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Lecture Notes on Math 833 – Stochastic PDEs (Draft) March 6, 2020 Hao Shen University of Wisconsin-Madison, US, Email: [email protected] Contents 1 Stochastic heat equation with additive noise 1 1.1 Explicit solution ............................... 1 White noise. Scaling invariance. Solution via heat kernel. Moments. Fourier transform. 1.2 Properties of solution ............................. 4 Besov regularity. Wavelets and H ¨ older regularity. Gaussian free field as invariant measure. 1.3 Examples and challenges of nonlinear SPDEs ................ 9 2 Stochastic heat equation with multiplicative noise 12 2.1 Itˆ o solution .................................. 12 Itˆ o integral. Existence and uniqueness of solution. 2.2 Weak solution ................................. 14 3 Φ 4 equation 16 3.1 Local mild solution in d =1 ......................... 16 Fixed point argument. 3.2 Important heuristics: perturbation theory ................... 17 Wick theorem. Feynman diagrams. 3.3 Φ 4 equation in d =2 ............................. 17 4 (Controlled) rough paths 18 1 Stochastic heat equation with additive noise 1.1 Explicit solution From the discussion of stochastic heat equation (SHE) – simple yet a prototype of Stochastic PDE (SPDE) – we can appreciate what problems, properties and tools we can explore for SPDEs in general. SHE with additive space-time white noise ξ reads t u u + ξ. (1.1) 1
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Page 1: Lecture Notes on Math 833 – Stochastic PDEs (Draft)hshen3/SPDE.pdf · 2 STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE White noise. We call a random distribution 1 if it is a continuous

Lecture Notes on Math 833 – Stochastic PDEs (Draft)

March 6, 2020

Hao Shen

University of Wisconsin-Madison, US, Email: [email protected]

Contents1 Stochastic heat equation with additive noise 1

1.1 Explicit solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1White noise. Scaling invariance. Solution via heat kernel. Moments. Fourier transform.1.2 Properties of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Besov regularity. Wavelets and Holder regularity. Gaussian free field as invariant measure.1.3 Examples and challenges of nonlinear SPDEs . . . . . . . . . . . . . . . . 9

2 Stochastic heat equation with multiplicative noise 122.1 Ito solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Ito integral. Existence and uniqueness of solution.2.2 Weak solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Φ4 equation 163.1 Local mild solution in d = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 16Fixed point argument.3.2 Important heuristics: perturbation theory . . . . . . . . . . . . . . . . . . . 17Wick theorem. Feynman diagrams.3.3 Φ4 equation in d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 (Controlled) rough paths 18

1 Stochastic heat equation with additive noise

1.1 Explicit solution

From the discussion of stochastic heat equation (SHE) – simple yet a prototype of StochasticPDE (SPDE) – we can appreciate what problems, properties and tools we can explore forSPDEs in general. SHE with additive space-time white noise ξ reads

∂tu = ∆u+ ξ . (1.1)

1

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2 STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE

White noise. We call η a random distribution 1 if it is a continuous linear map f 7→ η(f )from C∞c (the space of compactly supported smooth test functions) into the space of squareintegrable random variables on some fixed probability space (Ω,P).

A white noise ζ onD ⊂ Rd is a random distribution, for which ζ(f )f∈C∞c is a collectionof centered joint Gaussians on a fixed probability space (Ω,P) with covariance given by (“L2

property”)

E[ζ(f )ζ(g)] =

∫D

f (x)g(x)dx .

This is often formally written as E[ζ(x)ζ(y)] = δ(x− y) where δ is the Dirac distribution. 2

A space-time white noise ξ is a white noise on a space-time domain Λ ⊂ R× Rd with

E[ξ(f )ξ(g)] =

∫Λ

f (t, x)g(t, x)dtdx , or E[ξ(t, x)ξ(s, y)] = δ(t− s)δ(x− y) . (1.2)

One example of white noise on R+ is dBdt

, the derivative of Brownian motion; in this case theabove “L2 property” is just the Ito isometry.

Cylindrical Wiener process. The space-time white noise can be also constructed asthe time derivative of the so called “cylindrical Wiener process”, an infinite dimensionalgeneralization of Brownian motion. We briefly describe this construction. For a separableHilbert space H (e.g. L2(Td) if we are interested in white noise on Td), let (ei)i∈N be anorthogonal basis. We then define

W (t) =∑k∈N

bk(t)ek (1.3)

where bkk∈N are independent standard Brownian motions. One has

E[〈W (t), f〉H〈W (t), g〉H] =∑k,`

Ebk(t)b`(s)〈ek, f〉H〈e`, g〉H = (t ∧ s)∑k

〈ek, f〉H〈ek, g〉H

= (t ∧ s)〈f, g〉H .

So W has the same covariance as 1D Brownian motion, but has the “L2 property” in spatialdirection. A same calculation with W replaced by its time derivative yields δ(t− s)〈f, g〉H .

The subtlety in this construction is that W is not an element in H when H is infinitedimensional, as can be seen from E〈W (t),W (t)〉H = t

∑k〈ek, ek〉 = ∞. In fact W can

be constructed as a process in a larger Hilbert space H ′ ⊃ H via for instance 〈f, g〉H′ =∑k

1k2〈f, g〉H , so that E〈W (t),W (t)〉H′ = t

∑k

1k2〈ek, ek〉H <∞. (In general, it suffices to

have ι : H → H ′ Hilbert-Schmidt i.e. ιι∗ trace class; and H is the Cameron-Martin spacefor H ′ with a Gaussian measure of covariance ιι∗ in the language of Malliavin calculus.)

Scaling. Scaling or “dimension analysis” will be important in our course. For δ on Rd,one can check (by testing against f ∈ C∞c ) that λdδ(λx) = δ(x). Heuristically, this means

1Here a distribution refers to a generalized function, as opposed to “probability distribution”.2Recall its definition δa(f ) def

= f (a) and δ def= δ0. The notation δ(x) is formal, but it will be convenient to

use this notation and write for instance∫δ(x− y)f (y)dx = f (x). The formal definition of ζ in terms of δ is

“justified” by E[ζ(f )ζ(g)] =s

E[ζ(x)ζ(y)]f (x)g(y)dxdy =sδ(x− y)f (x)g(y)dxdy =

∫f (x)g(x)dx.

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STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE 3

that its “scaling dimension” [δ] = [x]−d. According to this heuristic, the space-time whitenoise ξ then has “scaling dimension” [ξ] = [t]−

12 [x]−

d2 . In fact, one has

λk0+k1

2 ξ(λk0t, λk1x) law= ξ(t, x) (k0, k1) ∈ N2 ,λ ∈ R+ ,

which can be shown by testing against f .The equation (1.1) also has a scaling invariance. For invariance, the variables t and x

have to be scaled diffusively (i.e. parabolically) (t, x) → (λ2t, λx). A heuristic dimensioncounting shows that if the three terms in (1.1) have the same scaling dimension [∂tu] =

[∆u] = [ξ] = [t]−12 [x]−

d2 = [x]−

d+22 , then [u] = [x]−

d−22 . More precisely, if

u(t, x) def= λ

d−22 u(λ2t, λx) (1.4)

then it satisfies ∂tu(t, x) = ∆u(t, x) + ξ(t, x) where ξ(t, x) := λd+22 ξ(λ2t, λx) law= ξ(t, x).

Remark 1.1 Finding the “scaling dimensions” for various objects here is useful for the discussion oftheir approximations. For example, ε−d1|x|≤ε/2 is the right approximation for δ as ε→ 0, as can bechecked by testing against f ∈ C∞c . For a white noise ξ in Rd, a central limit theorem can be shown:namely for a smooth, not necessarily Gaussian, mean zero random field ζ with bounded moments,ε−

d2 ζ(ε−1x) converges to ξ in law.

