Dipartimento di Matematica
Corso di Laurea Magistrale in Matematica
Energy solutions for the stationary
Kardar-Parisi-Zhang equation
Tesi di Laurea Magistrale
Relatore
Prof. Franco Flandoli
Controrelatore
Prof. Marco Romito
Candidato
Francesco Grotto
Contents
1 Introduction 5
1.1 White Noise and Singularity . . . 5
1.2 Defining Solutions to KPZ . . . 6
1.3 Preliminaries and Notation . . . 7
2 The infinite-dimensional Ornstein-Uhlenbeck process 9 2.1 The Ornstein-Uhlenbeck Process . . . 9
2.2 Heuristic remarks on Burgers’ drift . . . 11
2.3 Gaussian Analysis . . . 13
3 Controlled Processes and Energy Solutions 19 3.1 Regularisation estimates . . . 19
3.2 Burgers’ drift . . . 23
3.3 Energy Solutions . . . 27
4 The Hairer-Quastel invariance principle 29 4.1 Boltzmann-Gibbs principle . . . 29
4.2 Approximating equations and existence of energy solutions . . . 33
4.3 Convergence to energy solutions . . . 37
A Stochastic Analysis 39 A.1 Pathwise Stochastic Calculus . . . 39
A.2 Gaussian Spaces and White Noise . . . 40
A.3 The Support of White Noise Measure . . . 41
A.4 Chaos decomposition . . . 43
A.5 Invariant measures and reversibility . . . 45
Chapter 1
Introduction
Let us consider the following non-linear one-dimensional stochastic partial differential equa-tion, called the Kardar-Parisi-Zhang equaequa-tion,
∂th = ν∆h + λ(∂xh)2+
√
Dξ. (KPZ) This equation has been introduced in a celebrated Physics paper in 1986, [KPZ], to give a universal description of growing interfaces fluctuations in one dimension, that is, on the real line R or the 1-dimensional torus T. A vast Physics literature covers the topic; for a general review we refer to the notes [Q12]. From a mathematical point of view, the equation is ill-posed and can not be understood by means of classical SPDEs techniques. Indeed, it is difficult even to give a meaningful definition of solution, and only very recent developments in the theory of singular SPDEs have been able to tackle the problem. The aim of the present thesis is to review one of these new approaches, namely, Energy Solutions.
In the following sections we provide an overview on: mathematical ill-posedness of (KPZ), physical motivations and universality conjectures, the Cole-Hopf solutions and the first con-vergence result ever obtained, some remarks on the literature regarding (KPZ) and a list of the results contained in the thesis. The last section of this introductive chapter is dedicated to notation.
1.1
White Noise and Singularity
Let us begin by defining the objects involved in the equation (KPZ), which we will study on the torus T = R/2πZ. First of all, ξ is a space-time white noise on T, which means that it is the (unique in law) Gaussian process indexed by L2([0, T ] × T) with covariance
E[ξ(φ)ξ(ψ)] = hφ, ψiL2([0,T ]×T), ∀φ, ψ ∈ L2([0, T ] × T),
where we have fixed a time horizon T > 0. We refer the reader to Appendix A and the refer-ences therein for equivalent definitions and the properties of white noise we will use through-out the thesis. The noise ξ can be viewed as a space-time random distribution, so we will look for solutions of (KPZ) which are stochastic processes with trajectories C([0, T ],S0(T)), whereS0(T) is the space of Schwarz distributions (see [Ru]). The derivatives ∂x, ∆ = ∂x2are
thus understood in distributional sense. As for the time derivative, since the white noise ξ defines, by Wt(ϕ) = ξ(1[0,t]ϕ), ϕ ∈ L2(T), a cylindrical Wiener process on L2(T) (see again
Appendix A), we can interpret the equation in Ito’s stochastic differential sense. Finally, we assume ν, D > 0 and λ ∈ R.
Now that we have clarified the meaning of quantities involved in (KPZ), we are ready to see why, in fact, it does not make sense as it is written. Consider the following stochastic Burgers’ equation,
∂tu = ν∆u + λ∂xu2+
√
D∂xξ, (SBE)
which is formally equivalent to (KPZ) by means of the change of variables ut= ∂xht. There
are many reasons to expect that (SBE) leaves invariant a multiple of the space white noise η, whose definition is perfectly analogous to the one of ξ, only on L2
(T). The precise invariant measure is pD/2νη. Just like ξ, η can be seen as a random distribution: more precisely η turns out to be an Hα-valued random variable for all the Sobolev spaces with α < 1/2.
Hence, it is clear that if we assume that utis a stationary process with white noise marginals,
which is what we expect from a stationary solution of (SBE), then the drift term ∂xu2t does
not make sense, for it is well known that we can not multiply such distributions. The same issue appears in the case of (KPZ), where we expect the measure of brownian sheet (the anti-derivative of white noise) to be invariant.
1.2
Defining Solutions to KPZ
Before any further rigorous mathematical discussion, it is good to review some of the un-derlying physical concepts. As we have said above, (KPZ) was proposed as a universal model for fluctuations of growing interfaces: let us state here, somewhat informally, the two universality conjectures which motivate the study of (KPZ).
Conjecture 1 (Strong Universality.). In the scaling limit with exponents 1-2-3, the fluctu-ations of any 1 + 1-dimensional model ˜h of a growing interface are the same of solutions h to (KPZ): lim λ→∞λ −1˜h(λ2x, λ3t) − ˜C λt = lim λ→∞λ −1h(λ2x, λ3t) − C λt.
Conjecture 2 (Weak Universality.). For any “natural” family of antisymmetric growing interface models hε, with ε a parameter measuring antisymmetry such that the propagation
speed is proportional to √ε, as ε → 0 there exist constants Cε ' ε−1 such that the scaling
limit with exponents 1-2-4 of √
εhε(ε−1x, ε−2t) − Cεt
converges to solutions of (KPZ).
With “natural” we mean, for instance, the same discrete model with an increasing number of sites, or a model on an interval which extends to the whole line R in the limit. For more precise formulations and discussion about these conjectures we refer to [Q14] and [QS].
Both the conjectures assume a notion of solution to (KPZ), which is all but obvious as we have seen. From a mathematical point of view, however, the universality conjectures do are useful, since they allow us to understand whether a definition of solution is sensible: indeed, once we have mathematically defined what a solution is, if we are able to prove universality for some models in our setting, we have shown the definition to be “physical”. Of course, all such definition must turn out to be equivalent.
The first of those “good” notions of solution ever devised was the Cole-Hopf solution of (SBE) used by Bertini and Giacomin in [BG] to prove weak universality for an interact-ing particle system density field. They exploited the fact that, in a purely formal sense, the transformation Zt = exp λν∂x−1ut maps (SBE) to the multiplicative stochastic heat
equation
∂tZ = ∆Z + Zξ (mSHE)
which, in dimension 1, is well-posed if interpreted in Ito’s sense. Thus one can define solutions to (SBE) applying to the solution of (mSHE) the inverse transformation. In [BG],
however, the proof of convergence relied of a peculiar characteristic of the model, which allowed to perform an exponential transformation, hence the difficulty of any generalisation. The major mathematical breakthrough were the results obtained by Hairer in [Ha] and later (with Quastel) in [HQ], where existence and uniqueness, along with an invariance principle were derived for (KPZ) by means of the theory of Regularity Structures, for which Hairer was awarded the Fields Medal in 2014.
We will be concerned with a different approach to (KPZ), introduced by Gon¸calves and Jara ([GJ10] and [GJ14]) and later refined by Gubinelli and Jara ([GJ]): the concept of Energy Solutions will provide a notion of solution to the stationary (KPZ) by means of a martingale problem. Our setting will thus be somewhat more familiar than Hairer’s Regularity Structures, however it will require stationarity, which is the main drawback of the theory. Nonetheless, Energy Solutions have already proved to be a useful tool in the study of KPZ universality, and convergence of particle models to these solutions has been rigorously established (examples being [DGP], [FGS], [Go],[GJS]), whereas Regularity Structures look difficult to apply to particle problems. The only universality result in the latter case being [HQ], which considers SDEs not coming from particle systems. General reviews on this thesis’ matter can be found in [G16] and the recent paper [GP6].
Let us recall for the sake of completeness a third approach to singular equations such as (KPZ), namely the Paracontrolled Distributions. See [G04], [G10], [GIP], [GP4] and the very recent review [GP5].
1.3
Preliminaries and Notation
It is convenient to reduce our equations (KPZ) and (SBE) to standard parameters: by transforming ut7→ r 2ν Dut/ν, ξ(10,tφ) 7→ √ νξ(10,t/νφ),
we obtain the same equations with ν = 1 and D = 2, which assumed from now on. The parameter λ remains free, but we often set λ = 1, since the generic case does not differ much. Of course the linear case λ = 0 is another matter.
Most of our notation is standard and comes from the references we refer to. Let us only clarify the use of a couple of symbols.
