Quasilinear Stochastic Partial Differential Equations

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Quasilinear Stochastic Partial

Differential Equations

Owen Hearder

May 2019

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Declaration

The work in this thesis is my own except where otherwise stated.

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Acknowledgements

I would like to thank my supervisor Pierre Portal for guiding me through my thesis. His dedication to teaching and providing useful feedback has greatly helped.

I would also like to acknowledge Kelly, Frederick, Dean, Jack and Diclehan for making working in the office enjoyable. Without them, studying would have been very boring. Great Work.

I would also like to thank my family who supported me while I studied. Thank you to the ANU MSI for giving me this opportunity to study.

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Abstract

In this thesis we present a method of showing the existence and uniqueness of solutions to a quasilinear stochastic partial differential equation of divergence form. This document acts as a guide to the paper [8], where this method was first used.

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Notation and terminology

We collect here some notation which may not be introduced during the thesis.

Notation

supp(ψ) denotes the support of a real-valued function. N does not include zero.

Md×d(R) is the set of all real-valued d×d matrices.

sgn(x) denotes the sign function, which outputs−1 or 1 and is unde-fined at 0.

σ(A) is the σ-algebra generated by a collection of sets A.

σ(Xi|i∈I) is theσ-algebra generated by a collection of random variables.

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Chapter 1 Introduction

The goal of this thesis is to show the existence and uniqueness of solutions to the quasilinear stochastic partial differential equation

du(t) = −div(B(u(t)))dt+ div(A(u(t))∇u(t))dt+σ(u(t))dW(t) x∈_Td_{, t}_∈_{[0, T}_]

u(0) =u0.

(1.1) Here, B : _R → _Rd _{is a (flux) function and} _A _:

R → Md×d(R) is a uniformly positive definite bounded matrix. The stochastic termW is a cylindrical Wiener process with values inL2(_Td) andσ is a Hilbert-Schmidt operator valued map on L2(_Td). The precise definitions of the terms in (1.1) will be given in chapter 4.

Quasilinear stochastic PDE’s occur in applications such as the stochastic Navier-Stokes equation for which there is a complete answer to existence and uniqueness of solutions. This is contrary to the deterministic case. These equa-tions also appear in describing phase flows in porous media. The stochastic term in each of these applications can represent random perturbations in the system or even a lack of information about the parameters of the system.

This thesis acts as a guide to [8], where the method of showing existence and uniqueness uses the Itˆo formula on the L1 norm. We aim to present a detailed account of the proofs given in [8], particularly in the cases where the arguments weren’t immediately obvious upon first reading. We also aim to provide the background needed to understand the problem statement and method of proving it.

In chapter 2 we collect a number of mathematical results and definitions to be used later in the thesis. Some of these topics include Bochner integration, function spaces, the heat semigroup and deterministic PDE results.

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In chapter 3 we detail the construction of the stochastic integral with respect to a cylindrical Wiener process. For the Hilbert space-valued case we do this in detail. We briefly outline the construction for Banach space-valued processes. We then state a generalized Itˆo formula and present existence, uniqueness and regularity results of two specific stochastic PDEs. The existence of solutions to stochastic PDEs with locally monotone coefficients is the starting point for the existence proof in chapter 5.

There are no proofs in these first chapters. All results are referenced from textbooks and papers. The focus of this thesis is describing the arguments of [8]. In chapter 4 we state the problem of existence and uniqueness of solutions to (1.1). We state the conditions on the coefficients, used in [8], and define what a solution to (1.1) is. These will be L2 _solutions.

Chapter 5 contains the existence proof and is broken up into 4 parts. The first part shows existence ofL2_{solutions to regularized versions of (1.1) with a uniform} energy estimate. The second part shows these solutions have Lp _{estimates. The}

third part proves an L∞ estimate on the gradient of the solutions. The final part then shows these solutions to the regularized version of (1.1) are in fact approximations to a solution to (1.1). Due to the uniform energy estimate these are approximations in the weak sense, but with the regularity estimates, these are shown to be approximations with respect to the norm.

In chapter 6 we prove the uniqueness of solutions to (1.1). Here we need the compactness of _Td _{to be able to dominate the} _L1 _{norm with the} _L2 _norm.

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Chapter 2 Preliminaries

This chapter contains a collection of mathematical results to be used in the proof of existence and uniqueness of solutions to (1.1). Some notation and definitions are introduced as well.

In this thesis, everything is assumed to be real-valued.

2.1 Measure Theory, Probability Spaces and

Stochas-tic Processes

This section introduces notation and definitions related to measure theory and integration and probability theory. For a detailed treatment of measure theory, see [13]. For a detailed treatment of probability theory, see [5]. For a detailed treatment of stochastic processes see [12].

We say (Ω,F,_P) is a probability space if it is a measure space with_P(Ω) = 1. Here Ω is the underlying set, F the σ-algebra of subsets and _P the probability measure. We say such a space is complete if for every A ⊂ Ω such that A ⊂ F where F ∈ F and µ(F) = 0, we have A ∈ F. We will exclusively be working with complete probability spaces.

A measurable function X : Ω→_Ris called a random variable. If X ∈L1_(Ω) then we use the notation

E(X) :=

Z

Ω

X(ω)d_P(ω).

As a convention, when writing down random variables, the dependence on the parameter ω ∈Ω is usually not written and is implicitly assumed to be there.

Given two measure spaces (M1,A1, µ1) and (M2,A2, µ2), there is a canonical way to define the cartesian product of these measure spaces. The associated

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measure space is denoted (M1 × M2,A1 ⊗ A2, µ1×µ2). In this thesis we will make use of Tonelli’s theorem without mention.

Theorem 2.1 (Tonelli’s Theorem). Suppose f : M1 × M2 → R is measurable and non-negative. Then

Z

M1×M2

f =

Z

M1

Z

M2

f(x, y)dµ2(y)dµ1(x) =

Z

M2

Z

M1

f(x, y)dµ1(x)dµ2(y).

In most cases we will be applying it to non-negative functions defined on [0, T]×Ω, usually represented by the norm of another function, to interchange the order of integration: _ERT

0 =

RT

0 E.

In this thesis, we will be working on a finite time interval [0, T]. A filtration, (Ft)t∈[0,T] on (Ω,F,P) is a family of σ-algebras such that Fs ⊂ Ft⊂ F for every

0≤s≤t≤T. We call a filtration normal if 1. A∈ F0 for every A∈ F such that P(A) = 0,

2. and for every t∈[0, T),we have Ft=

T

t<s≤T

Fs.

As we will see in the next chapter on stochastic integration, we will be working with normal filtrations on a complete probability space.

A stochastic process on [0, T]×Ω is a function u : [0, T]×Ω→ _R such that u(t) : Ω → _R is a random variable for each t ∈ [0, T]. We sometimes write (u(t))t∈[0,T] to denote this process as an indexed collection of random variables.

We say a stochastic process is (Ft)t∈[0,T]-adapted ifu(t) is Ft-measurable for

each t∈[0, T].

We say a stochastic process is continuous if

P({ω ∈Ω|u(·, ω) : [0, T]→R is continuous}) = 1. (2.1) We sometimes say that u is pathwise continuous.

We say two stochastic process X and Y are independent if for every n ∈ _N

and t1, ..., tn∈[0, T] we have

P(X(t1)≤x1, ..., X(tn)≤xn, Y(t1)≤y1, ..., Y(tn)≤yn)

=_P(X(t1)≤x1, ..., X(tn)≤xn)·P(Y(t1)≤y1, ..., Y(tn)≤yn),

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2.2. BOCHNER INTEGRATION 5

2.2 Bochner Integration

This section outlines the basic definitions and properties of integration of Banach space valued functions. For a comprehensive exposition of this topic, see [10].

Suppose we have a measure space (M,A, µ) and a Banach spaceX. We may assume X is a separable Banach space, since in our applications, the function spaces we look at are separable. This will simplify matters when defining the Bochner integral. A function f :M →X is measurable if f−1_(B)_{∈ A} _{for every} B ∈ B(X), the Borel σ-algebra of X.

We now construct the Bochner integral.

A simple function f :M → X is one of the form f(s) =

N

X

n=1

1An(s)xn,

for some x1, ..., xN ∈ X and A1, ..., AN ∈ A. We define the Bochner integral of

such a simple function as

Z

M

f dµ :=

N

X

n=1

µ(An)xn.