Solution via heat kernel. If ξ was a function, the linear equation (1.1) with initialcondition u(t, ·) = u0 (where u0 is deterministic) has the following explicit solution

u(t, x) =

∫Rd

∫ t

0

P (t− s, x− y)ξ(s, y)dsdy +

∫RdP (t, x− y)u0(y)dy (1.5)

where P is the heat kernelP (t, x) = (4πt)−

d2 e−

|x|24t . (1.6)

We call this the mild solution to (1.1). 3 It remains to give a suitable meaning to the firstterm in (1.5) (this term is sometimes called “stochastic convolution”). We assume u0 = 0for simplicity. Then, since ξ is centered Gaussian and (1.5) is linear in ξ, u should also be acentered Gaussian distribution.

The subtlety is the singularity of P at (t, x) = 0, so that P (t− ·, x− ·) is not necessarilyL2. A simple calculation using (1.2) shows

E[u(t, x)u(t, x)] =

∫Rd

∫ t

0

P (t− s, x− y)P (t− s, x− y)dsdy .

In particular the “variance”

E[u(t, x)2] =

∫Rd

∫ t

0

P (t− s, x− y)2dsdy =

∫ t

0

1

(8π(t− s)) d2

∫Rd

e−|x−y|22(t−s)

(2π(t− s)) d2dyds

=

∫ t

0

(8π(t− s))−d2ds <∞ if and only if d = 1 .

3The mild solution is a strong solution. The notion of “strong” and “weak” solutions in SPDE is differentfrom those in PDE. In SPDE, a strong solution assigns each realization of ξ a function u as in (1.5), whereas aweak solution solves a process that has the required law, which we discuss later.

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4 STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE

This could also follow 4 by Fourier transform (in x) and Parseval’s theorem:∫ t

0

|P (t− s, k)|2ds =

∫ t

0

e−2(t−s)|k|2ds =1− e−2t|k|2

2|k|2

which is integrable as |k| → ∞ if and only if d = 1.This means that u should not have a pointwise value when d > 1. On the other hand u is a

bona fide random distribution; in fact u(f ) has variance ‖∫

Rd+1 P (t− ·, x− ·)f (t, x)dtdx‖2L2

which is always finite for f ∈ C∞c . Below we describe the regularity of u as a distribution,but we first give another way of solving the equation using Fourier transform.

Solution via Fourier transform. For a Fourier analysis, assuming for simplicity thatour underlying space is the torus Td. Recall that for any real function u =

∑k∈Zd u(k)eikx,

the Fourier coefficients satisfy u(k) = u(−k). The cylindrical Wiener process is given byW (t) =

∑k∈Zd βk(t)e

ikx where, since eikx are complex, βkk∈Zd are independent complexvalued Brownian motions (whose real and imaginary parts are independent, each being astandard Brownian motion divided by

√2), with βk = β−k. Applying (spatial) Fourier

transform to (1.1), we get ∂tu = −k2u+ ξ, and this is actually a system of decoupled 5 SDEs

du(k) = −k2u(k)dt+ dβk . (1.7)

The solution is now given by

u(t, k) =

∫ t

0

e−(t−s)|k|2dβk(s) + e−t|k|2

u0(k) .

Focusing on the random term by assuming u0 = 0, then u is centered Gaussian by Gaussianityof u and linearity of Fourier transform. Its covariance is given by (for k1, k2 6= 0)

E[u(t, k1)u(t,−k2)] = 1k1=k2

∫ t

0

e−2(t−s)|k1|2ds = 1k1=k2

1− e−2t|k1|2

2|k1|2(1.8)

where we have used Ito isometry.

1.2 Properties of solution

Besov Space regularity. Let χ, % ∈ C∞c be nonnegative radial functions on Rd, such thatsuppχ is contained in a ball and supp% is contained in an annulus, satisfying “partition ofunity” χ(z) +

∑j>0 %(2−jz) = 1 for all z ∈ Rd, with supp(χ)∩ supp(%(2−j·)) = 6# for j ≥ 1

and supp(%(2−i·))∩ supp(%(2−j·)) = 6# for |i− j| ≥ 2. See [] (Proposition 2.10) for existenceof such functions. We will write %−1 = χ and %j = %(2−j·) for j ≥ 0.

The Littlewood–Paley blocks are now defined as

∆ju = F−1(%jFu) for j ≥ −1.

Then one has u =∑

j>−1 ∆ju. For p, q ∈ [1,∞] we define

Bαp,q =

u ∈ S ′(Rd) : ‖u‖Bαp,q =

( ∑j≥−1

(2jα‖∆ju‖Lp)q)1/q

<∞,

4This could be also “guessed” by dimension counting: [P ] = [t]−d/2 = [x]−d, so [P 2] · [t] · [x]d > 0 ifand only if d = 1.

5decoupled except for the constraint u(k) = u(−k)

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STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE 5

with the usual interpretation as `∞ norm in case q =∞. The space Bαp,q does not depend on

(χ, %). We write Cα = Bα∞,∞. For α ∈ R+\Z, it can be shown that Cα are identical to the

Holder spaces. We write utdef= u(t, ·).

Lemma 1.2 Let γ = −d−22

. For any ε > 0, δ ∈ (0, 1), p ∈ N we have

E‖ut − us‖pBγ−δ−εp,p

≤ C|t− s|δp/2 .

From this, together with Kolmogorov continuity theorem, and a continuous Besov imbed-

ding Bαp,p → B

α− dp

∞,∞, by taking p large enough we have6

u ∈ C([0, T ], Bγ−ε∞,∞) a.s. ∀ε > 0 . (1.9)

To illustrate the main idea, we assume that the underlying space is the torus Td. We will onlyprove a simpler version of Lemma 1.2:

E‖u(t, ·)‖pBαp,p ≤ C ∀α < γ . (1.10)

Proof. By definition, ‖u‖pBαp,p =∑

j≥−1 2jαp‖∆ju‖pLp . Note that if we started by trying tobound moments of Bα

∞,∞ it would be inconvenient to deal with expectation of supremum;however now with p <∞ we only need to compute

E‖∆jut‖pLp = E∫

Td

∣∣∣ ∑k∈Zd

%j(k)ut(k)eikx∣∣∣pdx

Since u (and thus u since Fourier transform is linear) is Gaussian, the above is bounded by

≤ C

∫Td

E[∣∣∣ ∑

k∈Zd%j(k)ut(k)eikx

∣∣∣2] p2dx .Now everything boils down to a second moment calculation of Gaussian: by (1.8)

E∣∣∣ ∑k∈Zd

%j(k)ut(k)eikx∣∣∣2 =

∑k1,k2∈Zd

%j(k1)%j(k2)E[ut(k1)ut(−k2)]eik1x−ik2x

=∑k∈Zd

%j(k)2 1− e−2t|k|2

2|k|2 2−2j2jd = 2j(d−2)

where denotes ‘bounded above and below up to proportional constants’, since %j issupported on an annulus of width 2j (thus it contains ∼ 2jd terms). Here 1 − e−2t|k|2 isasymptotically 1 as k →∞. The summability then requires∑

j≥−1

2jαpE‖∆ju‖pLp ∑j≥−1

2jαp(2j(d−2))p2 <∞ ⇔ α < γ = −d− 2

2

as required by (1.10).