• With ., ' and analogues we denote (in)equalities up to multiplicative constants. Those constants will not depend on relevant quantities involved in the statements, however we will sometimes point out the dependence of the constant on, say, the number p, writing .p.
• The brakets h·, ·i denote inner products of Hilbert spaces or duality couplings between topological dual spaces. The correct meaning will be clear from the context or made clear with a subscript. Since we are going to deal with both real and complex objects, to simplify notation we assume from now on that h·, ·i is to be considered as a sesquilinear form. For instance, we mean
hf, giL2(T)=
Z
T
f (x)g(x)∗dx,
with g∗ the complex conjugate, even if f , g or both are real, in which case our no-tation does not cause problems anyway. In the case of smooth functions φ ∈S and distributions ρ ∈S0, we write
The thesis is organised as follows. The next chapter analyses the linear part of (SBE), the Ornstein-Uhlenbeck equation: the results developed there will allow us to give meaning to the quadratic Burgers’ nonlinearity for a suitable class of processes in Chapter 3, at the end of which we will be able to give our definition of solution to (SBE). Moreover, the good properties of Ornstein-Uhlenbeck will provide a regularisation estimate which is going to be instrumental in all the subsequent arguments. The last chapter deals with an invariance principle for (SBE), that is, we will prove the convergence of (classical) solutions of a large class of SDEs to our energy solutions, thus giving an example of model satisfying the weak universality conjecture in our sense of solution. This will also imply existence of energy solutions.
Chapter 2
The infinite-dimensional
Ornstein-Uhlenbeck process
It is quite natural to begin our study of (SBE) from its linear part, in the hope that the behaviour of solutions to the full equation will somehow be a “perturbation” of its linear part. So let us ignore the nonlinear drif term in (SBE) and consider the equation
∂tX = ∆X +
√ 2∂xξ,
X0= χ,
(OU) commonly known as the (infinite-dimensional) Ornstein-Uhlenbeck equation. The theory concerning this equation is very classical: whether one interprets the space-time white noise in the sense of Walsh’s martingale measures (see [Wa]) or in the Hilbert space theory of [DPZ], a complete description of solutions and their properties is available. We will use the latter setting, which will be outlined in the first section of this chapter: see Appendix A for basic definitions and precise statements, and the aforementioned references for a complete treatment.
In the second section we will see how Burgers’s drift does make sense in the linear case, that is, we will show that thanks to the regularisation properties of (OU), the functional ∂xu2can be defined as a space-time distribution. Motivated by this fact, we will also develop
some heuristic considerations about the assumptions under which a stationary white noise process can be squared to define Burgers’ drift. The third section collects the Gaussian analysis tools we will need to deal with such processes in the next chapters.
Let us fix the Fourier basis on T, ek(x) = (2π)−2eikxfor k ∈ Z; we are going to use some
basic analysis on T, for which we refer to any standard textbook in analysis. To lighten notations, let us write Hα= HαT, S = S (T) and so on.
2.1
The Ornstein-Uhlenbeck Process
Let us cast (OU) in a Hilbert setting starting from the noise term. A slight modification of the arguments in Appendix A allows us to understand √2∂xξ as a generalised Wiener
process1W˜ton H−1, and as a standard Wiener process in Hα for α < −3/2, which are the
Sobolev spaces in which H−1 has a Hilbert-Schmidt embedding. The trace class covariance operator of ˜Wtin Hαis given by
Ehh ˜Wt, ekiHαh ˜Ws, ehiHα
i
= (t ∧ s)hQek, ehiHαek, Qek = 2k2hki2α.
1We reserve the symbol W to cylindrical Wiener processes, which ˜W is not.
Let us assume α < −3/2 and set our equation in the Hilbert space Hα, rewriting (OU) in the following usual notation,
dXt= ∆Xtdt + d ˜Wt,
X0= χ.
(2.1) The Laplace operator ∆ acting from D(∆) = Hα+2⊂ Hα to Hα, and the heat semigroup
Ston Hαgenerated by ∆ are given by their well known Fourier representation.
Let Ft be the filtration generated by ˜Wt, χ be an F0-measurable Hα-valued random
variable, then an Hα-valued predictable process with (Bochner) integrable trajectories is
called a weak solution of (OU) if
hXt, yiHα= hχ, yiHα+
Z t
0
hXs, ∆∗yiHαds + h ˜Wt, yiHα
for any t ∈ [0, T ] and each y ∈ D(∆∗) (or, equivalently, each test function in S ), almost surely. There exists a unique, probabilistically strong, weak solution to (OU), given by
Xt= Stχ +
Z t
0
St−sd ˜Ws, (2.2)
which is a centred Gaussian process with covariance operator given by Qtek = hki2α
1 − e−2k2tek.
Remark 2.1. Since we are using {ek}k∈Zas a basis for all the Hα, the representations of Q
and Qtdepend on the chosen α, but the relation between them does not, and is given by
Qt=
Z t
0
SrQSr∗dr.
Remark 2.2. Since {ek}k∈Z is not a normalised basis for h·, ·iHα, to avoid bothersome hkiα
factors, we can use instead the coupling h·, ·i onS0×S . We can thus decompose ˜Wt on
the Fourier modes ek in terms of the complex Brownian motions βk = ξ(1[0,t]e−k),
˜ Wt= X k∈Z ikβtkek, E[βtkβ h s] = 2(t ∧ s)δk+h=0,
the series converging in Hα, and write the solution as hXt, e−ki = e−k 2t hχ, e−ki + Z t 0 ike−k2(t−s)dβsk. (2.3) In fact, we can and will consider the Ornstein-Uhlenbeck process in S0, but let us keep track of Sobolev regularity in this paragraph.
The solution of (OU) defines a time homogeneous Markov process on Hα, with transition
semigroup Pton Cb(Hα) satisfying the Feller property, given explicitly by
Ptφ(x) =
Z
Hα
φ(y)N (Stx, Qt)(dy), x, y ∈ Hα.
Since Qthas uniformly bounded trace, there exists a unique invariant measure (see Appendix
A), and by explicit computation it is easy to see that it is the law of a space white noise η, which is supported by H−1/2−.
In fact, by Kolmogorov’s criterion and Gaussian hypercontractivity, one can prove that for any β, γ > 0 such that β + γ < 1/2 the processR0tSt−sd ˜Wshas almost surely trajectories
in Cβ([0, T ], D ((−∆)γ)). Thus the best space regularity H−1/2−, the same we have just found in the stationary case, still does not allow us to define Xt2pointwise, since we are not
dealing with functions. We will discuss further this problem in the next paragraph, but first let us recall a few more properties of Xtas a Markov process.
As remarked above, Pt is a semigroup of linear maps from Cb(Hα) to itself. We did
not state anything about its continuity, and in fact the problem of choosing a topology on Cb(Hα) to make Pt strongly continuous (so that one can study its infinitesimal generator)
is somewhat subtle: a solution in the Gaussian case, which is ours, can be found in [GPZ]. Nonetheless, with standard tools, we can derive how the generator acts on smooth functions. Since we want to study functions of Xt, Ito’s formula is going to be instrumental.
How-ever, we are not dealing with (analytically) strong solutions, which do not exist, so we can not apply the usual Ito formula for functions F ∈ C2
b(Hα), since it would read
dF (Xt) = hFx(Xs), ∆XsiHαds +
1
2Tr(Fxx(Xs)Q)ds + hFx(Xs), d ˜WsiHα, (2.4) and this does not make sense because Xt, a space white noise in the stationary case, does not
belong to D(∆) almost surely. To get around the problem one may consider approximations of the unbounded operator ∆ (in the sense of Yosida), or regularise the noise by projecting on finite Fourier modes. The latter strategy is equivalent to consider cylindrical smooth functions of the form
F : Hα→ R, F (x) = f (ˆx−n, . . . , ˆxn),
where ˆxn is the n-th Fourier coefficient of x and f ∈ Cb∞(R
n). The fact that equation (2.4)
holds for such functions suggests that the drift term operator L0F (x) := hFx(x), ∆xiHα+
1
2Tr(Fxx(x)Q)
acts on cylindrical functions as the generator of Xt. To support this intuition, by means
of Ito’s formula (and an approximation procedure), one can prove the following result. For v : [0, T ] × Hα
→ R, consider the backward Kolmogorov equation,
∂tv(t, x) = L0v(t, x), t > 0, x ∈ D(∆),
v(0, x) = φ(x), x ∈ Hα. (2.5)
We say that v is a strict solution if it is jointly continuous, twice Fr´echet differentiable in x with jointly continuous and bounded derivatives, Gateaux differentiable in t and it satisfies the equation for any x ∈ D(∆), t ≥ 0.
Proposition 2.1. If φ ∈ C2 b(H
α), equation (2.5) has a unique strict solution, which is given
by
v(t, x) = E[φ(Xtx)] = Ptφ(x), t ≥ 0, x ∈ Hα,
where Xt is the solution of (OU) starting from the deterministic data x ∈ Hα.