We see from this definition that the integral is X valued.

We then establish that every measurable function is the µ-almost everywhere limit of simple functions. This is always true if X is separable. This is not true in general.

We then define a measurable function, f : M → X as Bochner integrable if there exist a sequence of simple functions, (fn)∞n=1, such that

lim

n→∞

Z

M

||f −fn||Xdµ= 0,

where these integrals are real-valued integrals. In this case, we define

Z

M

f dµ= lim

n→∞

Z

M

fndµ,

where this limit exists and is independent of the sequence (fn)∞n=1.

We have the following characterisation of Bochner integrable functions. A measurable function f :M →X is Bochner integrable if and only if

Z

M

||f||Xdµ <∞,

where this integral is a real-valued integral.

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Theorem 2.2. Letf :M → X be Bochner integrable and letT be a closed linear operator with domain D(T) ⊂ X with values in a Banach space Y. Suppose f takes values in D(T) µ-almost everywhere and the µ-almost everywhere defined function T f : M →Y is Bochner integrable. Then f is Bochner integrable as a D(T)-valued function, R

Mf dµ∈D(T), and

T

Z

M

f dµ=

Z

M

T f dµ.

In this thesis we apply this theorem to bounded operatorsT ∈ L(X, Y), which certainly satisfy theorem 2.2 with D(T) =X.

We now introduce the function spaces Lp₍_M_;_{X) for} _p _∈ _[1,_∞_{]. We say}

f :M →Xandg :M → Xare equivalent iff(s) =g(s) forµ-almost everywhere s ∈ M. As usual we do not make the distinction between a function and it’s equivalence class in most cases. We then have the following definition:

Definition 2.3. For p ∈ [1,∞) we define Lp₍_M_;_{X) as the linear space of all}

(equivalence classes of) measurable functions f :M →X for which

Z

M

||f||p_Xdµ < ∞,

where this integral is a real-valued integral. We give Lp₍_M_;_{X) the norm}

||f||Lp₍_M_;_X₎ :=

Z

M

||f||p_Xdµ

1_p

.

We defineL∞(M;X) as the linear space of all (equivalence classes of) measurable functions f : M → X for which there exists a real number r ≥ 0 such that µ({||f||X > r}) = 0. We give L∞(M;X) the norm

||f||L∞₍_M_;_X₎:= inf{r ≥0|µ({||f||_X > r}) = 0}.

With this definition, it follows Lp₍_M_;_{X) is a Banach space for} _p _∈ _[1,_∞_].

It is also true that if X is separable, µ(M) < ∞ and there exists a sequence (An)∞n=1 ⊂ A such that (An)n∞=1 generates A, then Lp(M;X) is separable. A specific example we will apply this to, later on in chapter 5, is to the space L1_{([0, T}_];_{H) for some separable Hilbert space} _{H. Since}_B_{([0, T}_{]) is generated by} a countable collection of subsets, we see L1_{([0, T}_];_{H) is separable.}

We now state properties about the dual spaces ofLp₍_M_;_{X). We have when}

X is the separable dual of another Banach space and (M,A, µ) is σ-finite, then for every p∈[1,∞) we have an isometric isomorphism

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2.3. FUNCTIONS OVER _TD AND FUNCTION SPACES 7 where 1

p+

1

q = 1. Hence in the specific example above, we have (L

1_{([0, T}_];_H))∗ ₌ L∞([0, T];H).

2.3 Functions over

_T

d

and Function Spaces

In this section we discuss functions defined on _Td _{and various function spaces.}

For more details on integral order Sobolev spaces, H¨older spaces and the space H−1_{, see [4]. For more details on fractional order Sobolev spaces, see [14].}

We may view _Td _{using the standard manifold construction, which is useful}

when talking about properties of the heat semigroup (see section 2.4). We may also view _Td _{as the set [0,}_1]d _⊂

Rd with periodic boundary conditions. This means that a function f : _Td _→

R can be identified with a 1-periodic function ˜

f :_Rd _→

R. This means for everya∈Rd and j = 1, ..., d ˜

f(a+ej) = ˜f(a),

where in this case the ej represents an element of the standard basis for Rd.

Sometimes we may need to talk about a distance on_Td_{. When we say}_{x, y} _∈

Td we treat them as elements of [0,1]d_{. We then define the distance between}_x _and

y on _Td _{to be}

|x−y|_Td = inf{|x−y−z| | for z ∈_Zd}. (2.2)

When talking about integration over_Td, we use the Lebesgue measure on the set [0,1]d. Indeed, for a non-negative function, f, on _Td we define

Z

Td f :=

Z

[0,1]d

˜ f(x)dx,

and if f ∈L1₍

Td) we define the integral off overTd in the same way. For k ∈ _N ∪ {0} and p ∈ [1,∞], we denote Hk,p₍

Td) to be the Sobolev space with differentiability order k and integrability order p. In the following chapters we are only concerned with the cases k = 0 and k = 1. By definition H0,p₍

Td) = Lp(Td). The norm on H1,p(Td) is given by

||u||p_H1,p₍

Td) =||u||

p Lp₍

Td)+||∇u||

p Lp₍

Td), (2.3)

where we define the Lp _{norm of}_∇_u _{to be the} _Lp _{norm of} _|∇_u_|_.

For a ≥ 0 and p ∈ (1,∞), we define the fractional Sobolev space Ha,p₍

Td) (otherwise known as the Bessel potential spaces) by

Ha,p(_Td) ={f ∈Lp(_Td)| F−1 _{(1 +}_|_ξ_|2₎a₂_F_f

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whereF is the Fourier transform on Lp₍

Td). Whena∈N∪ {0}, this definition is consistent with the original definition of the integral order Sobolev spaces. These spaces also arise as interpolation spaces of the integral order Sobolev spaces in the sense [Hk,p₍

Td), Hk+1,p(Td)]θ =H(1−θ)k+θ(1+k),p(Td). The details of this statement are beyond the scope of this thesis and is only mentioned for the interested reader.

We now state the definition of H−1,2₍

Td) and present some properties. The spaceH−1 _:=_H−1,2₍

Td) is defined as the dual space of H1 :=H1,2(Td). For f ∈H−1 _{we define}

||fkH−1 := sup{|_H−1hf, ui_H1| |u∈H1,||u||H1 ≤1}, (2.4)

where _H−1h·,·i_H1 is the pairing between H

−1 _and _H1_{. We have the following} properties:

Proposition 2.4. We have the inclusion H1 _⊂ _L2₍

Td) ⊂ H−1 where if f ∈ L2₍

Td)⊂H−1 then

H−1hf, ui_H1 =hf, ui_L2₍

Td). The proof of proposition 2.4 is in [4].

We also define the H¨older spaces. For a compact set D⊂_Rk_{, for some}_k _∈

N, and Banach space X we define the space Cλ(D;X), for λ ∈ (0,1), to consist of functions u:D→X such that

||u||_Cλ₍_D_;_X₎ := sup x∈D

||u(x)||X + sup x,y∈D

x6=y

||u(x)−u(y)||X

|x−y|λ <∞. (2.5)

If instead the target space is _R, we define C1+λ_{(D), for} _λ _∈ _(0,_{1), to consist of}

functions u:D→_R such that

||u||_C1+λ₍_D₎:= sup x∈D

|u(x)|+ sup

x∈D

|∇u(x)|+ sup

x,y∈D x6=y

|∇u(x)− ∇u(y)|

|x−y|λ <∞. (2.6)

We also state Poincar´e’s inequality.

Theorem 2.5 (Poincar´e’s Inequality). There exists a K > 0, depending on d, such that

||u||2

H ≤K(||u||2L1₍

Td)+||∇u|| 2

H) (2.7)

for every u∈H1_.

Since_Td _{has finite measure, the}_L1 _{norm is bounded by the} _L2 _{norm and we} see (2.7) implies

||u||2

H ≤K(||u||

2

H +||∇u||

2

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2.4. THE HEAT SEMIGROUP ON L2(_TD) 9 In light of (2.3) and (2.8), we see

− ||∇u||2

H ≤ −

1 1 +K||u||

2

H1 +

K 1 +K||u||

2

H. (2.9)

This inequality will be used later on in chapter 5 to show local monotonicity conditions.

2.4 The Heat Semigroup on

L

2

(

_T

d

)

This section states important properties of the heat semigroup on L2₍

Td). For details, see [7] and [6].