6more precisely, we can find a version of u which is continuous in t

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6 STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE

Holder regularity. In (1.9) we view u as a process in time taking values in a Besovspace. Here we explore the other viewpoint, which is, viewing u as a random distributionover space-time, and we measure regularity of u in real space (rather than Fourier). We againassume that the underlying space is Td. Write Λ = [0, T ] × Td. We first introduce somenotation. For z ∈ Λ we define a parabolic distance ‖z‖ =

√|t| + |x|. For ϕ ∈ Crc (Λ) with

some r > 0, and λ ∈ (0, 1), we define

ϕλ(s,y)(t, x) def= λ−(d+2)ϕ(λ−2(t− s), λ−1(x− y)) (1.11)

namely ϕ is re-centered to (s, y) and parabolically rescaled by λ.For α ∈ R+\Z+, let Cαs be the completion of C∞c under

‖f‖Cαs =∑|k|<bαc

supz∈Λ

|Dkf (z)|+ supz,z∈Λ

|Dkf (z)−Dkf (z)|‖z − z‖α−bαc

where |k| = 2k0 + k1 for (k0, k1) ∈ N × Nd. When α < 0, the space Cαs is defined as thecompletion of C∞c with respect to

‖f‖Cαs = supλ∈(0,1)

supz∈Λ

supϕλ−α|f (ϕλz )| (1.12)

where supϕ is over all functions ϕ which have ‖ϕ‖Cr0 ≤ 1 for r0 = −bαc and supported ina unit ball. For α < 0 it can be shown that Cαs is essentially equivalent to the Besov spaceBα∞,∞, but with respect to the parabolic distance over space-time.

For the space-time white noise ξ, a simple second moment calculation using (1.2) showsthat

E[ξ(ϕλz )2] =

∫Λ

(ϕλz (w))2dw . λ−2(d+2)∫‖w−z‖≤λ

dw = λ−2(d+2)λd+2 = λ−(d+2) (1.13)

where we applied a brutal bound on ϕλz (w) using (1.11), and . stands for ≤ up to a pro-portional constant that is uniform in λ, z, ϕ. This calculation is consistent with the scalingdimension discussed around (1.4).

Our goal here is to prove that ξ ∈ Cαs for any α < −d+22

, by showing E‖ξ‖pCαs < ∞,similarly as we’ve done for Besov space in (1.10). The challenge is to take expectation ofthe supremum over infinitely (uncountably) many λ, z, ϕ on RHS of (1.12). The theory ofwavelets allows us to simply deal with countably many of them, and thereby replace thesupremum by sum, making it easier to take expectation. Here’s a quick tour to wavelets.

Wavelets. Let Λndef= ((2−2nZ)× (2−nZ)d) ∩ Λ. Given ϕ and n ∈ Z+ we write

ϕn(s,y)(t, x) def= 2

(d+2)n2 ϕ(22n(t− s), 2n(x− y)) . (1.14)

Note that the difference between the notation ϕλ and ϕn is that ‖ϕλ‖L1 stays constant asλ→ 0 whereas ‖ϕn‖L2 stays constant as n→∞. Here is an important theorem in wavelets.

Theorem 1.3 Fix r > 0. There exist ϕ ∈ Crc and a finite collection Ψ = ψ of Crc functionson Λ, such that ϕ0

zz∈Λ0 ∪ ψnz n∈Z+,z∈Λn form an orthonormal basis of L2(Λ).

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STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE 7

We skip the proof of this theorem, but only briefly explain the idea behind it, for L2(R)instead of L2(Λ) for simplicity. Daubechies proved that given r > 0, there exists ϕ ∈ Crc (R)such that 7

1. For each k ∈ Z,∫

R ϕ(x)ϕ(x+ k)dx = 1k=0;

2. there exist “structure constants” ak such that ϕ(x) =∑

k∈Z akϕ(2x+ k).

In view of property 1. we define Vn = spanϕnx : x ∈ Λn ⊂ L2(R). Property 2. then showsthat V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vn ⊂ Vn+1 ⊂ · · · . Writing Vn+1 = Vn ⊕ Vn, it turns out thatthere exists ψ(x) =

∑k∈Z bkϕ(x+ k) ∈ Crc for some constants bk such that Vn = spanψnx :

x ∈ Λn. Therefore, we have an L2 decomposition L2(R) = V0 ⊕ V0 ⊕ V1 ⊕ V2 ⊕ · · · , withan orthonormal basis as in Theorem 1.3.

We can characterize Cαs for α < 0 by the above wavelet basis, in the same spirit as wedefine distribution spaces using Fourier coefficients once we have the L2 Fourier basis.

Theorem 1.4 Let α < 0. f ∈ Cαs if and only if

|f (ψnz )| . 2−(d+2)n

2−nα |f (ϕ0

y)| . 1 (1.15)

where the proportional constants in . are uniform in n ∈ Z+, z ∈ Λn, y ∈ Λ0, ψ ∈ Ψ.

Proof. We only prove necessarity (sufficiency is harder). Since |f (ϕλz )| . λα for anyλ ∈ (0, 1), z ∈ Λ and ϕ ∈ Crc with unit support, in particular it holds for the ϕ and ψ inTheorem 1.3, for λ = 2−n, and z ∈ Λn. Noting the difference between the definitions of ϕλ

and ϕn, with λ = 2−n one has the following identity to translate between the two notation

f (ψλz ) = 2n(d+2)

2 f (ψnz ) (1.16)

from which (1.15) follows. We refer to [Hai14, Proposition 3.20] for a complete proof.

Thanks to Theorem 1.4, we can now bound E‖ξ‖pCαs :

E‖ξ‖pCαs = E[

supψ

supn≥0

supz∈Λn

(2(d+2)n

2+nαξ(ψnz ))p

].∑ψ

∑n≥0

∑z∈Λn

2(d+2)np

2+nαpE[ξ(ψnz )p]

.∑n≥0

2n(d+2)2(d+2)np

2+nαp (1.17)

where ψ ranges in the finite collection Ψ ∪ ϕ given in Theorem 1.3. This is finite if andonly if α < −d+2

2− d+2

p. Therefore, given α = −d+2

2− ε with ε > 0, one can choose p large

enough so that −ε < −d+2p

, which ensures E‖ξ‖pCαs <∞. 8 Here, an important input of the

7It is easy to find a discontinuous one, e.g. the Haar wavelet, but finding one in Crc is very nontrivial.8A student in the class asked that given α = −d+2

2 −ε with a small ε > 0, why it happens that E‖ξ‖pCαs <∞for a sufficiently large pwhile the bound (1.17) would be∞ for a small p′ (say p′ = 1), which seems to contradictwith the general fact ‖X‖Lp′ ≤ ‖X‖Lp for p′ < p. The explanation is that when E‖ξ‖pCαs <∞ for a sufficiently

large p we do have E‖ξ‖p′

Cαs<∞ for any p′ ≤ p, but this can’t be seen from (1.17). The bound (1.17) is only

an upper bound, where we replaced supz∈Λnby∑z∈Λn

, causing a factor 2n(d+2); if we did not have this factor

2n(d+2), we would be able to conclude that∑n≥0 2

(d+2)np2 +nαp <∞ if and only if α < −d+2

2 (for any p). This“loss of sharpness” of course does not matter at all, since we only need to find some p so that E‖ξ‖pCαs <∞.

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8 STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE

bound is E[ξ(ψnz )p] . E[ξ(ψnz )2]p2 = ‖ψnz ‖

pL2 . 1, which is essentially the same calculation

as (1.13). In future, it will be more convenient to do this second moment calculation in theway as in (1.13), because one can just read off the regularity from the exponent of λ from theright hand side (and it is equivalent anyway, in view of (1.16)). The above estimate holdswith more generality and let’s write it as a lemma:

Lemma 1.5 If ζ is a Gaussian random distribution over space-time, such that E[ζ(ϕλz )2] .λ2α for some α < 0 then ζ ∈ Cαs for any α < α.

