Random walk and the heat equation

Chapter 1. Random Walk and Discrete Heat Equation

1.1. Simple random walk

1.2. Boundary value problems

1.3. Heat equation

1.4. Expected time to escape

1.5. Space of harmonic functions

1.6. Exercises

Chapter 2. Brownian Motion and the Heat Equation

2.1. Brownian motion

2.2. Harmonic functions

2.3. Dirichlet problem

2.4. Heat equation

2.5. Bounded domain

2.6. More on harmonic functions

2.7. Constructing Brownian motion

2.8. Exercises

Chapter 3. Martingales

3.1. Examples

3.2. Conditional expectation

3.3. Definition of martingale

3.4. Optional sampling theorem

3.5. Martingale convergence theorem

3.6. Uniform integrability

3.7. Exercises

Chapter 4. Fractal Dimension

4.1. Box dimension

4.2. Cantor measure

4.3. Hausdorff measure and dimension

4.4. Exercises.