We will need a regularizing operator on H :=L2₍

Td) to be able to regularize our equation (1.1). The existence of solutions to the regularization will be easier to show and they will be approximations to the solution of (1.1).

The regularizing operator we will use is the semigroup on H generated by the Laplacian on _Td_{. We will denote the semigroup as (P}

ε)ε>0. For each ε > 0, Pε ∈ L(H) with

||Pε||L(H) ≤1. (2.10) We may write for f ∈H

Pεf(x) = (Pε∗f)(x) =

Z

Td

Pε(x−z)f(z)dz, (2.11)

where Pε(z) is the heat kernel. The heat kernel is explicitly given by

Pε(x) =

X

m∈Zd

exp(−4π2|m|2ε) exp(2πim·x)). (2.12)

The following properties hold

Z

Td

Pε(x−z)dz = 1 for every x∈Td, (2.13)

||Pεf||L∞₍

Td) ≤Cε||f||H for every f ∈H, (2.14)

Z

Td

(Pε1(x−z)−Pε2(x−z))h(z)dz

≤C||h||_Cη₍

Td)(ε1−ε2)

αη

for every h∈Cη(_Td),

(2.15)

for some αη >0, where η ∈(0,1).

We also have the strong continuity of Pε. This means Pεf → f as ε → 0 in

the L2₍

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2.5 Inequalities

In this section we state some useful inequalities.

The following inequality follows from basic calculus:

Proposition 2.6 (Power Inequality). Let p∈[1,∞). Then there exists aCp >0

depending on p such that

N

X

n=1 an

!p

≤Cp N

X

n=1

ap_n, (2.16)

for every a1, ..., aN ≥0.

We also have Young’s inequality. The proof of which is found in [4].

Proposition 2.7 (Young’s Inequality). For every γ >0 and a, b∈_R, we have ab≤γa2+ 1

4γb 2

. (2.17)

We also have Gr¨onwall’s inequality, which will be useful to find uniform bounds on certain functions. A proof of the general result is found in [3].

Proposition 2.8 (Gr¨onwall’s Inequality). Let u ∈ L1([0, T]) and suppose there exist M1, M2 ≥0 such that

u(t)≤M1+M2

Z t

0

u(s)ds,

for every t∈[0, T]. Then for every t∈[0, T]

u(t)≤M1exp(M2T). (2.18)

2.6 Deterministic PDE Regualrity

This section covers two useful regularity theorems from deterministic PDE theory. We look at regularity of the solutions to the equation

∂tv(t, x) =divx(a(t, x)∇xv(t, x)) + divx(g(t, x)) +f(t, x)

v(0, x) =v0(x),

(2.19)

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2.6. DETERMINISTIC PDE REGUALRITY 11 We first define the spaceCα2,α([0, T]×_Td) forα >0. We sayf ∈C

α

2,α([0, T]×

Td) if

||f||_Cα

2,α := sup

(t,x),(s,y)∈[0,T]×_Td

(t,x)6=(s,y)

 

|f(t, x)−f(s, y)|

max{|t−s|12,|x−y|

Td}

α 

<∞. (2.20)

The expression in the denominator, which is raised to the power of α, is called the parabolic distance on [0, T]×_Td_{. We see that} _Cα₂,α _{is just a Holder space}

with respect to the parabolic distance, hence it is a Banach space. The following theorem is a restatement of Theorem 3.2 in [1]:

Theorem 2.9. Assume there exist µ, ν > 0 such that ν|ξ|2 _≤ _{a(t, x)ξ}_·_ξ _≤_µ_|_ξ_|2 for every (t, x) ∈ [0, T]×_Td_{. Also assume} _v

0 ∈ Cβ(Td) for some β > 0. Now suppose f ∈ Lr0_{([0, T}_] ×

Td) and g ∈ L2r0_{([0, T}_] ×

Td), then there exists an α∈(0, β] and a constant K1 >0 depending on d, µ, ν and r0 such that

1. There exists a weak solution,v, to (2.19) withv ∈L2([0, T];H1)∩C([0, T];H). 2.

||v||_Cα

2,α ≤K1 ||v0||Cα(Td)+||g||L2r0([0,T]×_Td₎+||f||_Lr₀_([0_,T_]_×

Td)

. (2.21)

We also have the higher regularity estimates. The following theorem is a restatement of Theorem 3.3 in [1]

Theorem 2.10. Let α ∈ (0,1). Assume a, g ∈ Cα2,α([0, T] × _Td) and f ∈

Lp_{([0, T}_]_×

Td) for some p≥ N1−α+2. If in addition we have v0 ∈C

1+α_{([0, T}_]_×

Td), then there exists a unique weak solution, v, to (2.19). Moreover there exists a constant K2 >0 such that

||v||

C1+2α,1+α ≤K2P(||a||C

α

2,α)(||v0||C1+α(Td)+||g||Cα2,α+||f||Lp([0,T]×Td)), (2.22) where P is a polynomial.

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2.7 Hilbert-Schmidt Operators

In this section we state the definiton of a Hilbert-Schmidt Operator. For more details, see [11].

SupposeU andV are separable Hilbert spaces. Let (ek)∞k=1be an orthonormal basis for U. We say a linear transformation T : U → V is a Hilbert-Schmidt operator if

||T||2_L₂₍_U,V₎ :=

∞

X

k=1

||T ek||2V <∞. (2.23)

We denote the space of such operators as L2(U, V), where it is endowed with the norm in (2.23). This norm is independent of the choice of orthonormal basis for U.

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Chapter 3 Stochastic Integration

This section will define the stochastic integral and state some of its properties. In the theory of stochastics, it is necessary to formulate how a stochastic process changes with respect to a change in the stochastic driving force. A simple example is a process which is described by

dX(t) =a(t)dB(t)

where the process X changes infinitesimally depending on the weight a times an increment in the Brownian motion driving the process. Intuitively this would translate to

dX(t)

dB(t) =a(t).

Unfortunately there is no way to make sense of the left hand side of this equation, since the Brownian itself is nowhere differentiable. Instead, we give meaning to the integral equation

X(t) =X(0) +

Z t

0

a(s)dB(s).

We first define the stochastic integral over Hilbert space valued processes. The method presented follows the exposition in [11].

We then briefly describe the construction of the stochastic integral over Ba-nach space valued processes and introduce theγ-radonifying operators. For more details, see [16].

We then state some useful results concerning the stochastic integral, such as the Burkh¨older-Davis-Gundy inequality and the Itˆo formula.

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We then present existence and regularity results of relatively simple stochastic PDE’s.

For this chapter, we will be working on a fixed finite time interval, [0, T], and a complete probability space (Ω,F,_P).

3.1 Stochastic Integration on Hilbert Spaces

For this section we letU and H be separable Hilbert spaces. All of the following results are taken from [11].

For a Q∈ L(U) (the bounded linear operators fromU to itself), we say it is symmetric if Q=Q∗ (the adjoint). We say Q is non-negative if for every f ∈U we have hQf, fi_U ≥0. For such a Q, we define the trace of Qto be

tr(Q) :=

∞

X

k=1

hQek, eki_U, (3.1)

where (ek)∞k=1 is an orthonormal basis for U. The trace is independent of the choice of this orthonormal basis.

3.1.1 Wiener Processes and Filtrations

We now want to define Gaussian measures on (U,B(U)), whereB(U) is the Borel σ-algebra of U.

Definition 3.1. A probability measure,µ, on (U,B(U)) is called Gaussian if for every u∈U

ˆ µ(u) :=

Z

U

exp(ihu, vi_U)dµ(v) = exp

ihm, ui_U − 1

2hQu, uiU

,

for some m ∈ U and some non-negative, symmetric, finite trace operator Q ∈ L(U). Moreoverµis uniquely determined by m andQ. We callm the mean and Q the covariance and we write µ=N(m, Q).

We now want to define the stochastic process we will be integrating against. It will be called a standard Q-Wiener process.

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3.1. STOCHASTIC INTEGRATION ON HILBERT SPACES 15 1. W(t,·) is a U-valued random variable on (Ω,F,_P),

2. W(0) = 0,

3. W(·, ω) is continuous on [0, T] for _P-almost everyω ∈Ω,

4. for every 0≤t1 < ... < tN ≤T, N ∈N, the random variables

W(t2)−W(t1), ..., W(tN)−W(tN−1) are independent;

5. the increments above have Gaussian laws:

P◦(W(t)−W(s))−1 =N(0,(t−s)Q) for every 0≤s ≤t≤T. We note that in the case of U =_R and Q the identity, we haveW is just the standard Brownian motion on [0, T], which we will denote as β.