2.2
Heuristic remarks on Burgers’ drift
As anticipated, we collect in this paragraph a somewhat informal motivation to our study of (OU). Our aim is to define ∂xXt2 (which is the ill-posed term of (SBE)) in the linear
case of (OU). Let us consider the stationary solution Xt of (OU), that is, Xthas the law
to multiply pointwise Xt2, and indeed, as revealed by a simple application of Borel-Cantelli
lemma, even the formal expansion hX2
t, eki =
X
m+n=k
hXt, emihXt, eni
diverges almost surely. Nonetheless, X2
t, or better its derivative ∂xXt2, does make sense as
a space-time distribution. We begin with a proof of this fact heavily relying on the fast decorrelation properties of (OU), namely we will use that
E [hXt, ekihXs, ehi] = δk+h=0e−k
2|t−s|
,
which easily descends from the explicit solution of stationary (OU). In the following we denote as usual with ˆXtk = hXt, e−ki the k-th Fourier coefficient of Xt, and we define
Jt=
Rt
0∂xX 2
sds as the formal series
Jt= Z t 0 ∂x X k∈Z ˆ Xskek !2 ds =X k∈Z ik X m+n=k Z t 0 ˆ XsnXˆsmekds. (2.6)
Proposition 2.2. Equation (2.6) defines almost surely a H−1/2−-valued process with C1/2−
trajectories. Thus, we can define ∂xXt2 = ∂tJt where the time derivative is taken in the
distributional sense.
Proof. Let us bound the second moment of Jt (formally, we should truncate the series,
perform our calculations and then pass to the limit), E[| ˆJtk|2] = k2 Z t 0 Z t 0 X m+n=k E[ ˆXsnXˆr−n]E[ ˆXsmXˆr−m]dsdr (2.7) = k2 Z t 0 Z t 0 X m+n=k e−(n2+m2)|s−r|dsdr (2.8) . k2t X m+n=k Z ∞ 0 e−(n2+m2)rdr = k2t X m+n=k 1 m2+ n2, (2.9)
where this last term is easily estimated (for k 6= 0) by X m+n=k 1 m2+ n2 . Z R dx x2+ (k − x)2 . |k| −1,
so we conclude that E[|Jtk|2] . |k|t, which gives us the desired space regularity. The H¨older
time regularity now follows by an analogous estimate of moments of the increments, Gaussian hypercontractivity and Kolmogorov criterion (see Appendix A).
Of course, one can not hope to carry out this proof in the nonlinear case of (SBE), since there is no reason to expect such decorrelation properties. Let us then sketch very briefly how in fact we can prove Proposition 2.2 using weaker properties of Xt.
The idea, often referred to as the Ito-Tanaka trick (see for instance [FGP]), is to use the Ito formula (2.4),
F (Xt) − F (X0) − MtF =
Z t
0
L0F (Xs)ds (2.10)
(for cylindrical F as above), where MF
t is a martingale depending on F . The existence of
function F such that the right-hand side equals Jt, then we can derive the bounds for Jt
that we need from standard martingale estimates of the right-hand side. A few remarks are in order:
• Jt is (claimed to be) a distribution-valued process, whereas F is real-valued, but the
problem is easily solved by considering single Fourier coefficients ˆJk t =
Rt
0(∂\xX 2 t)kds;
• the required F , which should solve L0F (Xs) = \(∂xXt2)k is hardly going to be
cylin-drical, so we should extend L0 to a wider class of functions or use an approximation
procedure to perform rigorous calculations;
• while estimates of the boundary term in (2.10) are easy to obtain once we have F explicitly, to bound the martingale term (by means of Burkholder-Davis-Gundy in-equality) the hypotesis of stationarity becomes fundamental;
• finally, it is not obvious how to find an adequate F .
Thus, to carry out this programme we need to restrict to the stationary case and to develop further our knowledge of white noise Gaussian analysis, with particular attention to the operator L0: this is going to be the content of the next paragraph.
We will not, however, conclude the proof in the linear case: instead we will develop a detailed a posteriori approach to more general processes controlled by (OU). Indeed, since we understand that stationarity and an adequate Ito formula is all we need to make sense of Burgers’ drift, we might as well study directly a wide class of processes possessing those properties. Unfortunately, modifying the dynamics of the process can and will change the structure of the drift term. To avoid this problem we need to introduce additional hypotesis, namely we will consider reversible processes with generators made of a symmetric part coinciding with L0 and an antysimmetric part. By time reversal, the antysimmetric
part of the drift in Ito’s formula, as well as the boundary terms in (2.10), will vanish, thus granting the very same estimates of the linear case.
The informal arguments we have discussed so far can be justified in the following way: consider Galerkin approximants of (SBE),
∂tuN = ∆uN+ ∂x(ΠNuN)2+
√
2∂xΠNξ
u0= ΠNη,
(2.11) where ΠN is the projection on the first N Fourier modes; these equations are easily shown
to be well-posed, they still preserve the space white noise measure, the stationary solution is in fact reversible and we have an Ito formula of the form
dF (uNt ) = (L0+ BN)F (uNt )dt + dM N t ,
where L0 is as above and the additional term ?? is antysimmetric. The programme we
outlined above can thus be carried out (see [G16] for details), and the resulting estimates provide enough compactness to consider limit points sharing the same properties of the ap-proximants. In this sense, it is natural to postulate those properties (white noise invariance, reversibility, Ito’s formula) for solutions to (SBE). The a posteriori results of Chapter 4 will include the claimed convergence of Galerkin approximants.
2.3
Gaussian Analysis
Let Xtbe the stationary (OU) process as above, considered as aS0-valued process: from now
for cylindrical functions of Xt that we have used informally in the previous paragraphs. It
is important to keep in mind the following clarifications about the probability space. Remark 2.3. Let (Ω, F ) the measurable space on which the space-time white noise ξ driving (OU) and the initial space white noise distribution η are defined as, respectively,S0(R+×T)
andS0(T)-valued random variables, and remember that they are assumed to be indepen-dent. We have seen that at each time t ≥ 0 the law of Xt is a space white noise: its
dependence on η and ξ is clarified in the forthcoming proposition.
We denote Lp(η) = Lp(Ω, Law(η)) (and similar spaces) the space of σ(η)-measurable real random variables with finite p-moments (for p = 0 we just assume measurability). Since Law(Xt) = Law(η), for F ∈ L0(η) we can define the composition F (Xt) as follows: by
Doob’s theorem there exists a Borel mapping ΨF : S0 → R such that F = ΨF ◦ η, so we
set F (Xt) = ΨF◦ Xt.
Proposition 2.3 (Mehler’s formula). Fix t ≥ 0, then there exists a space white noise η0 independent of η such that
Xt= et∆η + I − e2t∆
1/2 η0,
where the operators are all defined by means of their usual Fourier representations. Proof. By (2.2), for ϕ ∈S , we have that
Yt(ϕ) = Xt(ϕ) − et∆η(ϕ) = −
√
2ξ1[0,t](·)e(t−·)∆∂xϕ
,
(the · denoting a time variable) is aS0-valued Gaussian variable independent of η, and its covariance is
E[Yt(ϕ)Yt(ψ)] = h I − e2t∆ ϕ, ψiL2(T),
so it is clear that I − e2t∆−1/2Yt is a space white noise. In fact, we are simply checking
that the operator Qtdefined above on Hα concides with I − e2t∆.
Remark 2.3 leads us naturally to consider functionals F ∈ Lp(η) of Xt, to avoid the
com-plications of the theory on Cb(Hα). We refer to Appendix A and [Nu] for many definitions
and the basic analysis on this space, let us just recall here some notation. For F ∈ L2(η), by J
nF we denote the projection on the n-th Wiener chaos Hn, in the
decomposition L2(η) =L
n≥0Hn. Since η is a white noise, JnF = In(fn) ∈ Hn, where the
kernel fn∈ L2(Tn) is a symmetric function and
In(Fn) =
Z
Tn
fn(x1, . . . , xn)η(dx1) . . . η(dxn),
(see Appendix A for a precise definition). Finally, C denotes the dense subspace in Lp(η) of
cylindrical functions
F :S0→ R, F (ρ) = f (ρ(φ1), . . . , ρ(φn)),
where φ1, . . . , φn ∈ S are finite test functions and f ∈ Cp∞(Rn) is a smooth function of
polynomial growth at infinity.
Remark 2.4. Cylindrical functions do not have a unique representation, and whenever we make a statement about such functions we should check its independence of a particular representative. The reader may interpret any such following statement as independent of the representation, and can check by himself all the details we omit for the sake of brevity. We are now ready to define what we might call the Markov semigroup of Xtin L2(η).