Proof. With λ = 2−n we have |E[ξ(ψnz )p]| . E[ξ(ψnz )2]p/2 . (2−(d+2)n2−2αn)p/2, so in (1.17)the summability condition reads n(d + 2) + (d+2)np

2+ nαp − (d+2)np

2− nαp < 0, which is

α < α− (d+ 2)/p. We then choose p large as above.

Gaussian free field as invariant measure. For each k 6= 0, (1.7) is an Ornstein-Uhlenbeck process, which has the (1-dimensional complex) Gaussian measure N (0, 1

|k|2 ) as

an invariant measure, namely e−12|k|2|u(k)|2/Z (where Z is the suitable normalization). 9

The Gaussian free field Φ on Td is a random distribution, with Φ(0) = 0 (namelyΦ(1) =

∫Td Φ = 0) which is given by

Φ =∑

k∈Zd\0

ak|k|eik·x (1.18)

where ak are independent complex standard Gaussians, except for a constraint ak = a−k,namely Re(ak), Im(ak) ∼ N (0, 1

2) independent s.t. E|ak|2 = 1. Since

E|Φ(k)|2 =E|ak|2

|k|2=

1

|k|2

the Gaussian free field is invariant for the dynamic (1.7). Alternatively, one can view theGaussian free field as a centered Gaussian random distribution with “covariance” (−∆)−1 inthe following sense: for any f, g ∈ C∞c (Td) with

∫Td f =

∫Td g = 0,

E[Φ(f )Φ(g)] = E∑k 6=0

Φ(k)f (k)∑6=0

Φ(`)f (`) =∑k 6=0

f (k)g(−k)|k|2

= 〈f, (−∆)−1g〉L2(Td) .

This leads to the formal notation of Gaussian free field in some literature Φ ∼ e−12

∫(∇Φ)2dx.

We refer to [She07, Section 2] for a more systematic discussion on Gaussian free field (via“abstract Wiener space” approach and “Gaussian Hilbert space” approach).

The stationary solution to the SHE (1.1) then reads

ut(k) =

∫ t

−∞e−(t−s)|k|2dβk(s) or u(t, x) =

∫ t

−∞

∫TdP (t− s, x− y)ξ(s, y)dsdy

9The invariance of this Gaussian measure for the SDE dX = −|k|2Xdt + dB can be checked, forinstance by Kolmogorov forward equation which states that the probability density p(t, x) for Xt satisfies∂tp = ∂2

xp− ∂x(−|k|2xp), and p(x) = e−12 |k|

2|x|2/Z indeed satisfies this PDE (both sides vanish), namely Xt

can have density p(x) for all t.

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STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE 9

Holder regularity for stationary solution. Recall the parabolic distance ‖z‖ definedabove. We have the following result about the degree of singularity of heat kernel at origin:

|P (z)| . ‖z‖−d for ‖z‖ ≤ 1

2

where the proportional constant in . is independent of z. To show this, recall the formula(1.6) for P (t, x); when

√t ≥ |x|, one has

P (t, x) = (4πt)−d2P (1, x/

√t) . |t|−

d2 e−

|x|24t . |t|−

d2 . ‖z‖−d

and when√t ≤ |x|, one has P (t/x2, 1) = |x|d

(4πt)d2e−|x|24t . |x|d

(4πt)d2

( |x|2

4t)−d/2 . 1, so

P (t, x) = |x|−dP (t/x2, 1) . |x|−d . ‖z‖−d .

Another useful result is:

Lemma 1.6 If f and g are functions on space-time and |f (z)| . ‖z‖α, |g(z)| . ‖z‖β whereα, β ∈ (−d− 2, 0) and α + β + (d+ 2) < 0 then |(f ∗ g)(z)| . ‖z‖α+β+(d+2).

Proof. (f ∗ g)(z) =∫f (z − w)g(w)dw. Let r def

= ‖z‖/2. We consider three regimes inspace-time. First, ‖w‖ < r, in which case ‖z − w‖ > C‖z‖ for some C > 0, so we get∫‖w‖<r |f (z − w)||g(w)|dw .

∫‖w‖<r ‖z‖

α‖w‖βdw . ‖z‖α+β+(d+2). The second regime is........

We can now prove u ∈ Cαs for any α < −d−22

. Proof: TO TYPE...

1.3 Examples and challenges of nonlinear SPDEsWe briefly review a number of physically important nonlinear equations, and discuss thechallenge to define the meaning of a solution to nonlinear SPDEs driven by very singularnoises. The common difficulty in defining their solutions is lack of regularity; as we willsee, this is related to the so called “ultraviolet divergence” in physics. We restrict ourselvesto examples that are built on top of the stochastic heat equation, and recall that its solutionu ∈ Cαs for any α < −d−2

2, that is, more singular as d becomes higher.

Kardar-Parisi-Zhang (KPZ) equation. The equation, proposed by [KPZ86], modelsinterface growth, which is ubiquitously found in nature, where each point of the interfacerandomly grows up or drops down over time, with a trend to locally smooth the interface out(the effect of ∂2

xH), and the growth depends in a nonlinear way in the slope (the effect of(∂xH)2):

∂tH = ∂2xH + (∂xH)2 + ξ.

In d = 1, the solution to the linear part is below Holder 12, and we can not expect the

nonlinearity to improve regularity; thus (∂xH)2 does not have any classical meaning. Thewell-posedness of the KPZ equation in one spatial dimension was first solved in [Hai13]. Theproblem is more severe when d ≥ 2.

Stochastic heat equation with multiplicative noise (mSHE). Let f be a continuousfunction, consider

∂tu = ∆u+ f (u)ξ . (1.19)

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10 STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE

The specialization f (u) = u, i.e.∂tu = ∆u+ uξ (1.20)

has a significant connection to the KPZ equation: one can formally check that if H solvesKPZ, then the Hopf-Cole transform u := eH solves (1.20). Other choices of f such asf (u) = α

√u(1− u) arise in modeling population dynamics and genetics.

A classical result known as Young’s theorem states that the multiplication f · g can beclassically defined if the sum of their Holder regularities is positive (thinking of

∫BdB as a

counter-example). In d = 1, since u is expected to have Holder regularity below 12, and ξ

below −32, the multiplication uξ lacks classical meaning. It turns out that the Ito solution

theory successful for stochastic ordinary differential equations can extend to this SPDE 10;this was summarized in for instance the lecture notes [Wal86], and we will discuss this inSection 2.

Nonlinear parabolic Anderson model (PAM). The equation reads

∂tu = ∆u+ f (u)ζ (1.21)

where f is a continuous function and ζ is a noise which typically is assumed to be whitein space, but constant in time (i.e. “spatial white noise”). This models the motion of massthrough a random media. One can prove (say, using Lemma 1.5), that ζ has Holder regularitybelow −d

2, so when d ≥ 2, one encounters the same difficulty to define its solution as for

mSHE. We refer to [GIP15, HL18] and references therein for well-posedness results ind = 2, 3.

PAM (especially the linear case f (u) = u) is a simple model which exhibits intermittencyover long time; for the study of long time behavior, one often considers the spatial-discreteequation with ζ being independent noises on lattice sites, see for instance the reviews [CM94]and [K16] for further discussion and references regarding long time behaviors of PAM.

Stochastic Navier-Stokes equation. This is a vector valued equation:

∂t~u+ ~u · ∇~u = ∆~u−∇p+ ~ζ, div ~u = 0 (1.22)

where p is the pressure, ~ζ is a d-vector valued noise. For instance, when each component of ~ζis taken as an independent space-time white noise, it models motion of fluid with randomnessarising from microscopic scales, and in this case one has the same difficulty as above ford ≥ 2; we refer to [DPD02, ZZ15] for well-posedness results.