We now want to define a normal filtration such that W is adapted to it. Let W be a standard Q-Wiener process and define

N :={A∈ F |_P(A) = 0}, F0

t :=σ(W(s)|0≤s≤t)

˜

F0

t :=σ(F

0

t ∪ N)

Ft:=

\

t<s≤T

˜

F0

s for t ∈[0, T) and FT := ˜FT0.

(3.2)

Then (Ft)t∈[0,T] is a normal filtration and W is (Ft)t∈[0,T]-adapted. Moreover W(t)−W(s) is independent of Fs for every 0≤s≤t ≤T.

3.1.2 Martingales

We introduce the notion of a martingale. We do this because the stochastic integral we define will be a martingale. This fact is vital in the proof of existence and uniqueness of solutions to (1.1).

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Proposition 3.3. Let X : Ω → H be a Bochner integrable random variable. Let G ⊂ F be a sub σ-algebra. Then there exists a unique G-measurable random variable _E(X|G) : Ω →H such that

Z

A

Xd_P=

Z

A

E(X|G)dP, (3.3)

for every A∈ G.

With this proposition we may now define what a martingale is.

Definition 3.4. Let M : [0, T]×Ω → H be an H-valued stochastic process on (Ω,F,_P) with filtration (Ft)t∈[0,T].

M is called an Ft-martingale if

1. _E(||M(t)||H)<∞ for every t ∈[0, T],

2. M(t) is Ft-measurable for every t∈[0, T],

3. _E(M(t)|Fs) = M(s)P-almost surely for every 0≤s≤t ≤T.

We define the space M2

T(H) to consist of the Ft-martingales,M, such that

||M||M2

T(H) := sup t∈[0,T]

E ||M(t)||2

1

2 _<_∞_. _(3.4)

The space M2

T(H) is a Banach space.

From the definition of a martingale, we see that for every t∈[0, T]

E(M(t)) =E(E(M(t)|F0)) =E(M(0)). (3.5) this equality will be used to show the expectation of the stochastic integral is zero.

We now introduce stopping times and stopped martingales. See [10] for more details.

Definition 3.5. A measurable function τ : Ω →[0, T] is called a stopping time with respect to a filtration (Ft)t∈[0,T] if {τ ≤t} ∈ Ft for every t ∈[0, T].

We introduce the notations∧t := min{s, t}. We have the following property of martingales:

Proposition 3.6. If M is an Ft-martingale and τ : Ω → [0, T] is a stopping

time with respect to (Ft)t∈[0,T], then the process (M(t∧τ))t∈[0,T] is also an Ft

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3.1. STOCHASTIC INTEGRATION ON HILBERT SPACES 17

3.1.3 The Stochastic Integral

We are now prepared to define the stochastic integral.

We let W be a standard Q-Wiener process with the associated normal filtra-tion (Ft)t∈[0,T].

We first define the integral on elementary functions, then define it on the completion of the elementary functions with respect to an appropriate norm.

Definition 3.7. An L(U, H)-valued process Φ : [0, T]×Ω → L(U, H) is called elementary with respect to (Ft)t∈[0,T] if there exist 0 = t0 < ... < tk =T, k ∈ N

such that

Φ(t) =

k−1

X

m=0

Φm1(tm,tm+1](t) (3.6)

for every t ∈ [0, T], where for each 0 ≤ m ≤ k −1, Φm : Ω → L(U, H) is Ftm

-measurable (with respect to B(L(U, H))) and only takes on a finite number of values in L(U, H).

We define Int(Φ)(t) :=

Z t

0

Φ(s)dW(s) :=

k−1

X

m=0

Φm(W(tm+1∧t)−W(tm∧t)) (3.7)

for every t ∈[0, T].

The set of all such elementary processes is denoted E.

We now have the property that the stochastic integral is in fact an Ft

-martingale:

Proposition 3.8. For every Φ ∈ E we have Int(Φ) ∈ M2

T(H). In other words

Int :E → M2

T(H).

In light of (3.5) and the fact W(0) = 0, we have

E(Int(Φ)(t)) =E(Int(Φ)(0)) = E(0) = 0. (3.8) We now want to endow E with a norm such that Int : E → M2

T(H) is an

isometry. Once this is done we may define Int on the completion of E, E, with respect to this norm.

First we state a technical lemma about bounded operators:

Lemma 3.9. If Q ∈ L(U) is non-negative and symmetric, then there exists a unique non-negative symmetric Q12 ∈ L(U) such that Q

1 2 ◦Q

1 2 =Q.

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We then define the norm onE by

||Φ||2_T :=_E

Z T

0

||Φ(s)◦Q12||2

L2(U,H)ds

. (3.9)

With this norm it follows for every Φ∈ E ||Φ||T =

Z ·

0

Φ(s)dW(s)

_M2

T(H)

. (3.10)

This equality is called the Itˆo isometry.

Therefore we can complete E with respect to this norm to get a mapping Int : E → M2

T(H). We would like an explicit representation of E.

Theorem 3.10. We have an explicit representation of E. We first define the σ-algebra

PT :=σ(Y : [0, T]×Ω→R|Y is left continuous and adapted to (Ft)t∈[0,T]). (3.11) Then

E =L2[0, T]×Ω;L2

Q12(U), H

, (3.12)

where measurability is with respect to PT and B(L2(Q

1

2(U), H)).

Once we extend the definition of the stochastic integral to Wiener processes with Q being the identity, we would then have Q12 also being the identity and

Q12(U) =U.

3.1.4 Extension to Cylindrical Wiener Processes

We want to extend our definition of the stochastic integral to when the increments ofW have the identity as the covariance. The previous construction doesn’t apply and in fact we don’t have a definition of anId-Wiener process since tr(Id) =∞. We first want to define a linear operator J : U → U such that J ∈ L2(U, U) andJ is one-to-one. We call such a map a Hilbert-Schmidt embedding. Explicitly we may define

J(u) =X

k=1 1

k hu, ekiUek, (3.13) where (ek)∞k=1 is an orthonormal basis for U. This defines a Hilbert-Schmidt embedding.

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3.2. STOCHASTIC INTEGRATION ON BANACH SPACES ANDγ-RADONIFYING OPERATORS19

Proposition 3.11. Let (ek)∞k=1 be an orthonormal basis of U and βk, k ∈N, be a family of independent real-valued Brownian motions. Define Q= J◦J∗, then Q∈ L(U) and Q is non-negative, symmetric with finite trace. The series

W(t) :=

∞

X

k=1

βk(t)J ek (3.14)

converges inM2

T(U). MoreoverW in (3.14) defines a standardQ-Wiener process

on U.

We also have Φ∈L2(U, H) if and only if Φ◦J−1 ∈L2(Q

1

2(U), H).

We call such a process, W, a cylindrical Wiener process with identity as the covariance.

With this proposition in mind we may now define the stochastic integral with respect to a cylindrical Wiener process,W. For every Φ∈L2_{([0, T}_]_×_Ω;_L

2(U, H)), measurable in the sense of theorem 3.10, we define

Z t

0

Φ(s)dW(s) :=

Z t

0

Φ(s)◦J−1dW(s), (3.15) where the second integral is with respect to the standard Q-Wiener process in (3.14).

We still have the property that the stochastic integral over a cylindrical Wiener process is a martingale.

3.2 Stochastic Integration on Banach Spaces and

γ

-radonifying Operators

In this section, we introduce theγ-radonifying operators and briefly outline stoc-ahstic integration over Banach space-valued processes. This is needed since the Burkh¨older-Davis-Gundy inequality (to be stated in the next section) is stated with this framework in mind. For more details on the construction of the stochas-tic integral, see [16]. For more details on γ-radonifying operators, see [15].

For this section we have U is a separable Hilbert space and E is a separable Banach space.

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the stochastic integral is a Banach space isomorphism. Then we may complete the space of simple processes with respect to this norm.

In the Banach space valued setting, there isn’t an obvious analogue to the Hilbert-Schmidt operators. The space of operators we want instead is the space of γ-radonifying operators, γ(U, E).

