Contents

Chapter 1. Prelude 1

Chapter 2. Wiener integrals 9

2.1. White noise 9

2.2. Stochastic convolutions 11

2.3. Brownian sheet 12

2.4. Fractional Brownian motion 15

Chapter 3. A linear heat equation 19

3.1. A non-random heat equation 19

3.2. The mild solution 22

3.3. Structure theory 22

3.4. Approximation by interacting Brownian particles 28

3.5. Two or more dimensions 30

3.6. Non-linear equations 30

Chapter 4. Walsh–Dalang integrals 33

4.1. The Brownian filtration 33

4.2. The stochastic integral 34

4.3. Integrable random fields 37

Chapter 5. A non-linear heat equation 39

5.1. Stochastic convolutions 40

5.2. Existence and uniqueness of a mild solution 44

5.3. Mild implies weak 50

Chapter 6. Intermezzo: A parabolic Anderson model 53

6.1. Brownian local times 53

6.2. A moment bound 56

Chapter 7. Intermittency 63

7.1. Some motivation 63

7.2. Intermittency and the stochastic heat equation 66

7.3. Renewal theory 67

7.4. Proof of Theorem 7.8 69

Chapter 8. Intermittency fronts 71

8.1. The problem 71

8.2. Some proofs 72

Chapter 9. Intermittency islands 79

9.1. The existence and size of tall islands 79

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