Proposition 2.4. The following definitions are equivalent on L2(η):
1. let η0 be a space white noise independent of η, ΨF as above, and put for F ∈ Lp(η)
with p ≥ 1, TfF = ELaw η 0h ΨF et∆η + I − e2t∆1/2 η0i,
which is well defined since the argument of ΨF is still a space white noise;
2. if F =P n≥0In(fn) ∈ L2(η), set TtF = X n≥0 In(et∆fn),
where the Laplacian ∆fn and et∆fn are their respective versions on Tn;
moreover, Tt is a semigroup of nonnegative contractions on Lp(η) for p ≥ 1, symmetric on
L2(η), and if F
tis the filtration generated by ξ(1[0,t]·), it holds for every t, s ≥ 0, F ∈ L1(η),
E[F (Xt+s)|Fs] = TtF (Xs). (2.12)
Proof. It is easy to see that both formulas define semigroups of linear contractions on their domains, to see that they coincide on L2(η) we only need to check it for Wiener’s exponentials
F = expη(h) − 12khk2L2
, h ∈ L2(T) (the reproducing kernel). Starting from the first definition we get the second,
ELaw η0 exp et∆η + I − e2t∆1/2 η0−1 2khk 2 L2 (2.13) = exp et∆η +1 2h I − e 2t∆ h, hi L2− 1 2khk 2 L2 (2.14) = exp et∆η −1 2 et∆h 2 L2 (2.15) =X n≥0 et∆h n L2Hn et∆η ket∆hk L2 =X n≥0 1 n!In(e t∆h⊗n), (2.16)
where Hn is the n-th Hermite polynomial and we have used the chaos decomposition of the
exponential function F =P
n≥0 1 n!In(h
⊗n).
Nonnegativity in Lp(η), p ≥ 1, and symmetry in L2(η) are easy consequences of the definitions, it only remains the Markov property. We prove it for F ∈ C, without loss of generality F (ρ) = f (ρ(h)), and conclude by density in L1(η): by Proposition 2.3 we have
E[F (Xt+s)|Fs] = E[f (Xt+s(h))|Fs] = E f Xt+s(h) − et∆Xs(h) + et∆Xs(h) Fs = ELaw η0hf I − e2t∆1/2 η0(h) + et∆Xs(h) i = TtF (Xs).
The operator L0 defined in the previous section on cylindrical functions (for a strictly
smaller subclass of C, in fact) turns out to be the infinitesimal generator of the semigroup Tt.
1. L0 : Dp(L0) ⊂ Lp(η) → Lp(η) is the generator of Tt on Lp(η), p ≥ 1, so L0 is closed
on its dense domain Dp(L0);
2. for F =P
n≥0In(fn) ∈ D(L0) ⊂ L2(η), where D(L0) is the dense subspace
D(L0) = F ∈ L2(η) : fn ∈ Hsym2 (T n )∀n ∈ N and X n≥0 n! kfnk2H2(Tn)< ∞ , (2.17) define L0F =Pn≥0In(∆fn) ∈ L2(η), with ∆fn the n-dimensional Laplacian operator
in weak sense;
3. L0 is the closure in Lp(η), p ≥ 1, of the (closable) operator defined on cylindrical
functions F ∈ C, F (ρ) = f (ρ(φ1), . . . , ρ(φn)), by L0F (ρ) = n X i=1 Fi(ρ)ρ(∆φi) + n X i,j=1 Fi,j(ρ)h∂xφi, ∂xφjiL2(T), (2.18)
where Fi(ρ) = ∂if (ρ(φ1), . . . , ρ(φn)) and similarly Fij; in particular C is a core for L0.
Moreover, from the first definition it follows that −L0 is non-negative on Lp(η) and, if
considered on its domain in L2(η), it is symmetric.
Proof. Let us begin with the equivalence of the first two definitions on L2(η): we call L0
the one of the second definition, the other is referred to as the generator of Tt. We have to
show that F ∈ L2(η) belongs to (2.17) if and only if the limit lim
t→01t(TtF − F ) exists in
L2(η) and in this case it coincides with L
0F . If F belongs to (2.17), E " TtF − F t − L0F 2# =X n≥0 n! et∆f n− fn t − ∆fn 2 L2Tn ≤ 2X n≥0 n! kfnk 2 H2Tn,
where the middle term is a sum of objects converging to 0 for t → 0, so the right-hand estimate implies by dominated convergence that the whole series has to vanish in the limit t → 0. Conversely, if limt→01t(TtF − F ) = G in L2(η), JnG = In(gn) = lim t→0 TtJnF − JnF t = limt→0In et∆f n− fn t ,
and since In is an isometry between Hn and L2sym(Tn), we easily conclude that gn= ∆fn.
To show that the first two definitions are equivalent to the third, we only need to check that the L0 they define, which is closed, acts on cylindrical functions as in (2.18). We begin
considering functionals of the form F (η) = Hn(η(φ)) with Hn the n-th Hermite polynomial
as above and φ ∈S such that kφkL2T= 1. Applying the properties detailed in Appendix
A, we compute
Hn0(η(φ))η(∆φ) + Hn00(η(φ))h∂xφ, ∂xφiL2T
= nHn−1(η(φ))η(∆φ) − n(n − 1)Hn−2(η(φ))hφ, ∆φiL2T
= nIn−1(φ⊗(n−1))I1(∆φ) − n(n − 1)In−2(η(φ))hφ, ∆φiL2T
= nIn(φ⊗(n−1)∆φ) = In(∆φ⊗n).
Since in this case F = In(φ⊗n), the thesis holds for such F . The case F = In(φ1⊗ · · · ⊗ φn),
φi ∈S descends from polarisation, and then by density one obtains the general case F =
An important role will be played by the quadratic form associated to L0and the related
Sobolev-like spaces we now define. Let F, G ∈ C be cylindrical functions, represented on the same φ1, . . . , φn∈S respectively by f, g ∈ Cp∞(Rn), and set
E(F, G)(ρ) = 2
n
X
i=1
Fi(ρ)Gj(ρ)h∂xφi, ∂xφjiL2(T),
where Fi(ρ) = ∂if (ρ(φ1), . . . , ρ(φn)) and analogues as above. A trivial computation shows
that this is equivalent to say
E(F, G)(ρ) = L0(F G)(ρ) − F (ρ)L0G(ρ) − G(ρ)L0F (ρ).
We will write E (F ) = E (F, F ), so to consider E both as a symmetric bilinear operator and a quadratic form (symmetry is clear from the definition). The latter point of view allows us to define some convenient spaces.
Definition 2.1. Let p ≥ 1, for F ∈ C set
|||F |||p1,p= Eh|E(F )|p/2i, kF kp1,p= E [|F |p] + |||F |||p1,p;
these are easily shown to be a seminorm and a norm respectively, and we call W1,p the completion of C in Lp(η) with respect to k·k1,p.
The homogeneous version ˙W1,pis obtained by taking the quotient of W1,pover the closed
subspace of functions with |||F |||1,p= 0, and it is a Banach space with norm simply given by |||·|||1,p.
Set now, for F ∈ L2(η),
|||F |||2W˙−1,2= sup G∈C 2hF, GiL2(η)− kGk1,2 ;
consider the quotient of the subspacen|||F |||2−1,2< ∞o over n|||F |||2−1,2= 0o, and take its completion ˙W−1,2 with respect to |||·|||
−1,2.
Remark 2.5. W1,2 is a Hilbert space with inner product
hF, Gi1,2= E[F G] + E[E (F, G)];
˙
W1,2 is also a Hilbert space, and its inner product is just E[E (F, G)]. Finally, the space
˙
W−1,2is another Hilbert space: its inner product can be written explicitly by polarisation.
Let ek(x) = (2π)−n/2eik·x be the Fourier basis on Tn, and let us define, say inS0(Tn),
the operators ∇ek = ikek and its inverse ∇−1. Observe that their respective domains
are homogeneous Sobolev spaces ˙H1
(Tn) and ˙H−1
(Tn), which coincide with their
non-homogeneous counterparts on zero-mean distributions.
Then, for p = 2, the norms we have defined above and the corresponding Sobolev-like spaces can be expressed in terms of the chaos expansion as follows (see [GP1]).
Lemma 2.1. For F =P n≥0In(fn) ∈ L2(η), we have |||F |||21,2 =X n≥0 n! k∇fnk2L2(Tn), |||F ||| 2 −1,2= X n≥0 n! ∇−1fn 2 L2(Tn). (2.19)
Thus, recalling that (see Appendix A) kF k2L2(η)=
P n≥0n! kfnk 2 L2(Tn), we have that ˙ W1,2= X n≥0 In(fn) : fn ∈ ˙Hsym1 (T n),X n≥0 n! kfnk2H˙1(Tn) , (2.20) and the analogous expressions for non-dotted W, H and index −1.
