding walks on the hypercubic lattice d the method is based on loop erasure and restoration and does not require exact enumeration data our bounds are best for highd and in fact agree with the first four terms of the 1 d expansion for the connective constant the bounds are the best to date for dimensionsd 3 but do not produce good results in two dimensions ford 3 4 5 and 6 respectively our lower bound is within 2 4 0 43 0 12 and 0 044 of the value estimated by series extrapolation

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

blowup for the heat equation with a noise term in this paper we study blow up of the equation u t u xx u gamma dot w tx where dot w tx is a two dimensional white noise field and where dirichlet boundary conditions are enforced it is known that if 3 2 then the solution exists for all time in this paper we show that if is much larger than 3 2 then the solution blows up in finite time with positive probability we prove this by considering how peaks in the solution propagate if a peak of high mass forms we rescale the equation and divide the mass of the peak into a collection of peaks of smaller mass and these peaks evolve almost independently in this way we compare the evolution ofu to a branching process large peaks are regarded as particles in this branching process offspring are peaks which are higher by some factor we show that the expected number of offspring is greater than one when is much larger than 3 2 and thus the branching process survives with positive probability corresponding to blowup in finite time

asymptotics of knotted lattice polygons we use monte carlo methods to investigate the asymptotic behaviour of the number and mean square radius of gyration of polygons in the simple cubic lattice with fixed knot type let be the number of n edge polygons of a fixed knot type in the cubic lattice and let be the mean square radius of gyration of all the polygons counted by if we assume that where is the growth constant of polygons of knot type and is the entropic exponent of polygons of knot type then our numerical data are consistent with the relation where is the unknot and is the number of prime factors of the knot if we assume that then our data are consistent with both and being independent of these results support the claims made in janse van rensburg and whittington 1991a 24 3935 and orlandini et al 1996 29 l299 1998 topology and geometry in polymer science ima volumes in mathematics and its applications berlin springer