Remark 1.7 We remark that while this course focuses on singular noises, when modelinglarge scale random stirring of the fluid, the noise ~ζ is often assumed to be smooth (called“colored noise” in contrast with white noise), and in fact the most important case is that theequation is driven by only a few number of random Fourier modes. In these situations thelong-time behavior is of primary interest, and various dynamical system questions such asergodicity and mixing are studied. There is a vast literature on this topic, and we only referto the book [KS12] and the survey articles [Mat03, Fla08, Kup10].

Parisi-Wu stochastic quantization. This refers to a large class of singular SPDEs arisingfrom Euclidean quantum field theories defined via Hamiltonians (or actions, energy etc.).

10so this gives a roundabout meaning to the KPZ equation, via the Hopf-Cole transform, but it was not clearin what sense logu solves KPZ until [Hai13].

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STOCHASTIC HEAT EQUATION WITH ADDITIVE NOISE 11

They were introduced by Parisi and Wu in [PW81]. Given a HamiltonianH(Φ) which is afunctional of Φ, one considers a gradient flow ofH(Φ) perturbed by space-time white noiseξ:

∂tΦ = −δH(Φ)δΦ

+ ξ . (1.23)

Here δH(Φ)δΦ

is the variational derivative of the functionalH(Φ); for instance, whenH(Φ) =12

∫(∇Φ)2dx is the Dirichlet form, δH(Φ)

δΦ= −∆Φ and (1.23) boils down to the stochastic

heat equation (1.1). Note that Φ can be also multi-component fields, with ξ being likewisemulti-component. The famous Φ4 equation

∂tu = ∆u− u3 + ξ

also arises from this procedure withH(Φ) =∫

12(∇Φ)2 + 1

4Φ4dx.

The significance of these “stochastic quantization equations” (1.23) is that given a Hamil-tonianH(Φ), the formal measure

1Ze−H(Φ)DΦ (1.24)

is formally an invariant measure11 for Eq. (1.23). Here DΦ is the formal Lebesgue measureand Z is a “normalization constant”. We emphasize that (1.24) are only formal measuresbecause, among several other reasons, there is no “Lebesgue measure” DΦ on an infinitedimensional space and it is a priori not clear at all if the measure can be normalized. Thesemeasures arise from Euclidean quantum field theories. In their path integral formulationsquantities of physical interest are defined by expectations with respect to these measures. Thetask of constructive quantum field theory is to give precise meaning or constructions to theseformal measures, see the book [Jaf00].

Exercises

1. Provide a complete proof of Lemma 1.2 (rather than its simplified version (1.10)).

2. Provide a complete proof to Theorem 1.4.

3. Let Hα be the Sobolev (Hilbert) space. With similar arguments as in the proof for(1.9), show that u ∈ C([0, T ], H−

d−22−ε) a.s. for any ε > 0.

4. Prove that the Gaussian free field given by (1.18) a.s. belongs to the Sobolev (Hilbert)

space H−d−22−ε, or B

− d−22−ε

∞,∞ , for any ε > 0.

5. Let ζ be the spatial white noise in (1.21), that is, ζ is constant in time, and Eζ(x)ζ(y) =δ(x − y). Prove using similar criteria as in Lemma 1.5 that ζ has Holder regularitybelow −d

2.

11Being invariant means that if the initial condition of (1.23) is random with “probability law” given by(1.24), then the solution at any t > 0 will likewise be distributed according to this same “probability law”. Forreaders familiar with stochastic ordinary differential equations, one simple example is given by the Ornstein-Uhlenbeck process dXt = − 1

2Xtdt+ dBt where Bt is the Brownian motion, and its invariant measure is the

(one-dimensional) Gaussian measure 1√2πe−

X2

2 dX .

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12 STOCHASTIC HEAT EQUATION WITH MULTIPLICATIVE NOISE

2 Stochastic heat equation with multiplicative noise

∂tZ = ∆Z + Zξ . (2.1)

We call Z a mild solution if

Z(t, x) =

∫Rd

∫ t

0

P (t− s, x− y)Z(s, y)ξ(s, y)dsdy +

∫RdP (t, x− y)u0(y)dy (2.2)

In this, we must have that∫ t

0

∫R P

2(t− s, x− y)E[Z2(s, y)]dyds <∞ for the Ito integrals tomake sense and be finite.

2.1 Ito solution

Ito integral. Recall from Section 1 that the space-time white noise is the time derivative ofthe cylindrical Wiener process, which is a Brownian motion in each 1-dimensional projection.The Ito integral w.r.t. white noise (or rather w.r.t. the cylindrical Wiener process) is definedanalogously to the 1-dimensional case.

Given ϕ ∈ C∞c (R) we can define∫

R+×R 1(0,t](s)ϕ(x)ξ(x, s)dxds, as in Section 1. This isa Brownian motion in t with variance

∫ϕ2(x)dx, since by a similar calculation as in below

(1.3), the correlation of this integral at time t = t1 and t = t2 equals (t1 ∧ t2)∫ϕ2(x)dx.

Let F0 = 6# and for each t > 0 define Ft to be the σ–field generated by∫R+×R

1(0,s](u)ϕ(u)ξ(u, x)dxdu : 0 ≤ s ≤ t, ϕ ∈ C∞c (R).

It is clear that Ft is a filtration. As the next step, we consider “piece-wise constant” processes.Let S be the set of functions of the form

f (t, x, ω) =n∑i=1

Xi(ω)1(ai,bi](t)ϕi(x),

where Xi is a bounded Fai–measurable random variable and ϕi ∈ C∞c (R). For f ∈ S , define∫R+×R

f (t, x)ξ(t, x)dxdt =n∑i=1

Xi

∫R+×R

1(ai,bi](t)ϕi(x)ξ(t, x)dxdt.

It is easy to check that the integral is linear and satisfy Ito isometry from L2(R+ ×R×Ω) toL2(Ω), that is

E

[(∫R+×R

f (t, x)ξ(t, x)dxdt)2]

=

∫R+×R

E[f 2(t, x)]dxdt.

Let P be the sub–σ–field of B(R+ ×R)×F generated by S . Let L2(R+ ×R×Ω,F ,P)be the space of square integrable P–measurable random variables f (t, x, ω). These will be theintegrators. It is important to note that these are non–anticipating in the sense that f (t, x, ω)only depends on the information Ft up to time t. This is analogous to the distinction betweenIto and Stratonovich integrals in the one–dimensional case. The construction of the stochasticintegral will be defined through the isometry and approximation.

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STOCHASTIC HEAT EQUATION WITH MULTIPLICATIVE NOISE 13

Lemma 2.1 S is dense in L2(R+ × R× Ω,F ,P)

Proof. Same as one–dimensional case.

Thus, if f ∈ L2(R+ × R × Ω,F ,P) there exist fn ∈ S wuch that fn converges to f inf ∈ L2(R+ × R× Ω,F ,P). By the isometry,

In(ω) :=

∫R+×R

fn(t, x, ω)ξ(t, x)dxdt

is a Cauchy sequence in L2(Ω,F ,P). Hence there is a limit point I ∈ L2(Ω,F ,P) which isdefined to be the stochastic integral

∫R+×R f (t, x)ξ(t, x)dxdt. This is linear in f and the Ito

isometry holds.

Existence and uniqueness of mild solution. We will work with L2(Ω,F ,P) boundedinitial data and solutions:

Definition 2.2 A function z0(·) is L2(Ω,F ,P) bounded if supx∈R E[z0(x)2] < ∞ and aspace-time function Z(·, ·) is L2(Ω,F ,P) bounded if supt∈[0,T ]

x∈RE[Z(t, x)2] <∞.

