An Introduction to Stochastic PDEs

1 An Introduction to Stochastic PDEsJuly 24, 2009 Martin HairerThe University of Warwick / Courant InstituteContents1 ..22 Some Motivating model for a random string (polymer) .. Stochastic Navier-Stokes equations .. Stochastic heat equation .. have we learned? ..63 Gaussian Measure bounds on Gaussian measures .. Cameron-Martin space .. of Gaussian measures .. Wiener processes and Stochastic integration ..244 A Primer on Semigroup continuous semigroups .. with selfadjoint generators .. semigroups .. spaces ..395 Linear SPDEs / Stochastic and space regularity.

2 Behaviour .. in other topologies ..556 Semilinear solutions .. inequalities and Sobolev embeddings .. equations .. Stochastic Navier-Stokes equations ..721 IntroductionThese notes are based on a series of lectures given first at the University of Warwick in spring 2008and then at the Courant Institute in spring 2009. It is an attempt to give a reasonably self-containedpresentation of the basic theory of Stochastic partial differential equations, taking for granted basicmeasure theory, functional analysis and probability theory, but nothing else.

3 Since the aim wasto present most of the material covered in these notes during a 30-hours series of postgraduatelecture, such an attempt is doomed to failure unless drastic choices are made. This is why many2 SOMEMOTIVATINGEXAMPLES important facets of the theory of Stochastic PDEs are missing from these notes. In particular, wedonottreat equations with multiplicative noise, we donottreat equations driven L evy noise, wedonotconsider equations with rough (that is not locally Lipschitz, even in a suitable space)nonlinearities, we donottreat measure-valued processes, we donotconsider hyperbolic or ellipticproblems, we donotcover Malliavin calculus and densities of solutions, etc.

4 The reader who isinterested in a more detailed exposition of these more technically subtle parts of the theory mightbe advised to read the excellent works [DPZ92b, DPZ96, PZ07, PR07, SS05].Instead, the approach taken in these notes is to focus on semilinearparabolicproblems drivenbyadditivenoise. These can be treated as Stochastic evolution equations in some infinite-dimen-sional Banach or Hilbert space that usually have nice regularising properties and they alreadyform (in my humble opinion) a very rich class of problems with many interesting , this class of problems has the advantage of allowing to completely pass under silencemany subtle problems arising from Stochastic integration in infinite-dimensional AcknowledgementsThese notes would never have been completed, were it not for the enthusiasm of the attendants ofthe course.

5 Hundreds of typos and mistakes were spotted and corrected. I am particularly indebtedto David Epstein and Jochen Vo who carefully worked their way through these notes when theywere still in a state of wilderness. Special thanks are also due to Pavel Bubak who was runningthe tutorials for the course given in Some Motivating A model for a random string (polymer)TakeN+ 1particles with positionsunimmersed in a fluid and assume that nearest-neighboursare connected by harmonic springs. If the particles are furthermore subject to an external forcingF, the equations of motion (in the overdamped regime where the forces acting on the particle aremore important than inertia, which can also formally be seen as the limit where the masses of theparticles go to zero) would be given bydu0dt=k(u1 u0)+F(u0) ,dundt=k(un+1+un 1 2un)+F(un) ,n= 1.

6 ,N 1,duNdt=k(uN 1 uN)+F(uN).This is a primitive model for a polymer chain consisting ofN+ 1monomers and without self-interaction. It does however not take into account the effect of the molecules of water that wouldrandomly kick the particles that make up our string. Assuming that these kicks occur randomlyand independently at high rate, this effect can be modelled in first instance by independent whitenoises acting on all degrees of freedom of our model. We thus obtain a system of coupled stochas-tic differential equations:du0=k(u1 u0)dt+F(u0)dt+ dw0(t) ,dun=k(un+1+un 1 2un)dt+F(un)dt+ dwn(t) ,n= 1.

7 ,N 1,duN=k(uN 1 uN)dt+F(uN)dt+ dwN(t).SOMEMOTIVATINGEXAMPLES3 Formally taking the continuum limit (with the scalingsk N2and N), we can infer thatifNis very large, this system is well-described by the solution to a stochasticpartial differentialequationdu(x,t)= 2xu(x,t)dt+F(u(x,t))dt+dW(x,t) ,endowed with the boundary conditions xu(0,t)= xu(1,t)= 0. It is not so cleara prioriwhatthe meaning of the termdW(x,t) should be. We will see in the next section that, at least on aformal level, it is reasonable to assume thatEdW(x,t)dtdW(y,s)ds= (x y) (t s). The precisemeaning of this formula will be discussed The Stochastic Navier-Stokes equationsThe Navier-Stokes equations describing the evolution of the velocity fieldu(x,t) of an incom-pressible viscous fluid are given bydudt= u (u )u p+f,( )complemented with the (algebraic) incompressibility condition divu= 0.

8 Here,fdenotes someexternal force acting on the fluid, whereas the pressurepis given implicitly by the requirementthat divu= 0at all it is not too difficult in general to show that solutions to ( ) exist in some weak sense,in the case wherex Rdwithd 3, theiruniquenessis an open problem with a $1,000,000prize. We will of course not attempt to solve this long-standing problem, so we are going torestrict ourselves to the cased= 2. (The cased= 1makes no sense since there the conditiondivu= 0would imply thatuis constant. However, one could also consider the Burger s equationwhich has similar features to the Navier-Stokes equations.)

9 For simplicity, we consider solutions that are periodic in space, so that we viewuas a functionfromT2 R+toR2. In the absence of external forcingf, one can use the incompressibilityassumption to see thatddt T2|u(x,t)|2dx= 2 T2trDu(x,t) Du(x,t)dx 2 T2|u(x,t)|2dx,where we used the Poincar e inequality in the last line (assuming that T2u(x,t)dx= 0). There-fore, by Gronwall s inequality, the solutions decay to0exponentially fast. This shows that energyneeds to be pumped into the system continuously if one wishes to maintain an interesting way to achieve this from a mathematical point of view is to add a forcefthat is randomlyfluctuating.

10 We are going to show that if one takes a random force that is Gaussian and such thatEf(x,t)f(y,s)= (t s)C(x y) ,for some correlation functionCthen, provided thatCis sufficiently regular, one can show that( ) has solutions for all times. Furthermore, these solutions do not blow up in the sense that onecan find a constantKsuch that, for any solution to ( ), one haslim supt E u(t) 2 K,for some suitable norm . This allows to provide a construction of a model for homogeneousturbulence which is amenable to mathematical The Stochastic heat equationIn this section, we focus on the particular example of the Stochastic heat equation.