Lectures on Diffusion Problems and Partial Differential Equations

The Heat Equation......Page 7
Kolmogorov's Theorem......Page 17
The One Dimensional Random Walk......Page 21
Construction of Wiener Measure......Page 25
Generalised Brownian Motion......Page 37
Markov Properties of Brownian Motion......Page 39
Reflection Principle......Page 45
Blumenthal's Zero-One Law......Page 59
Properties of Brownian Motion in One Dimension......Page 63
Dirichlet Problem and Brownian Motion......Page 69
Stochastic Integration......Page 77
Change of Variable Formula......Page 97
Extension to Vector-Valued Itô Processes......Page 101
Brownian Motion as a Gaussian Process......Page 107
Equivalent For of Itô Process......Page 111
Itô's Formula......Page 123
Solution of Poisson's Equations......Page 135
The Feynman-Kac Formula......Page 139
An Application of the Feynman-Kac Formula..........Page 145
Brownian Motion with Drift......Page 153
Integral Equations......Page 161
Large Deviations......Page 167
Stochastic Integral for a Wider Class of Functions......Page 193
Explosions......Page 201
Construction of a Diffusion Process......Page 207
Uniqueness of Diffusion Process......Page 217
On Lipschitz Square Roots......Page 229
Random Time Changes......Page 235
Cameron - Martin - Girsanov Formula......Page 243
Behaviour of Diffusions for Large Times......Page 251
Invariant Probability Distributions......Page 259
Ergodic Theorem......Page 281
Application of Stochastic Integral......Page 287
Language of Probability......Page 289
Kolmogorovs Theorem......Page 293
Martingales......Page 296
Uniform Integrability......Page 311
Up Crossings and Down Crossings......Page 313
Bibliography......Page 322