Theorem 2.3 For L2(Ω,F ,P) bounded initial data z0 there exists a unique L2(Ω,F ,P)bounded mild solution to mSHE.

Proof of uniqueness. Assume Z and Z ′ solve SHE with the same initial data. Let u = Z−Z ′,hence u(0, x) ≡ 0. Define

f (t, x) = E[u(t, x)2] f (t) = supxf (t, x)

and note f (t) <∞ by L2(Ω,F ,P) boundedness. Thus, by Ito isometry,

f (t) = supx∈R

f (t, x) = supx∈R

∫ t

0

∫R

E[(z(y, s)− z′(y, s))2

]· p2(t− s, x− y)dyds

≤ supx∈R

∫ t

0

∫Rf (s, y)p2(t− s, x− y)dyds

≤ c

∫ t

0

f (s)ds√t− s

for some constant12 c > 0 whose value is not important for our purposes. Note that betweenthe second and third line we have evaluated the x–independent integral in y, thus explainingwhy we have dropped the supremum. Hence,

f (t) ≤ c

∫ t

0

f (s)ds√t− s

.

Iterate to get

f (t) ≤ c2

∫ t

0

∫ s

0

f (u)duds√

(s− u)(t− s)= c2

∫ t

0

f (u)∫ t

u

ds√(s− u)(t− s)

du. (2.3)

By Exercise 1, we can show that f (t) ≡ 0 hence Z = Z ′.12Constants will generally be denoted by c or C and can change line to line.

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14 STOCHASTIC HEAT EQUATION WITH MULTIPLICATIVE NOISE

Proof of existence. We will use Picard iteration. Let z0(t, x) ≡ 0 and define progressivelymeasurable approximations

zn+1(t, x) =

∫Rp(t, x− y)z0(y)dy +

∫ t

0

∫Rp(t− s, x− y)zn(s, y)ξ(s, y)dyds.

Then zn(t, x) := zn+1(t, x)− zn(t, x) satisfies

zn+1(t, x) =

∫ t

0

∫Rp(t− s, x− y)zn(s, y)ξ(s, y)dyds.

Hence, as before, via the Ito isometry

E[(zn+1(t, x))2

]=

∫ t

0

∫Rp2(t− s, x− y)E

[(zn(s, y))2

]dyds.

Definefn(t) = sup

s∈[0,t]x∈R

E[(z(s, x))2

]and note that f 0(t) <∞ by hypothesis. Then,

fn+1(t) = supu∈[0,t]x∈R

E[(zn+1(t, x))2

]= sup

u∈[0,t]x∈R

∫ u

0

∫Rp2(u− s, x− y)E

[(z(s, y))2

]dyds

≤ supu∈[0,t]x∈R

∫ u

0

∫Rp2(u− s, x− y)fn(s)dyds

≤ supu∈[0,t]

c

∫ u

0

fn−1(s)ds√u− s

≤ c

∫ t

0

fn−1(s)ds√t− s

.

Between the second and third lines we evaluated the x–independent integral in y. Theinequality in the last line can be seen by applying a change of variables so the integral is from0 to 1, and then using the fact that fn−1(s) is an increasing function.

As before we may iterate this once more and then change the order of integration. Wethus find that fn+1(t) ≤ c

∫ t0fn−1(u)du. By Exercise 2 this goes to zero as n → ∞ hence

proving that the zn(·, ·) form a Cauchy sequence in L2(Ω,F ,P). The limit point is z(·, ·). Itis clear, by convergence of stochastic integrals, that z solves the mild form of SHE.

2.2 Weak solutionThe solution as discussed in Section 1, such as the mild solution given by (1.5), is calledstrong solution, in the sense that we are given a probability space Ω, each realization of ξ(ω)(ω ∈ Ω) is mapped to a solution u(ω). Strong solutions could sometimes be inconvenient forinstance when proving certain results on convergence in law to the solutions from discretesystems in probability theory or statistical physics. A weak solution only seeks for a process(which could live on a different probability space) which has the “right law”. 13

Recall that if M is a continuous square-integrable (E(M2t ) < ∞ for all t) martingale

then the quadratic variation 〈M〉t is the unique non-decreasing stochastic process such that

13The notion of “weak solution” in SPDE is totally different from “weak solution” in classical PDE.

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STOCHASTIC HEAT EQUATION WITH MULTIPLICATIVE NOISE 15

M2t − 〈M〉t is a martingale.14 As an example, if B is the standard Brownian motion, B2 − t

is a martingale and thus 〈B〉t = t. Levy’s theorem states that if X is continuous localmartingale, X0 = 0, 〈X〉t = t, then X is a standard Brownian motion. As another example,given ϕ ∈ C∞c (R), ∫

R+×R1(0,t](s)ϕ(x)ξ(s, x)dxds (2.4)

considered in Section 2.1 is a Brownian motion with variance ‖ϕ‖2L2 (its quadratic variation

is t‖ϕ‖2L2).

Just as the martingale characterization of the law of Brownian motion in Levy’s theorem,a weak notion of solution to the aSHE on Td

∂tu = ∆u+ ξ u(0) = u0 (2.5)

can be also defined in a similar way. To this end, given ϕ ∈ C∞(Td) writing (u, ϕ) for theevaluation of the distribution u ∈ S ′(T2) against ϕ, we have, by (2.5)

(u(t), ϕ) = (u0, ϕ) +

∫ t

0

(u(s),∆ϕ)ds+

∫ t

0

(ϕ, ξ(s))ds

and the last term is precisely (2.4), that is, a Brownian motion with quadratic variation t‖ϕ‖2L2 .

This motivates the following definition.

Definition 2.4 v ∈ C(R+,S ′(T2)) is called a weak solution to (2.5) if

Mϕ(t) def= (v(t), ϕ)− (u0, ϕ)−

∫ t

0

(v(s),∆ϕ)ds

Γϕ(t) def= Mϕ(t)2 − t‖ϕ‖2

L2

(2.6)

are local martingales for any ϕ ∈ C∞(Td).

Note that v could be defined on a different probability space. We also say that v “solves amartingale problem (2.6)”.

Weak solutions can also be defined for mSHE

∂tZ = ∆Z + Zξ Z(0) = Z0 . (2.7)

The main modification is due to the fact that the last term integrating test functions hasdifferent quadratic variation.

Definition 2.5 v ∈ C(R+,S ′(T2)) is called a weak solution to (2.7) if Mϕ(t) defined as in(2.6) and

Γϕ(t) def= Mϕ(t)2 −

∫ t

0

(v(s)2, ϕ2)ds

are local martingales for any ϕ ∈ C∞(Td).

Theorem 2.6 If v is a weak solution to aSHE (resp. mSHE), then v has the same law as themild solution given by (1.5) (resp. Theorem 2.3).

14There is another notion of quadratic variation usually denoted by [M ]t, which is in general different from〈M〉t, but for continuous M they coincide.

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16 Φ4 EQUATION

Exercises

1. Show that the integral over s is equal to π so that f (t) ≤ πc2∫ t

0f (u)du. Iterate to show

f (t) ≤ (πc2)n+1

n!

∫ t

0

f (u)(t− u)ndu.

Since f (u) <∞ this shows that f (t) ≡ 0 hence Z = Z ′.

2. Iterating and using our a priori knowledge that f 0, f ′ <∞ show that

fn(t) ≤ (ct)n/2

(n/2)!.