We first define theγ∞ norm as follows. For a linear operator T : U → E we

define

||T||2

γ∞(U,E) := supE N X k=1

γkT hk

2 E , (3.16)

where the supremum is taken over all finite orthonormal systems{h1, ..., hN} ⊂U,

N ∈ _N and (γn)∞n=1 is a sequence of independent real-valued Gaussian random variables with mean zero and variance one. The norm is independent of the choice of Gaussian random variables. We define the space γ∞(U, E) as the space

of bounded linear operators with finite γ∞ norm. We define γ(U, E) to be the

completion of the space of finite rank operators F : U → E under the γ∞ norm

in γ∞(U, E).

Once this is established, the stochastic integral is defined on appropriate γ(U, E)-valued processe.

We have the following fundamental inequality, which is one half of the iso-morphism induced by the stochastic integral.

Theorem 3.12 (Burkh¨older-Davis-Gundy Inequality). Let p ∈ (0,∞) and φ : [0, T]×Ω→γ(U, E) be a measurable process, then there exists a Cp > such that

E sup

t∈[0,T]

Z t 0 φ(s)dW(s) p

≤Cp||φ||p_Lp_(Ω;_L2_([0_,T_];_γ₍_U,E₎₎₎

=CpE

Z T

0

||φ(s)||2

γ(U,E)

p

2!

.

(3.17)

In the case where E is a separable Hilbert space, then γ(U, E) = L2(U, E) with equal norms and the construction of the stochastic integral yields the same result as in section 3.1.

3.3 Properties of the Stochastic Integral

In this section we state some properties of the stochastic integral in the Hilbert space setting.

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3.3. PROPERTIES OF THE STOCHASTIC INTEGRAL 21 For stochastic integration, we have an analogue to theorem 2.2.

Proposition 3.13. Let Φ∈L2_{([0, T}_]_×_Ω;_L

2(U, H)) and L∈ L(H,H)˜ where H˜ is another separable Hilbert space. Then the process (L(Φ(t)))t∈[0,T] is an element of L2_{([0, T}_]_×_Ω;_L

2(U,H))˜ and L

Z t

0

Φ(s)dW(s)

=

Z t

0

L(Φ(s))dW(s) (3.18) P-almost surely for every t ∈[0, T].

We also have the following theorem which connects stochastic integration in Hilbert spaces to real-valued stochastic integration.

Proposition 3.14. A cylindrical Wiener process, W, can be written as W(t) =

∞

X

k=1

βk(t)ek, (3.19)

where (βk) is an independent family of real-valued Brownian motions and(ek)∞k=1 is an orthonormal basis ofU. The sum in (3.19) converges inL2(Ω;C([0, T];U)).

Let Φ∈L2([0, T]×Ω;L2(U, H)). Then with the above representation

Z t

0

Φ(s)dW(s) =

∞

X

k=1

Z t

0

Φ(s)(ek)dβk(s), (3.20)

where the sum converges in L2_(Ω;_{C([0, T}_];_H)).

We now state a definition which has many equivalent representations due to propositions 3.13 and 3.14.

Definition 3.15. Let Φ∈L2([0, T]×Ω;L2(U, H)) and φ∈H. Then we define

Z t

0

hΦ(s)dW(s), φi:=

Z t

0

hΦ(s), φidW(s), (3.21) where (hΦ(s), φi)s∈[0,T] ∈ L2([0, T]×Ω;L2(U,R)) and hΦ(s), φi is the mapping u7→ hΦ(s)u, φi.

Due to propositions (3.13) and (3.14) we have the equalities

Z t

0

hΦ(s)dW(s), φi=

∞

X

k=1

Z t

0

hΦ(s)ek, φidβk(s)

=

Z t

0

Φ(s)dW(s), φ

,

(3.22)

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3.4 Generalized Itˆ

o Formula

We state a fundamental theorem concerning solutions to a certain stochastic PDE. We are looking at solutions to the equation

du(t) = F(t)dt+ div(G(t))dt+ Φ(t)dW(t) u(0) =u0.

(3.23)

The following theorem is proved in [9].

Theorem 3.16. Let ψ ∈ C1₍

Td) and φ ∈ C2(R) with sup

x∈R

|φ00(x)| < ∞. We

as-sumeF, Gi ∈L2(Ω;L2([0, T];L2(Td)))fori= 1, ..., dandΦ∈L2(Ω;L2([0, T];L2(U;L2(Td)))). Suppose u is a weak solution to (3.23). This means that

u∈L2(Ω;C([0, T];L2(_Td)))∩L2(Ω;L2([0, T];H1,2(_Td))) and for every ξ ∈ C∞(_Td₎ _and _t _∈ _{[0, T}_] _{the following equality holds}

P-almost surely

hu(t), ξi=hu0, ξi+

Z t

0

hF(s), ξids

−

Z t

0

hG(s),∇ξids+

Z t

0

hΦ(s)dW(s), ξi,

(3.24)

where h·,·i is the inner product on L2(_Td).

Then, for every t∈[0, T] we have _P-almost surely

hφ(u(t)), ψi=hφ(u0), ψi+

Z t

0

hφ0(u(s))F(s), ψids

−

Z t

0

hφ00(u(s))∇u·G(s), ψids−

Z t

0

hφ0(u(s))G(s),∇ψids +

Z t

0

hφ0(u(s))Φ(s)dW(s), ψi

+1 2

∞

X

k=1

Z t

0

hφ00(u(s))(Φ(s)(ek))2, ψids,

(3.25)

where h·,·i is the inner product on L2(_Td).

3.5 SPDE’s with Locally Monotone Coefficients

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3.5. SPDE’S WITH LOCALLY MONOTONE COEFFICIENTS 23 We consider equations of the form

dX(t) = A(t, X(t))dt+B(t, X(t))dW(t)

X(0) = X0 (3.26)

where the coefficients A and B have the certain property of being locally mono-tone, which will be defined.

We fix T ∈ (0,∞) and a separable Hilbert space U. We will work on the finite time interval [0, T] and complete probability space (Ω,F,_P). We let W : [0, T]×Ω→U be aU-valued cylindrical Wiener process and let (Ft)t∈[0,T]be the associated normal filtration.

First we will be working with a Gelfand triple

V ⊂H ⊂V∗. (3.27)

This means H is a separable Hilbert space andV is a reflexive Banach space such that V ⊂H is dense and the inclusion map i:V →H is continuous.

We then assume

A: [0, T]×V ×Ω→V∗ and B : [0, T]×V ×Ω→L2(U, H) (3.28) are progressively measurable. This means for every t ∈ [0, T] these maps re-stricted to [0, t]×V ×Ω are B([0, t])⊗ B(V)⊗ Ft measurable.

We now assume the following conditions on the coefficients. Suppose there exists an α ∈ (1,∞), a β ∈ [0,∞), a θ ∈ (0,∞), a C0 ∈ R, a non-negative adapted process f ∈ L1_{([0, T}_]_×_{Ω) and a measurable locally bounded mapping} ρ:V →[0,∞) such that for every u, v, w ∈V and (t, ω)∈[0, T]×Ω:

The map λ7→_V∗hA(t, u+λv, ω), wi_V is continuous on _R, (H1)

2_V∗hA(t, u, ω)−A(t, v, ω), u−vi_V +||B(t, u, ω)−B(t, v, ω)||2_L

2(U,H) ≤(f(t, ω) +ρ(v))||u−v||2

H,

(H2)

2_V∗hA(t, v, ω), vi_V +||B(t, v, ω)||2_L

2(U,H) ≤C0||v||

2

H −θ||v|| α

V +f(t, ω) (H3)

||A(t, v, ω)||

α α−1

V∗ ≤(f(t, ω) +C0||v||αV)(1 +||v|| β

H). (H4)

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Definition 3.17. A continuous H-valued (Ft)-adapted process (X(t))t∈[0,T] is called a solution to (3.26) if

X ∈Lα([0, T]×Ω;V)∩L2([0, T]×Ω;H),

with α as in (H3) and (H4) and for every t∈[0, T] we have _P-almost surely

X(t) = X(0) +

Z t

0

A(s, X(s))ds+

Z t

0

B(s, X(s))dW(s),

where this equality holds in V∗. This makes sense, since anything that is V orH valued is also V∗-valued due to the Gelfand triple. Note that the first integral is a V∗-valued Bochner integral.

We now state the main theorem we will be using in our proof of existence.