Observe that the second point in Proposition 2.5 tells us that the domain D(L0) of L0in
L2(η) would correspond to the space W2,2 defined in analogy with (2.20). The next lemma establishes two expected relations between the objects we have defined so far. We will not give the proof, since in fact we are not going to use these facts; instead we refer to [GP1] and [KOL].
Lemma 2.2. W2,2 embeds continuously in ˙W1,2, and L
0 extends uniquely to an isometry
from ˙W1,2 into ˙W−1,2, with dense image. Moreover, W˙−1,2 is (identifiable with) the dual
Hilbert space of ˙W1,2.
We have now at hand all the tools we need to study functionals of the Ornstein-Uhlenbeck process. We conclude this section and the chapter with a rigorous version of the Ito for-mula for cylindrical functions of Ornstein-Uhlenbeck which we informally anticipated in the previous section.
Proposition 2.6 (Ito’s Formula). Consider a time dependent cylindrical function F (t, ρ) = f (t, ρ(φ1), . . . , ρ(φn)) with f ∈ C
1,2
b ([0, T ] × R
n) and fixed φ
1, . . . , φn. Then it holds, almost
surely for every t ∈ [0, T ],
F (t, Xt) = F (0, X0) +
Z t
0
(∂s+ L0)F (s, Xs)ds + MtF,
where MF
t is a martingale (with respect to the filtration generated by the driving noise ξ)
with quadratic variation given by MF t= Z t 0 E(F (s, Xs))ds. Proof. Since Bi
t= ∂xξ(1[0,t]φi), are identically distributed brownian motions satisfying
dXt(φi) = Xt(∆φi)dt + dBti, E[B i t, B
j
s] = h∂xφi, ∂xφiiL2,
a straightforward application of the classical Ito formula for n-dimensional diffusion processes gives us dF (t, Xt) = df (t, ρ(φ1), . . . , ρ(φn)) (2.21) = ∂tftdt + n X i=1 ∂ift· dXt(φi) + 1 2 n X i,j=1 ∂ij2ft· d [X(φi), X(φj)]t (2.22) = ∂tf dt + n X i=1 ∂if · Xt(∆φi)dt + 1 2 n X i,j=1 ∂2ijf · h∂xφi, ∂xφiiL2dt + n X i=1 ∂if · dBti (2.23) = (∂t+ L0)F (t, Xt)dt + n X i=1 ∂if · dBti. (2.24)
where we omitted the arguments of f for simplicity, and this concludes the proof since " Z · 0 n X i=1 ∂if (s, ρ(φ1), . . . , ρ(φn)) · dBsi # t = Z t 0 E(F (s, Xs))ds.
Remark 2.6. Since L0 and E are both defined on W2,2, as we have just shown, one might
wonder whether it is possible to extend Ito’s formula to the more general class of functionals C1([0, T ], W2,2). Since we are going to work for some more time at the level of cylindrical functions, we do not need this generality; moreover the possibility of such an extension is in fact non trivial.
Chapter 3
Controlled Processes and
Energy Solutions
The aim of this chapter is to give a sensible definition of (stationary) solutions to ∂tu = ∆u + ∂xu2+
√ 2∂xξ
u0= η,
(SBE) more specifically we want to understand in what sense a stationary process with space white noise marginals can satisfy (SBE). The first task is then to find a class of stationary processes with white noise marginals that allow us to rigorously define the Burgers’ drift, ∂xu2, let us
say generically as a space-time random distribution.
In the first section, coherently with the idea of studying (SBE) as a nonlinear perturba-tion of (OU), we define a class of processes controlled by (OU), postulating the key properties we outlined in Section 2.2. We then proceed to derive from such properties a fundamental regularising estimate which in Section 3.2 will finally allow us to define Burgers’ drift for this suitable class of processes, and thus to define the concept of Energy Solution to (SBE). The last section is devoted to a martingale formulation of this notion.
3.1
Regularisation estimates
In Section (2.2) we have discussed what should a “perturbation” of the linear equation (OU) look like to allow us to give a rigorous meaning to the nonlinear Burgers’ drift. There are no a priori reasons to postulate solutions of (SBE) to be semimartingales: indeed we do need an Ito formula for our processes, but much less structure is required to have it. We will use F¨ollmer pathwise theory (see [F¨o] and the review in Appendix A) to use Ito’s formula for continuous processes only having quadratic variation.
Let us assume a priori the following structure.
Definition 3.1 (Controlled processes). A couple (ut, At)t∈[0,T ]of processes with trajectories
in C0([0, T ],S0) is called a process controlled by Ornstein-Uhlenbeck (a controlled process for brevity’s sake) if the following hold:
1. (Invariant white noise) ut is stationary and its invariant measure is the space white
noise η;
2. (Zero energy perturbation) A0 = 0 and for each ϕ ∈S the process At(ϕ) has zero
quadratic variation (see Appendix A); 19
3. (Forward dynamics) ut and Atsatisfies, for t ∈ [0, T ] and any ϕ ∈S ,
ut(ϕ) = u0(ϕ) +
Z t
0
us(∆ϕ)ds + At(ϕ) + Mt(ϕ),
where Mt(ϕ) is a martingale with respect to the filtration Ft = σ(ut, At), with
quadratic variation [M (ϕ)]t= 2t kϕk 2 H1;
4. (Backward antysimmetric dynamics) the reversed processes ←−ut = uT −t and
←− At =
AT −t− AT, for which (1.) and (2.) trivially hold, satisfy the antysimmetric reversed
dynamics, for t ∈ [0, T ], ←−u t(ϕ) = ←−u0(ϕ) + Z t 0 ←−u s(∆ϕ)ds + ←− At(ϕ) + ←− Mt(ϕ),
where ←M−t(ϕ) is a martingale with respect to the reversed filtration
←−
Ft= σ(←−ut,
←− At),
with the same quadratic variation of Mt(ϕ).
Remark 3.1. It is immediate to see that in fact the martingales Mt(ϕ) and
←−
Mt(ϕ) are
stochastic integrators corresponding to space-time white noises. If we specify a priori a space-time white noise ξ with its associated filtration Ftand an F0-measurable space white
noise η, we call (ut, At)t∈[0,T ] a controlled process associated to (ξ, η) if it is a controlled
process such that u0= η and Mt(ϕ) =
√
2ξ(1[0,t]∂xϕ) for any ϕ ∈S , almost surely.
Remark 3.2. Note that if At= 0 we are back to the linear (OU). As in that case, since ut
has always the law of space white noise, we understand the composition F (ut) for F ∈ L0(η)
in the sense of Remark 2.3.
Remark 3.3 (The 0-th Fourier mode). Looking back to Chapter 2, it is clear that projecting the space S0 away from the 0-th Fourier mode, that is, considering only processes with
zero space average, would get us rid of the bothersome non equivalence of homogeneous and non-homogeneous Sobolev spaces on T and on (Ω, Law η). That projection is motivated in the case of (OU), since Xt(e0) has trivial dynamics due to the fact that ∂xξ(e0) =
0. In the case of (SBE) one could motivate the assumption ut(e0) = 0, equivalently the
redefinition of the space white noise by η(e0) := 0, because physically (SBE) should represent
a conservation law, and our theory has to take account of that. From now on, in this chapter, we assume that all distributions inS0, as well as allS0-valued variables and processes have 0 space average. Just like heuristics in Section 2.2, this assumption is going to be motivated a posteriori by the invariance result in Chapter 4.
As we anticipated, a major feature of controlled processes is that an Ito formula is available, in the very same form of the one we wrote in Chapter 2 for the Ornstein-Uhlenbeck process.
Proposition 3.1 (Ito’s Formula). Let (ut, At) be a controlled process, and consider a time
dependent cylindrical function F (t, ρ) = f (t, ρ(φ1), . . . , ρ(φn)) with f ∈ Cb1,2([0, T ] × Rn)
and fixed φ1, . . . , φn. Then it holds, almost surely for every t ∈ [0, T ],
F (t, ut) = F (0, u0) + Z t 0 (∂s+ L0)F (s, us)ds (3.1) + n X i=1 Z t 0 ∂if (s, us(φ1), . . . , us(φn))dAs(φi) + M F,u t , (3.2)
where MtF,uis a real martingale (with respect to the filtration generated by ut) with quadratic
variation given by MF t= Z t 0 E(F (s, us))ds.
Proof. We just combine the proof of Proposition 2.6 with Theorem A.1 in Appendix A and the various remarks therein to deal with the new term At, which, let us remark it, has zero
covariation with the semimartingale part since it has zero quadratic variation. Note that MtF,udoes not depend on At.
We are now ready to show, by means of the Ito-Tanaka trick, the fundamental estimate that has motivated heuristically in Chapter 2 our definition and which is going to be in-strumental in every following result. The forthcoming Proposition 3.2 collects two different formulations, whose characteristics will allow us to deal both when we know how to invert the operator L0 and when we do not. Indeed, solving explicitly the Poisson-like equation
L0H = F for a given F ∈ L2(η) can be very difficult in general, or even impossible, but one
can always invert λ − L0 for λ, and this is the way we will go round the problem in proving
the second estimate in Proposition 3.2.