3 Φ4 equation

3.1 Local mild solution in d = 1

∂tu = ∆u− u3 + ξ u(0) = u0 . (3.1)

u is called a mild solution if

u(t, x) =

∫Td

∫ t

0

Pt−s(x− y)ξ(ds, dy) + Ptu0(x)−∫

Td

∫ t

0

Pt−s(x− y)u(s, y)3dsdy (3.2)

where Ptu0(x) =∫

Td Pt(x− y)u0(y)dy is the solution to heat equation from initial conditionu0. We call u a local mild solution if there exists a stopping time τ s.t. (1) τ > 0 a.s. and (2)(3.2) holds a.s. for every t such that t < τ a.s. We call (u, τ ) a maximal local mild solution iffor any local mild solution (u, τ ), τ ≤ τ a.s.

Theorem 3.1 Let d = 1, α ∈ (0, 12), and u0 ∈ Cα. There exists a unique maximal local mild

solution (u, τ ) and u ∈ C([0, τ ), Cα). Moreover limt→τ ‖u(t)‖Cα =∞ a.s. on τ <∞.

The proof is based on very standard fixed point argument in classical PDE. We will oftenuse this kind of fixed point arguments later.

Proof. First, we have the following fact: since α > 0, ‖Ptu0‖Cα ≤ ‖u0‖Cα (“maximalprinciple” for heat equation). Given T > 0, let g ∈ C(R+, Cα) be given by

g(t, x) def=

∫Td

∫ t

0

Pt−s(x− y)ξ(ds, dy) + Ptu0(x) .

Define the fixed point map Mg,T : C([0, T ], Cα)→ C([0, T ], Cα) by

(Mg,Tu)(t) = −∫

Td

∫ t

0

Pt−s(x− y)u(s, y)3dsdy + g(t) .

A fixed point of Mg,T will be a mild solution. We will find such a (unique) fixed point byshowing that Mg,T is a contraction in a ball centered at g in C([0, T ], Cα), provided T > 0 issmall.

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Φ4 EQUATION 17

3.2 Important heuristics: perturbation theory3.3 Φ4 equation in d = 2

Consider the renormalized Φ4 equation

∂tuε = ∆uε − (u3ε − 3Cεuε) + ξε . (3.3)

Da Prato - Debussche argument. Write uε = Φε + vε, where

∂tΦε = ∆Φε + ξε .

The function vε can be thought of as the “remainder” after the zero-th order perturbation, andit satisfies

∂tvε = ∆vε −(v3ε + 3Φεv

2ε + 3(Φ2

ε − Cε)vε + Φ3ε − 3CεΦε

). (3.4)

In this subsection we aim to show:(1) Φ2

ε − Cε and Φ3ε − 3CεΦε converge in probability in Cαs for any α < 0. We will call

the limits :Φ2 : and :Φ3 : respectively. (The fact that their integrations against smoothfunctions have bounded second moment was essentially already seen in Section 3.2. Here weidentify the distribution space Cαs in which they converge.)

(2) When passing to the limit ε→ 0, we can find a local mild solution to the equation

∂tv = ∆v −(v3 + 3Φv2 + 3 :Φ2 : v+ :Φ3 :

)(3.5)

with classical PDE fixed point argument (similarly as in d = 1). We could then conclude thatu = Φ + v is the local solution to the renormalized Φ4 equation in two spatial dimensions –note that Φ is singular but is Gaussian and explicit, while v is more regular but implicit.

The above argument was first used by Da Prato and Debussche in [DPD03]Before proving the above two results, one might ask: why has the constant Cε in (3.3)

been precisely distributed to the “right places” in (3.4) to make the powers of Φε converge?We first give a better interpretation to the renormalization in this context.

Wiener chaos. The space-time white noise ξ as defined by (1.2) can be viewed as animbedding from L2 functions to L2 random variables:

L2(Rd+1) → L2(Ω) f 7→ ξ(f ) .

It preserves the L2 norm: E[ξ(f )2] = ‖f‖2L2 . Write H ⊂ L2(Ω) for the image of this

imbedding; elements of H are centered Gaussian random variables of the form ξ(f ). Let

Hn = p(Z1, · · · , Zk) | p ∈ R[z1, · · · , zk], deg p ≤ n, Z1, · · · , Zk ∈ H ,

where R[z1, · · · , zk] is the ring of real-coefficient polynomials of k unknowns. In plainwords, Hn contains random variables which are n-th order polynomials of Gaussians. Itsclosure Hn in L2(Ω) is called the n-th Wiener chaos. The orthogonal complement of Hn−1

in Hn, denoted by H :n : , is called the homogeneous chaos of order n. A useful fact is that ifZ ∈ H , then hn(Z) ∈ H :n : where hn is the n-th Hermite polynomial. (This is due to thefact that the Hermite polynomials form orthonormal basis of L2(Ω, 1√

2πe−x

2/2dx).)

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18 (CONTROLLED) ROUGH PATHS

4 (Controlled) rough paths

In Ito calculus, we define Ito integral by approximating measurable, adapted processes bysimple processes, and apply Ito isometry. It turns out that there is a different approach:roughly speaking, by postulating the simple integral

∫XdX one construct general integrals∫

Y dX for Y which locally “look like” X . This is called “rough path theory”. The truepower of “rough path theory” is that it gives an almost sure pathwise definition of stochasticdifferential equation; and the solution to the equation, when lifted onto the level of roughpaths, is continuous in the driven noise.

We denote by C = C([0, T ],Rn) the space of continuous functions, and by C2([0, T ],Rn)the space of continuous functions from R2 into Rn that vanish on the diagonal. We willoften omit [0, T ] and Rn in our notations. We define a difference operator δ : C → C2 byδXs,t = Xt −Xs. For an element X ∈ C2 we will mainly be interested in the rate at which itvanishes at diagonal ‖X‖β = sups 6=t

|Xs,t||s−t|β , rather than its regularity in each of its variables.

Rough paths. A rough path (X,X) consists of two parts: a continuous function X ∈C([0, T ],Rn), as well as a continuous “area process” X ∈ C2([0, T ],Rn×n) such that thealgebraic relations

Xij(r, t)− Xij(r, s)− Xij(s, t) = δX i(r, s)δXj(s, t) , (4.1)

hold for every triple of points (r, s, t) and every pair of indices (i, j). One should think of Xas an “auxiliary piece of information” which postulates the value of the quantity∫ t

s

δX i(s, r) dXj(r) def= Xij(s, t) , (4.2)

where we take the right hand side as a definition for the left hand side. The aim of imposing(4.1) is to ensure that Xij does indeed behave like an integral when considering it over twoadjacent intervals. We sometimes also call a rough path X instead of (X,X) for simplicity.For α ∈ (0, 1) we say (X,X) ∈ C α if (4.1) holds and

‖X‖α = sups 6=t

|δXs,t||s− t|α

<∞ , ‖X‖2α = sups 6=t

|Xs,t||s− t|2α

<∞ .

Example 4.1 Let n = 1, X be the Brownian motion,15 and Xs,t = 12(Bt − Bs)2 − c(t− s)

where c ∈ R. Then (X,X) is a rough path, and (X,X) ∈ C α for α < 12. In fact, we are here

just postulating that ∫ t

0

Bs dBs =1

2B2t − ct (4.3)

because (4.3) is equivalent (Exercise) with∫ tsδB(s, r) dB(r) = 1

2(Bt − Bs)2 − c(t− s). It

can be easily checked that (4.1) holds (Exercise).

Controlled rough paths. Given a rough path X taking values in Rn, we say that a pairof functions (Y, Y ′) is a rough path controlled by X if the “remainder term” R given by16

R(s, t) def= δY (s, t)− Y ′(s) δX(s, t) , (4.4)

15a typical sample path of Brownian motion16Here, Y (s, t), R(s, t) ∈ Rm and the second term is a matrix-vector multiplication. Note that this definition

does not depend on X thus we did not explicitly mention it.