Theorem 3.18. Suppose (H1), (H2), (H3) and (H4) hold wheref ∈Lp2([0, T]×

Ω)for some p≥β+ 2 and there exists a constant C such that for everyt ∈[0, T] and v ∈V

||B(t, v)||2_L₂₍_U,H₎ ≤C(f(t) +||v||2_H), ρ(v)≤C(1 +||v||α

V)(1 +||v|| β H).

(3.29)

Then ifX0 ∈Lp(Ω;H)isF0-measurable, there exists a unique solution(X(t))t∈[0,T] to (3.26) in the sense of definition 3.17.

3.6 Stochastic Regularity

In this section we state regularity results of solutions to the equation dz(t) =∆z(t)dt+ Ψ(t)dW(t)

z(0) =0. (3.30)

For brevity, we make the notation Ha,r _:=_Ha,r₍

Td).

We have the following proposition (adapted from Proposition 3.1 in [1]):

Proposition 3.19. Let a ≥ 0, r ∈ [2,∞) and Ψ∈ Lp_(Ω;_Lp_{([0, T}_];_{γ(H, H}a,r₎₎₎

be progressively measurable. This means Ψrestricted to Ω×[0, t] isFt⊗ B([0,

t])-measurable with respect to B(γ(H, Ha,r_{)). Then (3.30) has a unique solution,} _z,

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3.6. STOCHASTIC REGULARITY 25 1. If p ∈ (2,∞) and δ ∈ [0,1− 2

p), then for any λ ∈ [0,

1 2 −

1

p −

δ

2) we have z ∈Lp_(Ω;_Cλ_{([0, T}_];_Ha+δ,r₎₎ _and

E||z||p_Cλ_([0_,T_];_Ha+δ,r₎ ≤CLE||Ψ||p_Lp_([0_,T_];_γ₍_H,Ha,r₎₎. (3.31)

2. If p∈[2,∞) and δ∈(0,1), then z ∈Lp(Ω;Lp([0, T];Ha+δ,r)) and E||z||pLp_([0_,T_];_Ha+δ,r₎ ≤CLE||Ψ||

p

Lp_([0_,T_];_γ₍_H,Ha,r₎₎. (3.32)

We note that in [1], the proposition is stated for Dirichlet boundary conditions as opposed to the periodic boundary conditions suitable to our problem. In the main paper [8], they state proving the above for periodic boundary conditions is much simpler than the proof for Dirichlet conditions. To check this, one would need to check the semigroup of the laplacian on _Td _{has the same key}

proper-ties, as the laplacian with Dirichlet boundary conditions, used in the proof the proposition. For specifics, see the proof in [1] and the references therein.

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Chapter 4 Assumptions

For the next 3 chapters, we follow [8] in proving the existence and uniqueness of solutions to (1.1).

This chapter will define the conditions imposed on the equation (1.1) and what it means for a process to be a solution to it. We will also state and prove some properties of the equation from these assumptions.

4.1 Assumptions

We fix a T ∈(0,∞) and will be working on the finite time interval [0, T]. We fix d∈_N and work on the domain _Td_.

We now introduce the assumptions on B and A. We assume B :_R→_Rd _{is of class} _C1

b. This means the component functions,

Bi : R → R for i = 1, ..., d , are continuously differentiable and have a bounded derivative. We define the constant

CB := sup i=1,...,d

x∈_R

|B_i0(x)|<∞. (4.1)

We assume A:_R→Md×d(R) is of class Cb1. This means the component

func-tions, Aij :R→Rfor i= 1, ..., d and j = 1, ..., d , are continuously differentiable with bounded derivatives. We define the constant

C_A0 := sup

i=1,...,d j=1,...,d x∈R

|A0_ij(x)|<∞. (4.2)

We also assume A is uniformly positive definite and uniformly bounded. This means there exist constants δA >0 and CA >0 such that for every v ∈ Rd and

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x∈_R:

A(x)v·v ≥δA|v|2,

|A(x)v| ≤CA|v|.

(4.3)

For notational simplicity we write H := L2₍

Td), H1 := H1,2(Td), H−1 := H−1,2₍

Td) and Ha,r := Ha,r(Td). We note that H is an infinite dimensional separable Hilbert space.

We consider a fixed complete probability space (Ω,F,_P) and a cylindrical Wiener H-valued process W : Ω×[0, T] → H. We give (Ω,F,_P) the normal filtration, (Ft)t∈[0,T], associated toW. We let{ek}∞k=1 be an orthonormal basis of H such that

W(t) =

∞

X

k=1

βk(t)ek,

where (βk)∞k=1 is a sequence of independent Brownian motions.

We then assume for eachk ∈_Nthere are functionsσk :R→Rand a constant

Cσ >0 such that ∞

X

k=1

|σk(x)|2 ≤Cσ(1 +|x|2) for every x∈R, ∞

X

k=1

|σk(y1)−σk(y2)|2 ≤Cσ|y1−y2|2 for every y1, y2 ∈R.

(4.4)

Then, for every u ∈ H we define σ(u) : H → H by (σ(u)ek)(x) = σk(u(x)) for

each x∈_Td_.

We first check this is well defined and thatσ(u)∈L2(H, H). Indeed by (4.4)

||σ(u)||2_L₂₍_H,H₎ =

∞

X

k=1

||σ(u)ek||2H = ∞

X

k=1

Z

Td

|σk(u(x))|2dx

=

Z

Td

∞

X

k=1

|σk(u(x))|2dx

≤

Z

Td

Cσ(1 +|u(x)|2)dx

=Cσ(1 +||u||2H).

(4.5)

Hence σ(u)∈L2(H, H) and so it is well-defined and is a bounded operator. We note that the first inequality in (4.4) was omitted from [Hofmanova,Zhang]. These assumptions are crucial forσ to be well-defined. If only the second inequal-ity of (4.4) is assumed, then we may take the example σk(x) = k for which the

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4.2. DEFINITION OF SOLUTION 29 We make an extra assumption onσ. There exists aCγ >0 such that for every

u∈H and r∈[2,∞), we have

||σ(u)||γ(H,Ha,r₎ ≤

  

Cγ(1 +||u||Ha,r), for a∈[0,1]

Cγ 1 +||u||Ha,r +||u||a H1,ar

, for a∈(1,2).

(4.6)

Of all the assumptions (4.6) is the most restrictive. In fact in [8], they outline a possible way of showing existence without the assumption (4.6). This is beyond the scope of this thesis. For the method presented in this thesis, the assumption is vital in proving theorem 5.5.

4.2 Definition of Solution

The definition of solution we will be looking at is one that is weak in the PDE sense but strong in the probabilistic sense.

Before we define a solution we first view the coefficient of the dt term,−div(B(u(t)))+ div(A(u(t))∇u(t)), as an element ofH−1. For everyu∈H we defineF(u)∈H−1

as

H−1hF(u), vi_H1 =−hdiv(B(u)), vi_H − hA(u)∇u,∇vi_H. (4.7)

Symbolically we write −div(B(u)) + div(A(u)∇u) :=F(u).

Since we haveH ⊂H−1, in the sense of proposition 2.4, and||f||H−1 ≤ ||f||_H,

we may view each quantity, in (1.1), that take values in H to take values in H−1 instead.

We may now define what a solution is.

Definition 4.1. An (Ft)t∈[0,T]-adapted continuous process u: Ω×[0, T]→H is called a solution to (1.1), if

u∈L2(Ω×[0, T];H1) (4.8) and the following equality holds in H−1 for_P-almost surely ω ∈Ω:

u(ω, t) = u0(ω) +

Z t

0

F(u(ω, s))ds+

Z t

0

σ(u(s))dW(s)(ω), (4.9)

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Now we drop the ω’s and show why this is equivalent to the definition in [8]. The equation (4.9) means for every v ∈H1_{, we have}

P-almost surely

H−1hu(t), vi_H1 =_H−1hu0, viH1+

H−1

Z t

0

F(u(s))ds, v

H1

+

H−1

Z t

0

σ(u(s))dW(s), v

H1

.

(4.10)

By using theorem 2.2 and proposition 2.4, this is equivalent to

hu(t), vi_H =hu0, vi_H +

Z t

0 H

−1hF(u(s)), vi_H1ds

+

Z t

0

σ(u(s))dW(s), v

H

.

(4.11)

By the definition of F and (3.22), this is equivalent to

hu(t), vi_H =hu0, viH −

Z t

0

hdiv(B(u(s))), vi_Hds

−

Z t

0

hA(u(s))∇u(s),∇vi_Hds

+

Z t

0

hσ(u(s))dW(s), vi_H .