Lemma 3.1. For any F ∈ L2(η) and λ > 0 the function
Hλ,F(η) = − Z ∞
0
e−sλTsF (η)ds
belongs to D(L0) and solves (λ − L0)Hλ,F(η) = F (η). Moreover,
Hλ,F 2 1,2+ λ 2 Hλ,F 2 L2(η)≤ |||F |||−1,2 Hλ,F 1,2.
Proof. Since Tt, which we recall to be the generator of L0, is a contraction semigroup on
L2(η), the integral converges in Bochner’s sense in L2(η), and we can exchange the closed operator λ − L0 and the integral. We now use the explicit action of L0 and Tton the chaos
decomposition seen in Chapter 2: we have, for F = In(fn),
−(λ − L0)e−sλTsIn(fn) = In −(λ − ∆)e−s(λ−∆)fn = d dsIn(e −s(λ−∆)f n),
and thus we conclude by integrating in time. As for the estimate, it suffices to write
Hλ,F 2 1,2+ λ Hλ,F 2 L2(η)= hHλ,F, (λ − L 0)Hλ,FiL2(η) = hHλ,F, F i L2(η) ≤ |||F |||−1,2 Hλ,F 1,2, the last inequality given by Lemma 2.2.
Proposition 3.2 (Regularisation estimate). Let F ∈ Lp([0, T ], Dp(L0)), Dp(L0) the domain
of L0 in Lp(η), for p ≥ 1. Then for any controlled process (ut, At) it holds
E " sup t∈[0,T ] Z t 0 L0F (s, us)ds p# . Tp/2−1 Z T 0 |||F (s, ·)|||p1,pds. (3.3) Moreover, if F is only in L2([0, T ], L2(η)), E " sup t∈[0,T ] Z t 0 L0F (s, us)ds 2# . Z T 0 |||F (s, ·)|||2−1,2ds. (3.4)
Proof. Let us consider first the case of F cylindrical as in Proposition 3.1. Because of the backward dynamics assumption on (ut, At), the Ito formula applies also to ←u−t(with respect
to its filtration), and gives
F (T − T, ←−ut) = F (T − (Tt), ←−uT −t) + Z T T −t (∂s+ L0)F (T − s, ←−us)ds (3.5) + n X i=1 Z T T −t ∂if (T − s, ←−us(φ1), . . . , ←u−s(φn))d ←− As(φi) + ←− MF,uT −←M−F,uT −t, (3.6) where←M−F,uis a martingale with respect to the “backward” filtration, generated by ←u−t, and
quadratic variation
[←M−F,u]t=
Z t
0
E(F (T − s, ←−us))ds.
We add the two Ito formulas for ut and ←−ut, and obtain
Z t 0 L0F (s, us) + M F,u t + ←− MF,uT −←M−F,uT −t= 0.
Now we have gotten rid of the dependence on At and initial and final values, and the
conclusion follows straightforwardly from Burkholder-Davis-Gundy inequality. The general case F ∈ Lp([0, T ], D
p(L0)) is obtained by approximation: it suffices to show
that our cylindrical functions are dense in such space, and this is done by approximating with piecewise constant functions in time, then approximating the value on each step with a cylindrical function and finally mollifying in the time direction, this last passage leaving the approximation cylindrical in space since cylindrical functions are a linear subspace.
Let us now prove (3.2). To do so, fix an arbitrary cylindrical function H ∈ C and decompose
F = L0H − (L0H − F ),
so that by estimating separately E " sup t∈[0,T ] Z t 0 (L0H − F )(us)ds 2#1/2 ≤ Z T 0 E [(L0H − F )(us)] 1/2 ds (3.7) ≤ T kL0H − F kL2(η), (3.8) E " sup t∈[0,T ] Z t 0 L0H(us)ds 2#1/2 . T1/2|||H|||1,2, (3.9) (3.10) using in the latter the first version of our regularisation estimate, we obtain
E " sup t∈[0,T ] Z t 0 F (us)ds 2#1/2 ≤ T1/2|||H||| 1,2+ T kL0H − F kL2(η).
Plugging in the H = Hλ,F of Lemma 3.1 and using the inequality stated therein, T1/2 Hλ,F 1,2+ T L0Hλ,F − F L2(η)= T 1/2|||F ||| −1,2+ T λ 1/2|||F ||| −1,2.
Remark 3.4. If for each s ∈ [0, T ], F (s, η) ∈ L2(η) has a finite chaos expansion, so do L0F (s, η) and E (F (s, η)). Therefore, by Gaussian hypercontractivity (see Appendix A), for
p ≥ 1, E " sup t∈[0,T ] Z t 0 L0F (s, us)ds p# .pTp/2−1 Z T 0 |||F (s, ·)|||p1,2ds. (3.11) Remark 3.5. Since controlled processes are stationary, if F is independent of time we can improve our estimate as follows:
E " sup t∈[0,T ] Z t 0 L0F (s, us)ds p# . Tp/2|||F (·)|||p1,p. (3.12) We have dubbed our estimate as “regularising” since, strikingly, we are controlling norms of a second order differential operator with a lower order one. Indeed, whenever the right-hand side of the estimates makes sense, we are able to define the time integral on the left-right-hand side. This is what we are going to do in the next section: we will write in a purely formal way the Burgers’ drift ∂xu2tfor a controlled process as L0F (t, ut) for an adequate function F ,
so that controlling the W1,p norm of F will let us define ∂
xu2t as a space-time distribution
by means of the regularisation estimate. The calculations we are going to perform are a prototype for the more general estimates in Chapter 4.
3.2
Burgers’ drift
The results of this section extend Proposition 2.2 to controlled processes. We will deal with two different approximation procedures: the first one is the projection ΠN on the Fourier
modes with indeces k = 0, ±1, ±2, . . . , ±N , the other is the convolution by a smoothing kernel.
Definition 3.2. The family of functions θN
(x) = N θ(N x), for N ∈ N and θ ∈ S , are called an approximation of unity if θ is non-negative, even, it is supported by a small compact neighbourhood of 0 andR
Tθ(x)dx = 1.
Remark 3.6. It is well known that θN ∗ u converges to u inS0 if u ∈S0, in Hαif u ∈ Hα
for any α ∈ R, and as a continuous function if u is continuous (keep in mind that we work on T, which is compact).
Remark 3.7. Recall that the projection ΠN can be expressed as the convolution with
Dirich-let’s kernel: for ρ ∈S0
ΠNρ = DN ∗ ρ, DN(x) = sin
N +1 2 x
sin x2 .
It is well known that DN is not uniformly bounded in L1, more specifically one can prove
that DN
L1(T)∼ ln N , so D
N is not an approximation of unity.
Of course, the issues caused by the bad behaviour of Dirichlet’s kernel are compensated by the fact that Galerkin’s projection ΠN is extremely useful to deal with differential equations.
This is the reason why we are considering both the approximations given by ΠN and the
smoothing kernels θN in Proposition 3.3: we will need the different advantages of the two
approaches to deal with diverse situations.
The following proposition summarizes how, by means of the above approximations, it is constructed (a unique) Burgers’ drift as a space-time distribution.
Proposition 3.3. Let (ut, At) be a controlled process, N ≥ 1. Consider the smooth
approx-imantions of Burgers’ drift BN ∈ C1([0, T ],S ) defined by
BNt =
Z t
0
∂x(ΠNus)2ds. (3.13)
As N → ∞, the process BN converges in probability in C0([0, T ], H−1), and we define
Rt
0∂xu 2
sds as the limit. Moreover, for any approximation of unity θN(x) = N θ(N x) where
θ ∈S the approximants Btθ,N = Z t 0 ∂x θN∗ us 2 ds (3.14) converge to the same limitRt
0∂xu 2
sds in probability in C0([0, T ], H−1).
Proposition 3.3 will be consequence of the very general estimates in Lemma 3.2, but before we start our computations, we need to fix some notation and make a few remarks on the structure of Burgers’ drift.
Let us fix a symmetric zero-mean function θ ∈S , and define the approximated square of a distribution:
qθ(ρ)(x) = (θ ∗ ρ)2(x) − kθk2L2 = ρ(θ(x − ·)) − kθk
2
L2, ρ ∈S
0.