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(CONTROLLED) ROUGH PATHS 19

has better regularity properties than Y . Typically, we will assume that ‖Y ‖α < ∞ and‖Y ′‖α <∞ for some Holder exponent α ∈ (0, 1), but ‖R‖2α <∞, where ‖R‖2α is the normfor functions of two variables defined as above, namely |R(s, t)| . |s− t|2α.

One could heuristically think of (4.4) as similar with Taylor expansion for a smoothfunction

f (t)− f (s) = f ′(s)(t− s) +O(|t− s|2)

and just like smooth functions can be approximated by polynomials in t−s in a neighborhoodof s, (4.4) says that Y can be approximated byX locally. Thus Y ′ is often called the Gubinelliderivative.

The space of controlled rough paths controlled by X is denoted by D2αX endowed with

the semi-norm‖Y, Y ′‖X,2α = ‖Y ′‖α + ‖R‖2α .

D2αX is a Banach space under the norm |Y0|+ |Y ′0 |+ ‖Y, Y ′‖X,2α. On a fixed interval [0, T ]

the above norm on D2αX being finite implies that Y is α-Holder: indeed,

‖Y ‖α ≤ ‖Y ′‖∞‖X‖α + ‖R‖α ≤ C(1 + ‖X‖α)(|Y ′0 |+ ‖Y, Y ′‖X,2α) .

A priori there could be many distinct “derivative processes” Y ′ associated to a path Y .However, under certain conditions (so called “truely rough” [FH14, Section 6.2]) Y ′ is unique.For instance this is the case if X is a typical sample path of Brownian motion and if weimpose the bound ‖R‖β <∞ for some β > 1

2.

Example 4.2 If f is a smooth function, X = B, and consider Y := f (B), then

δY (s, t) = f (Bt)− f (Bs) = f ′(Bs)(Bt −Bs) + f ′′(Bs)(Bt −Bs)2 + · · ·

So we should choose Y ′(t) = f ′(Bt) then (Y, Y ′) is a rough path controlled by B.

Here is an interesting lemma, which says that composition of a controlled path with asmooth function is still a controlled path.

Lemma 4.3 If f is a smooth function, (Y, Y ′) ∈ D2αX , then (f (Y ), f (Y )′) ∈ D2α

X wheref (Y )′ def

= Df (Y )Y ′. Furthermore

‖f (Y ), f (Y )′‖X,2α ≤ Cα,T,f (1 + ‖X‖α)2(1 + |Y ′0 |+ ‖Y, Y ′‖X,2α)

Proof. (Sketch.) Using (Y, Y ′) ∈ D2αX it is clear that

f (Yt) = f (Ys) +Df (Ys)δYs,t +O(δY 2s,t) = f (Ys) +Df (Ys)(Y ′s δXs,t +RY

s,t) +O(δY 2s,t)

sof (Y )′ = Df (Y )Y ′ , Rf (Y )

s,t = Df (Ys)RYs,t +O(δY 2

s,t) .

From this and the assumed bounds on Y, Y ′, RY it’s then easy to prove the lemma. See[FH14, Lemma 7.3] for details.

Integrating controlled paths. It turns out that if (X,X) is a rough path taking values inRn and Y is a path controlled by X that also takes values in Rn, then one can give a naturalmeaning to the integral

∫ ba〈Yt, dXt〉, provided that X and Y are sufficiently regular. The

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20 (CONTROLLED) ROUGH PATHS

approximation Yt ≈ Ys + Y ′s δXs,t suggested by (4.4) shows that it is reasonable to definethe integral as the following limit of “second-order Riemann sums”:∫ b

a

〈Y (t), dX(t)〉 = lim‖P‖→0

∑[s,r]∈P

(〈Y (s), δX(s, r)〉+ trY ′(s) X(s, r)) , (4.5)

where P denotes a partition of the integration interval.

Theorem 4.4 Let (X,X) ∈ C α and (Y, Y ′) ∈ D2αX with a remainder R given by (4.4). Then,

provided that α > 1/3, (4.5) converges, and∣∣∣∫ t

s

〈δY (s, r), dX(r)〉 − trY ′(s) X(s, t)∣∣∣ ≤ Cn,α|t− s|3α(‖X‖α‖R‖2α + ‖X‖2α‖Y ′‖α) .

(4.6)

The basic idea of proof of Theorem 4.4 goes as follows. Recall that

Y (t)− Y (s) = Y ′(s)(X(t)−X(s)) +R(s, t)

When integrate the left hand side w.r.t. X , one only needs to define the integrations of thetwo terms on the right hand side w.r.t. X:

(1) The integration of X(t)−X(s) w.r.t. X is postulated by X, so as long as ‖X‖2α <∞ and ‖Y ′‖α < ∞ and 3α > 1 (i.e. the Young’s condition holds) one can integrateY ′(s)(X(t)−X(s)) w.r.t. X .

(2) Regarding R, since ‖X‖α <∞ and ‖R‖2α <∞, and α > 1/3, its integration againstX is negligible.

For detailed proofs, we refer to [FH14, Section 4.1-4.3].

Example 4.5 Take the above examples of Brownian motion, f a smooth function, we havewe can define the integral against B as∫ b

a

f (Bt) dBt = lim‖P‖→0

∑[s,r]∈P

(f (Bs)(Br −Bs) + f ′(Bs) X(s, r))

= lim‖P‖→0

∑[s,r]∈P

(f (Bs)(Br −Bs) + f ′(Bs)(1

2(Br −Bs)2 − c(r − s)))

(4.7)

Note that a typical sample path of Brownian motion is Holder α ∈ (1/3, 1/2), so this limitconverges by the above theorem. It is important to understand that

• The statement (X,X) ∈ C α needs probability (in particular Kolmogorov theoremleading to B ∈ Cα a.s. for α ∈ (1/3, 1/2))

• After we take (X,X) ∈ C α the convergence of the above Riemann sum is a purelydeterministic statement, and the limit above is deterministic (not “limit of randomvariables” as in the standard construction of Ito integrals).

To compare with Ito integral, we need to again think of the B in (4.7) as the Brownian motion(rather than a typical sample path of it), and when c = 1

2, the second term converges to zero

in probability in the limit, and this is exactly Ito integral.

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(CONTROLLED) ROUGH PATHS 21

Results such as Ito formula can be also proved, see [FH14, Chapter 5].

Solving differential equation driven by (X,X) in the space C αX . It will turn out that a

solution to an SDEdXt = f (Xt)dBt

will also be controlled by B. “Y controlled by a rough path X” means that “Y behaves likeB at small scales”. Indeed, in the above example, if one takes a small portion of the path Band a small portion of the path f (B), one can not really tell the difference between the twosmall portions (although the two paths may look very different at large scales, for instancef (B) has a drift.) It can be also interpreted as “Y can be locally approximated by B”, justlike Y ∈ C1 means “Y can be locally approximated by a linear / tangent function”:

Y (t)− Y (s) = Y ′(s)(X(t)−X(s)) +R(s, t) X(t) = t

where R(s, t) = C2. Here Y ′ is the derivative and R is the Taylor Remainder, as the notationsindicate.

In Ito’s approach, Under linear growth and Lipschitz conditions, for L2 initial data, wecan prove existence and uniqueness of adapted L2 solution. See standard book, such asOksendal’s book Section 5.2. This is essentially our approach for mSHE in Section 2.1.

Fixed point argument via rough paths: [FH14, Chapter 8]. Being typed......

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22 (CONTROLLED) ROUGH PATHS

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