(4.12)

Moreover, since C∞(_Td_{) is a dense subset of} _H1 _{we see that if (4.10) holds for} every v ∈ C∞(_Td_{), then it will hold for every} _v _∈ _H1_{. With an application} of integration by parts to the first integral of (4.12), this shows that (4.9) is equivalent to saying

hu(t), vi_H =hu0, vi_H +

Z t

0

hB(u(s)),∇vi_Hds

−

Z t

0

hA(u(s)∇u(s),∇vi_Hds

+

Z t

0

hσ(u(s))dW(s), vi_H,

(4.13)

P-almost surely for every v ∈ C∞(Td). This is the definition given in [8]. The reason for showing these equivalences is to show the notion of solution in [11] is compatible with (4.13).

4.3 Properties of the Coefficients

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4.3. PROPERTIES OF THE COEFFICIENTS 31

Proposition 4.2. Let A and B be as in section 4.1. Then for every x, y ∈ _R, ξ ∈_Rd_, _u_∈_H1 _and _v _∈_H:

|B(y)−B(x)| ≤d12C_B|y−x|, (4.14) |(A(y)−A(x))ξ| ≤d32C0

A|y−x||ξ|, (4.15)

div(B(u))∈H, (4.16)

Aij(v)∈H for every i= 1, ..., d and j = 1, ..., d, (4.17)

|A(u)∇u| ∈H. (4.18)

Proof. The inequality (4.14) follows from the mean value theorem from real anal-ysis and (4.1). Indeed if we define f : [0,1] → _Rd_{, t} _7→ _B_(x₊_t(y₋_{x)), then}

applying the mean value theorem to each coordinate function fj, j = 1, ..., d,

there exists cj ∈(0,1) such that

Bj(y)−Bj(x) = (y−x)B0j(x+cj(y−x)).

Therefore by (4.1)

|Bj(y)−Bj(x)| ≤CB|y−x|.

Therefore

|B(y)−B(x)|2 ₌

d

X

j=1

|Bj(y)−Bj(x)|2 ≤dCB2|y−x|

2_.

Taking the square root yields (4.14).

For (4.15) we apply the mean value theorem to the component functions Aij

and use the uniform bound (4.2) to get

|((A(y)−A(x))ξ|2 ₌

d

X

i=1

d

X

j=1

(Aij(y)−Aij(x))ξj

!2

≤

d

X

i=1

d

X

j=1

C_A0 |y−x||ξ|

!2

=d3(C_A0 )2|y−x|2_|_ξ_|2_.

Taking the square root yields (4.15).

For (4.16) we use the uniform bound (4.1) and calculate

|div(B(u))|2 ₌_|_B0

(u)· ∇u|2 _{≤ |}_B0

(u)|2_|∇_u_|2 _≤_dC2

B|∇u|

2_.

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For (4.17) we writeAij(v(x)) = (Aij(v(x))−Aij(0)) +Aij(0). Since constants

are square integrable over_Td_{we have}_A

ij(0) ∈H. We use the mean value theorem

to calculate

|Aij(v(x))−Aij(0)|2 ≤(CA0 )

2_|_v(x)_|2_.

Since the right hand side is integrable, the left hand side is also. Hence Aij(v)−

Aij(0) ∈H. SinceH is a vector space, Aij(v)∈H. From the above working, we

have the useful inequality

||Aij(v)||H ≤CA0 ||v||H + sup i=1,...,d j=1,...,d

|Aij(0)|. (4.19)

The statement (4.18) follows from (4.3) since

|A(u)∇u|2 _≤_C2

A|∇u|

2_.

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Chapter 5 Proof of Existence

This chapter is dedicated to proving the following theorem

Theorem 5.1. Let u0 ∈ Lm(Ω;C1+l(Td)) be F0-measurable for some l > 0 and every m ∈[2,∞). Under the assumptions of Chapter 4 there exists a solution to (1.1) in the sense of definition 4.1. Moreover, any solution to (1.1) satisfies the following energy inequality

CS :=E sup

0≤t≤T

||u(t)||2_H +

Z T

0

E||u(t)||2H1dt <∞. (5.1)

The method of proving this follows a number of complicated steps. First we regularize equation (1.1). We then show the coefficients of these regularized versions are locally monotone, hence we have unique L2 _{solutions to them (by} theorem 3.18) and we show they have a uniform energy estimate. We then showLp

estimates of these solutions. These estimates help us show estimates which areLp

in Ω andW1,∞ _in

Td. We then want to show these regularized solutions converge to a solution of (1.1). By the uniform energy estimate of the regularized solutions, we know they converge weakly to a process in some suitable topology. We then use the regularity estimates and stopping times to show this weak convergence implies strong convergence inL1 _{and that the limiting process is in fact a solution} to (1.1).

5.1 Regularizing the Equation

In this section, we regularize the equation (1.1) and show existence of solutions to these new equations.

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We define forε >0 and u∈H

Aε(u)(x) = [Pε(Aij(u))(x)] for x∈Td, (5.2)

where (Pε)ε>0 is the heat semigroup from section 2.4. This is well defined due to proposition 4.2 (4.17).

We then consider the regularized equation

duε(t) =−div(B(uε(t)))dt+ div(Aε(uε(t))∇uε(t))dt+σ(uε(t))dW(t)

uε(0) =u0.

(5.3)

The notion of solution to (5.3) we will be looking is the same as in definition 4.1. The only difference is A(u(t)) is replaced with A(u(t)). We now prove the

existence of solutions to (5.3) using Theorem 3.18.

Theorem 5.2. Let u0 ∈ L2(Ω, H) be F0-measurable. Under the assumptions of Chapter 4, for each ε >0 there exists a unique solution to (5.3). These solutions satisfy

CE := sup >0

E sup 0≤t≤T

||uε(t)||2_H +

Z T

0

E||uε(t)||2H1

<∞. (5.4)

Proof. We first show that for every ε >0, u∈H1,x∈_Td _and _ξ_∈

Rd

Aε(u)(x)ξ·ξ ≥δA|ξ|2,

|Aε(u)(x)ξ| ≤d

1 2C_A|ξ|.

(5.5)

By the (4.3) we know for every y∈_R

A(y)ξ·ξ≥δA|ξ|2,

|A(y)ξ| ≤CA|ξ|.

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5.1. REGULARIZING THE EQUATION 35 By definition of Aε we have by linearity of the integral and (5.6)

|Aε(u)(x)ξ|2 = d

X

i=1

d

X

j=1

Z

Td

Pε(x, z)Aij(u(z))dzξj

!2

=

d

X

i=1

Z

Td

Pε(x, z) d

X

j=1

Aij(u(z))ξjdz

!2

=

d

X

i=1

Z

Td

Pε(x, z)(A(u(z))ξ)idz

2

≤

d

X

i=1

Z

Td

Pε(x, z)|A(u(z))ξ|dz

2

≤

d

X

i=1

Z

Td

Pε(x, z)CA|ξ|dz

2

≤C_A2|ξ|2

d

X

i=1

(1)2 =dC_A2|ξ|2

(5.7)

Taking the square root gives the result. For the other inequality, we use (2.13) to get

Aε(u)(x)ξ·ξ =

Z

Td

Pε(x, z)A(u(z))ξ·ξdz ≥δA|ξ|2

Z

Td

Pε(x, z)dz =δA|ξ|2

We now want to frame our problem in terms of section 3.5. We see that H1 ⊂

H ⊂ H−1 is a Gelfand triple from proposition 2.4 and the fact ||h||H ≤ ||h||H1

for every h∈H1.

For each ε >0 we define a mapping Fε:H1 →H−1. It is defined by H−1hFε(u), vi_H1 =− hdiv(B(u)), vi_H − hAε(u)∇u,∇vi_H

=hB(u),∇vi_H − hAε(u)∇u,∇viH

(5.8)

for every u, v ∈ H1. The last equality follows from integration by parts. From here on h·,·i will denote the inner product on H.

We want to check if the equation (5.3) satisfies the conditions in section 3.5. The space U in section 3.5 is given by H in our problem. The coefficients of dt and dW(t) in (3.26) are given by

(t, v, ω)7→F(v) and (t, v, ω)7→σ(v). (5.9)

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L2(H, H) and F :H1 →H−1 are measurable. We note that this was not shown

in [8].