A key observation is that for a space white noise η we have qθ(η)(x) = I2 θ(x − ·)⊗2 ,
where I2 is the twice-iterated stochastic integral with respect to η (see Appendix A): this
follows easily from the fact that the second Hermite polynomial is x2−1. Such an expression of qθ allows us to apply the machinery we have developed in Section 2.3. The pointwise evaluation qθ(ρ)(x) for x ∈ T is not very easy to work with (ideally it corresponds to integrate against δx), and it is much preferable to consider instead the integrals
qθ,φ(ρ) = Z T (θ ∗ ρ)2(x)φ(x)dx − kθk2L2 Z T φ(x)dx, which in the white noise case become
qθ,φ(ρ) = I2(fθ,φ), fθ,φ(y, z) =
Z
T
θ(x − y)θ(x − z)φ(x)dx, whose Fourier coefficients on T2 are (using the symmetry of θ)
d fθ,φ
h,k=
√
2π ˆφh+kθˆhθˆk, h, k ∈ Z0. (3.15)
We will be interested also in the difference of two different approximations of the square, so for ζ ∈S with the same properties of θ we will write
qθ,ζ,φ(η) = qθ,φ(η) − qζ,φ(η) = I2(fθ,ζ,φ), f[θ,ζφh,k=
√
2π ˆφh+k ˆθhθˆk− ˆζhζˆk
, h, k ∈ Z0.
Lemma 3.2. For any T > 0, p ≥ 1, θ, φ ∈ S , with θ as above and controlled process (ut, At)t∈[0,T ], it holds E " sup t∈[0,T ] Z t 0 qθ,φ(us)ds p#1/p . T1/2 X h,k∈Z0 | dfθ,φ h,k|2 h2+ k2 1/2 . (3.16) Analogous estimates hold for ζ like θ and qθ,ζ,φ, \fθ,ζ,φ
Proof. First, let us note that, because of the explicit expression of L0on cylindrical functions,
the equation L0Hθ,φ(ρ) = qθ,φ(ρ) is easily solved by
Hθ,φ(ρ) = I2(gθ,φ), gdθ,φh,k= − d fθ,φ
h,k
h2+ k2, h, k ∈ Z0.
Since Hθ,φ(η) has a finite chaos expansion in L2(η) (indeed, it has only one term),
Propo-sition 3.2 and Remarks 3.4 and 3.5 apply and we get E " sup t∈[0,T ] Z t 0 qθ,φ(us)ds p#1/p .pT1/2 Hθ,φ(·) W1,2(η)' T 1/2 gθ,φ H1(T2), (3.17)
substituting the expression of gθ,φ in terms of fθ,φwe conclude. The estimate for qθ,ζ,φ is
completely analogous.
Remark 3.8. Note that the solution to L0Hθ,φ(ρ) = qθ,φ(ρ) can be written also as
Hθ,φ(ρ) = − Z ∞
0
Tsqθ,φ(ρ)ds,
which is the expression we derived in Lemma 3.1, but with λ = 0, which in this case does not create problems since we are dealing with cylindrical functions.
Proposition 3.3 is a direct consequence of the following corollary of Lemma 3.2. For a fixed controlled process (ut, At) and θ ∈S , it is convenient to define the (spatially smooth)
process Qθt(x) = Z t 0 qθ(us)(x)ds, and Qθt(φ) = hQ θ t, φiL2 = Z t 0 qθ,φ(us)ds, (3.18) for φ ∈S .
Corollary 3.1. If θN ∈S is an approximation of unity and DN is the Dirichlet kernel,
then QθN
t and QD
N
t converge for N → ∞ in Lp(Ω, P ), p ≥ 1 (where (Ω, P ) is the underlying
probability space) in C([0, T ], L2).
Moreover, the limit process is the same for every approximation of unity and the Dirichlet kernel, and we denote it by Qt=
Rt
0u 2 sds.
Proof. From Lemma 3.2 we get, for θ, ζ ∈S , E " sup t∈[0,T ] Q θ t(φ) − Q ζ t(φ) p#2/p . T X h,k∈Z0 | ˆφh+k|2 ˆ θhθˆk− ˆζhζˆk 2 h2+ k2 . (3.19)
Let us begin applying equation (3.19) to the Dirichlet kernel: for 0 ≤ M ≤ N we have E " sup t∈[0,T ] Q DM t (φ) − Q DN t (φ) p#2/p . T X h,k∈Z0 | ˆφh+k|2 1|h|,|k|≤N− 1|h|,|k|≤M 2 h2+ k2 ≤ T X |h|,|k|>M | ˆφh+k|2 h2+ k2 ≤ T X |`|>2M | ˆφ`|2 X |k|>M 1 k2+ (` − k)2 . T M −1kφk2 L2,
where in the last inequality we estimated the series with the integral Z
R
1|x|>Mdx
x2+ (` − x)2 . M −1,
uniformly in `. This concludes the desired convergence of QDtN.
As for the smoothing kernels θN, let us first note that from the uniform bound | cθN
k | ≤
θN L1= 1,
and the pointwise convergence of ˆθN
k to 1 for N → ∞, by dominated convergence equation
(3.19) gives convergence in C([0, T ],S0). The very same argument shows that, for any approximation of unity, the limit is the same as in the case of Galerkin’s approximations.
We are left to show that the convergence of QθN
t happens in C([0, T ], Hε), and not only
in the space distributions’ topology. The estimates are slightly more complicated than in the previous case. We will use the fact that since θN has small compact support, it can be
extended to a function on the whole line R, so we can consider its Fourier transform cθN. In
particular, we will use that ∂ξθc N(ξ) ∞≤ Z R |x|θN (x)dx . 1 N.
We start from the right-hand side of (3.19) applied to θM, θN: we bound θc M h θcMk − cθhNθckN = θc M h ( cθkM− cθkN) + cθkN( cθMh − cθNh) ≤ | cθM k − cθ N k | 2+ | cθM h − cθ N h| 2+ | cθM k − cθ N k || cθ M h − cθ N h|,
and consider only the first term of the last member, the other being of the same form. Recalling that cθM(0) = 1, by the mean value theorem we bound
X h,k∈Z0 | ˆφh+k|2| cθMk − cθ N k | 2 h2+ k2 ≤ X k∈Z0 | cθM k − cθ N k | 2 k2 X h∈Z0 | ˆφh+k|2 (3.20) . kφk2L2 Z R | cθM(ξ) − cθN(ξ)|2 ξ2 dξ = kφk 2 L2 Z R | cθM(ξ) − 1 + 1 − cθN(ξ)|2 ξ2 dξ (3.21) ≤ kφk2L2 Z R min ∂ξθc M ∞+ ∂ξθc N ∞, 2 |ξ| 2 dξ (3.22) . kφk2L2 Z R min 1 M, 1 |ξ| 2 dξ . kφk2L2M −1. (3.23)
This concludes the convergence of QθN
t , and thus the proof.
Remark 3.9. We did not insist on time regularity in our arguments, but it is not difficult to check that from our estimates and Kolmogorov’s continuity theorem the convergences in Corollary 3.1 and Proposition 3.3 actually take place, respectively, in Cα([0, T ], L2) and
Cα([0, T ], H−1), for all 0 ≤ α < 1/2.
Remark 3.10. We have contructed a continuous process Qt =R t 0u
2
sds ∈ L2(T) that, since
Dirac’s delta has regularity δx ∈ H−1/2−, cannot be evaluated in single points. In other
words, the above proof does not allow to take the pointwise limit of Qθ
t(x). One might
wonder if it is possible to trade some of the time regularity (see the above Remark) for more space regularity. However, since in Proposition 3.2 we need to use the Burkholder-Davis-Gundy inequality to bound martingales, we can not worsen the factor T1/2in our estimates, so no further space regularity can be obtained in that way. Nonetheless, we can conversely give up all the space regularity to obtain convergence in Cα([0, T ], L2) for 0 ≤ α < 3/4 in both Corollary 3.1 and Proposition 3.3. This can be done with a careful application of Kolmogorov’s criterion to the estimates above (see [GP1]).
We are now ready to give a proper definition of solution to (SBE), which is the content of the next section.
3.3
Energy Solutions
So far, we have defined the class of controlled processes and shown that they possess the nonlinear functional Rt
0∂xu 2
sds. Because of how we constructed it, it is natural to call
Rt
0∂xu 2
sds the Burgers’ drift for ut and give the following definition of solution to (SBE)
and, consequently, to (KPZ). In the following definition we reintroduce the parameter λ ∈ R we have set equal to 1 in the Introduction, since it will be convenient to have it at hand in the next chapters.
Definition 3.3. Let ξ be a space-time white noise, Ft its associated filtration and η an
F0-measurable space white noise. A (probabilistically) strong stationary solution to
∂tut= ∆ut+ λ∂x2ut+
√ 2∂xξ
is a controlled process (ut, At) associated to (ξ, η), such that At is indistinguishable from
λRt
0∂xu 2 sds.
A probabilistically weak stationary solution, or energy solution, is any controlled process with A = λR
0∂xu 2
sds as above, that is, the law of (ξ, η) is not specified a priori.