The measurability of σ is simple since it is Lipschitz continuous. Indeed for u, v ∈H1

||σ(u)−σ(v)||2

L2(H,H) =

∞

X

k=1

||(σ(u)−σ(v))ek||2H = ∞

X

k=1

Z

Td

|σk(u(x))−σk(v(x))|2dx

= Z Td ∞ X k=1

|σk(u(x))−σk(v(x))|2dx

≤Cσ

Z

Td

|u(x)−v(x)|2dx=Cσ||u−v||2H ≤Cσ||u−v||2H1.

The measurability of F also follows from the fact it is continuous. This is a little

more involved and we can only show pointwise continuity. Let v ∈H1_{. For every} u, w ∈ H1_{, define} _P

v(u, w) := |H−1hFε(u)−Fε(v), wiH1|. Then by proposition

4.2, (5.5) and (2.3)

Pv(u, w) =|hB(u)−B(v), wi − hAε(u)∇u−Aε(v)∇v, wi|

≤||B(u)−B(v)||H||w||H +| hAε(u)∇(u−v), wi |

+| h(Aε(u)−Aε(v))∇v, wi |

≤d12C

B||u−v||H||w||H +d

1 2C

A||∇(u−v)||H||w||H

+||(Aε(u)−Aε(v))∇v||H||w||H

≤d12C

B||u−v||H1||w||_H1 +d 1 2C

A||u−v||H1||w||_H1

+||(Aε(u)−Aε(v))∇v||H||w||H1.

(5.10)

We bound the last term using (2.14) and (4.2)

||(Aε(u)−Aε(v))∇v||2H =

Z Td d X i=1 d X j=1

Pε(Aij(u)−Aij(v))(x)∂xjv(x)

!2 dx ≤ Z Td d X i=1 d X j=1

Cε||Aij(u)−Aij(v)||H|∇v(x)|

!2 dx ≤ Z Td d X i=1 d X j=1

CεCA0 ||u−v||H|∇v(x)|

!2

dx

=C_ε2(C_A0 )2d3||∇v||2

H||u−v||2H

≤C_ε2(C_A0 )2d3||v||2

H1||u−v||2_H ≤C_ε2(C_A0 )2d3||v||2

H1||u−v||2_H1.

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5.1. REGULARIZING THE EQUATION 37 Taking the square root and combining with (5.10) yields

Pv(u, w)≤||w||H1||u−v||_H1(d 1

2(C_A+C_B) +C_εC0

Ad

3

2||v||_H1). (5.12)

Hence by (2.4)

||Fε(u)−Fε(v)||H−1 ≤(d 1

2(C_A+C_B) +C_εC0

Ad

3 2||v||

H1)||u−v||_H1, (5.13)

which shows the pointwise continuity of Fε. HenceFε is measurable.

As for the constants preceding the conditions (H1),(H2), (H3) and (H4), we take α= 2, β = 0, f to be a constant function to be determined later and θ and C0 to also be determined later. We also letρ(v) = C||v||2_H1 whereC is a constant

to be determined later. We let u, v, w∈H1_.

We first check (H1). We note that this was not done in [8]. This amounts to checking

λ 7→ _H−1hFε(u+λv), wi_H1

is continuous. Let (λn)∞n=1 ⊂Rbe a sequence which converges toλ∈R. We look at

H−1hFε(u+λnv), wi_H1 =−

Z

Td

div(B(u+λnv))(x)w(x)dx

−

Z

Td

A(u+λnv)(x)∇u(x)· ∇w(x)dx

−λn

Z

Td

A(u+λnv)(x)∇v(x)· ∇w(x)dx

(5.14)

and show this converges to _H−1hFε(u+λv), wi_H1. This follows from the

domi-nated convergence theorem. We first show the integrand of the 2nd term con-verges pointwise when n → ∞. We see

Aε(u+λnv)(x)∇u(x)·∇w(x) = d

X

i=1

d

X

j=1

Z

Td

Pε(x, z)Aij(u(z) +λnv(z))dz∂xju(x)∂xiw(x)

!2

.

Now we want to apply the dominated convergence theorem to take the limit as n → ∞ inside the integral. For fixed x, z ∈ _Td _{the limit of this integrand is}

Pε(x, z)Aij(u(z) +λv(z)) sinceAij is continuous. We see they are dominated by

an integrable function since

|Pε(x, z)Aij(u(z)+λnv(z))| ≤Pε(x, z)



|u(z)|+ sup

n∈_N

|λn||v(z)|+ sup i=1,...,d j=1,...,d

|Aij(0)|

 .

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Hence by the dominated convergence theorem

Aε(u+λnv)(x)∇u(x)· ∇w(x)→Aε(u+λv)(x)∇u(x)· ∇w(x) as n→ ∞.

(5.16) The integrand of the third term of (5.14) converges pointwise by the same argu-ment.

The integrand of the first term in (5.14) converges pointwise since B is con-tinuously differentiable. Now we show there is a dominating integrable function.

|div(B(u+λnv)(x)w(x)|=|B0(u+λnv)(x)||∇(u+λnv)(x)||w(x)|

≤d12C_B

|∇u(x)|+ sup

n∈N

|λn||∇v(x)|

|w(x)|. (5.17)

We also find the dominating integrable function for the integrand of the second term by using (5.5)

|Aε(u+λnv)(x)∇u(x)· ∇w(x)| ≤|Aε(u+λnv)(x)∇u(x)||∇w(x)|

≤d12C_A|∇u(x)||∇w(x)|.

(5.18)

The exact same argument applies to the third term of (5.14). Hence by the dominated convergence theorem

H−1hFε(u+λnv), wiH1 → _H−1hFε(u+λv), wiH1 as n → ∞, (5.19)

and (H1) is satisfied.

Now we want to show (H2). We have already shown

||σ(u)−σ(v)||2

L2(H,H)≤Cσ||u−v||

2

H. (5.20)

We now look to bound the other term. We defineM(u, v) := _H−1hFε(u)−Fε(v), u−vi_H1.

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inequal-5.1. REGULARIZING THE EQUATION 39 ity

M(u, v) =hB(u)−B(v),∇(u−v)i − hAε(u)∇(u−v),∇(u−v)i

− h(Aε(u)−Aε(v))∇v,∇(u−v),i

≤||B(u)−B(v)||H||∇(u−v)||H −δA||∇(u−v)||2H

+||(A(u)−A(v))∇v||H||∇(u−v)||H

≤d12C

B||u−v||H||u−v||H1−

δA

1 +K||u−v|| 2

H1 +

δAK

1 +K||u−v|| 2

H

+||(A(u)−A(v))∇v||H||u−v||H1 ≤d12C_B

1 4γ1

||u−v||2

H +γ1||u−v||2H1

− δA

1 +K||u−v|| 2

H1

+ δAK

1 +K||u−v|| 2

H +

1 4γ2

||(A(u)−A(v))∇v||H2 +γ2||u−v||2H1 ≤d12C

B

1 4γ1

||u−v||2

H +γ1||u−v||2H1

− δA

1 +K||u−v|| 2

H1

+ δAK

1 +K||u−v|| 2

H +

1 4γ2

C_ε2(C_A0 )2d3||v||2

H1||u−v||2_H +γ2||u−v||2H1

= d

1 2C_B

4γ1

+ δAK 1 +K +

C2

ε(C 0 A)2d3

4γ2

||v||2_H1

!

||u−v||2_H

−

δA

1 +K −d

1 2C

Bγ1 −γ2

||u−v||2

H1

(5.21)

So if we choose γ1 and γ2 small enough, such that ₁₊δA_K −d

1

2C_Bγ₁ −γ₂ > 0,

then

M(u, v)≤ d 1 2C_B

4γ1

+ δAK 1 +K +

C_ε2(C_A0 )2d3 4γ2

||v||2_H1

!

||u−v||2_H. (5.22)

Hence (H2) is satisfied since comnbining (5.20) and (5.22) yields 2M(u, v) +||σ(u)−σ(v)||2

L2(H,H) ≤

d12C_B

2γ1

+ 2δAK

1 +K +Cσ + C2

ε(C 0 A)2d3

2γ2

||v||2

H1

!

||u−v||2

H.

(5.23)

We now want to show (H3) is satisfied. We have already calculated in (4.5)

||σ(v)||2

L2(H,H) ≤Cσ(1 +||v||

2

Quasilinear Stochastic Partial Differential Equations