Remark 3.11. As already pointed out in the Introduction, the name “energy solution” orig-inated in [GJ14], to denote a slightly weaker concept (namely, time reversibility was not involved). The name hints to the fact that the crucial point is the definition of an “energy” functional for processes that can not even be squared. Being weaker then ours, the original concept of energy solution still appears as limit of the processes we will consider in the invariance result of Chapter 4, in particular since our kind of solutions exist, so do they. However, our uniqueness result does not apply to weaker formulations, for it uses all the assumptions we made. Uniqueness for the original energy solutions is still an open question. Before moving to the issues of existence, uniqueness and invariance of energy solutions, which are going to be the topics of the following chapters, let us see how our definition adapts to (KPZ). We just assume by definition that (stationary) solutions of (KPZ) are obtained as space derivatives of energy solutions to (SBE): our assumption is motivated a posteriori by the invariance result of the last chapter.
Definition 3.4. Let ξ be a space-time white noise, Ft its associated filtration and η an
F0-measurable space white noise. A (probabilistically) strong almost-stationary solution to
∂tht= ∆ht+ λ(∂xut)2+
√ 2ξ
is a pair (ht, Qt) of continuous distribution-valued processes, with Q0 = 0 almost surely,
such that
(ht, Qt) = (∂xut, ∂xAt),
where (ut, At) is an energy solution to (SBE), with the same parameter λ, associated to
(ξ, η).
As above, an energy solution is strong almost-stationary solution for which the law of the data (ξ, η) is not specified a priori.
Remark 3.12. The notation Qt is a strong hint to the functional of controlled process we
Remark 3.13. We called “almost-stationary” our solutions to (KPZ) for the following reason. At each fixed time ∂hthas to have the law of the space white noise, and this determines all
its Fourier modes except for the 0-th. This means that the law of d(ht)0can depend on time.
Indeed, this constitutes a flaw in the statement “energy solutions coincide with Cole-Hopf solutions”, since they happen to differ by a time-dependent, constant in space, drift, see [GP1].
Chapter 4
The Hairer-Quastel invariance
principle
In this chapter we will prove that energy solutions of (SBE) appear as the large scale limit of solutions to a wide class of non-singular SPDEs, namely
∂tv = ∆v + √ ε∂xF (v) + √ 2∂xξε u0= ηε, (4.1) where ξε and ηε are smooth approximations of the space-time and space white noise on T. As already remarked in the Introduction, this result, contained in [GP1], was inspired by its analogue in the context of Hairer’s regularity structures, see [HQ].
A precise formulation of our result, and further remarks about it, will be given in Section 4.2, and then proved rigorously in Section 4.3. Before that, we need however to obtain the adequate estimates to obtain tightness of the approximants. Our main tool will be the Boltzmann-Gibbs principle, a generalisation of the computations we performed in Section 3.2, which is the content of the following section.
In the following, we make use of all the notation we introduced so far, especially in Chapter 2, without further mention. We recall that we are still assuming that all functions and distributions on T have zero space average.
4.1
Boltzmann-Gibbs principle
Let us go back for a moment to the 1-dimensional Wiener chaos decomposition, to fix some useful notation. We denote by Γ the standard Gaussian law on R; analogously to Remark 2.3, the composition G(Γ) is well defined for any G ∈ L0(Γ). Since Hermite’s polynomials are an orthogonal basis of L2(Γ) (see Appendix A), the chaos expansion of G ∈ L2(Γ) can
be written as G(Γ) =X n≥0 cn(G)Hn(Γ), cn(G) = 1 n!E[G(Γ)Hn(Γ)]. (4.2) We will denote by JnΓG = cn(G)Hn(Γ) the projection of G on the n-th Wiener chaos of
L2(Γ). It is easy to check that
c0(G) = E[G(Γ)], c1(G) = E[ΓG(Γ)] c2(G) = 1 2E[Γ 2G(Γ) − G(Γ)], and so on. 29
Let us give right away the main result of this section, and collect all due remarks before the proof. Observe that if η is a space white noise on T, thenpπ/N ΠN0η(x) (which is a
function inS for x ∈ T varying) is a standard Gaussian real variable for any fixed x ∈ T. Proposition 4.1 (Boltzmann-Gibbs principle). Let (ut, At)t∈[0,T ] be a controlled process,
G ∈ L2(Γ), an almost everywhere differentiable function with G0 ∈ L2(Γ). Then
• (first order principle) for all k ∈ Z0, 0 ≤ s < t ≤ s + 1 and κ > 0, uniformly in N ≥ 0
integer, E Z t s * N π∂xΠ N 0G( p π/N ΠN0ur) − r N πc1(G)∂xΠ N 0ur, ek + dr 2 . |t − s|3/2−κk2E[G0(Γ)2]; (4.3) • (second order principle) for all k ∈ Z0, 0 ≤ s < t ≤ s + 1 and integers 0 ≤ M ≤ N ,
E Z t s * N π∂xΠ N 0G( p π/N ΠN0ur) − r N πc1(G)∂xΠ N 0 ur− c2(G)∂x(ΠM0 ur)2, ek + dr 2 . |t − s|k2 1M + π N(ln N ) 2 E[G0(Γ)2]. (4.4) Remark 4.1. From a physical point of view, the estimates above are implying that we can replace, in time average, local functions of “microscopic” fields ΠN
0 u (in the sense that the
space white noise is the “macroscopic” limit of its projections ΠN
0 ) such as G(pπ/N ΠN0ur),
with simple global functionals. This phenomenon is well understood in the theory of in-teracting particle systems (see [KOL]), and bears the name of Boltzmann-Gibbs principle, from which the name of our estimates.
Remark 4.2. It is not difficult to see that the part of Proposition 3.3 using (ΠN
0 us)2 as an
approximation for the squared process is a consequence of Proposition 4.1. Namely, one only has to apply the second order principle to G(Γ) = Γ2, for which c
0(G) = c2(G) = 1
and c1(G) = 0.
It is clear from the very structure of the estimates that our proof will be based on Proposition 3.2 just like Proposition 3.3. The strategy, as in that case, is to write the arguments of the integrals on the left-hand side as the image of some variable under L0,
to bound the W1,2-norm of such variable and then conclude by the regularising estimate.
We proceed in two steps, since as in the previous chapter inverting L0 requires some careful
analysis on L2(η), and we thus prefer to isolate the procedure in a preparatory lemma.
Lemma 4.1. Let G ∈ L2(Γ). For fixed φ ∈S (as usual with zero space average, ˆφ0= 0),
let Qφ be the cylindrical function in L2(η) defined by Qφ(η) = hG(pπ/N ΠN0η), φi =
Z
T
G(pπ/N ΠN0η(x))φ(x)dx, then the equation L0Hφ= Qφ is solved by the cylindrical function
Hφ(η) =X n≥0 cn(G) π N n/2 X 0<|k1|,...|kn|≤N ˆ φk1+···+kn (2π)(n−1)/2 Jn(η(ek1) · · · η(ekn)) k2 1+ · · · + k2n . (4.5)
Remark 4.3. As we already noted in the past chapters, we have the following good looking representation of the solution Hφ:
Hφ(η) = − Z ∞
0
G(pπ/N et∆ΠN0 η)dt.
Proof. First observe that since Hn(Γ) = Jn(Γn) (almost by definition, see Appendix A), we
can write Hn( p π/N ΠN0η) = Jn (pπ/N ΠN0η)n (4.6) =π N n/2 X 0<|k1|,...|kn|≤N ek1+···+kn (2π)n/2 Jn(η(ek1) · · · η(ekn)) , (4.7)
by expanding the n-th power inside Jn. Let us now expand
G(pπ/N ΠN0 η) =X
n≥0
cn(G)Hn(
p
π/N ΠN0 η),
so that plugging in (4.6), for any φ ∈S with ˆφ0= 0, we have
Qφ(η) = hG(pπ/N ΠN0η), φi (4.8) =X n≥0 cn(G) π N n/2 X 0<|k1|,...|kn|≤N ˆ φk1+···+kn (2π)(n−1)/2Jn(η(ek1) · · · η(ekn)) . (4.9)
From Lemma 2.1 it easily descends that for any 0 < |k1|, . . . |kn| ≤ N ,
L0Jn(η(ek1) · · · η(ekn)) = −(k
2
1+ · · · + k 2
n)Jn(η(ek1) · · · η(ekn)) ,
which, in view of (4.8), concludes.
Before we conclude the proof of Proposition 4.1, let us separate yet another computation. In the following lemma we bound functionals of controlled processes just like in Proposition 3.2, only in a much rougher way. More precisely, if in Proposition 3.2 we morally “gained a derivative”, here we control a functional with a term of the same order. Of course such an estimate holds for a way larger class of functionals.
Lemma 4.2. Let (ut, At)t∈[0,T ] be a controlled process and G as in Proposition 4.1. Then
for all k ∈ Z0, 0 ≤ s < t ≤ T and integer N ≥ 0,
E Z t s * N π∂xG( p π/N ΠN0ur) − r N πc1(G)∂xΠ N 0ur, ek + dr 2 . N |t − s|2k2E[G0(Γ)2]. (4.10)
