Lectures on Diffusion Problems and Partial Differential Equations
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Lectures on Diffusion Problems and Partial Differential Equations - By S. R. S. Varadhan. - Tata Institute of Fundamental Research, Bombay.
Contents
1 The Heat Equation 1
2 Kolmogorov?s Theorem 11
3 The One Dimensional Random Walk 15
4 Construction of Wiener Measure 19
5 Generalised Brownian Motion 31
6 Markov Properties of Brownian Motion 33
7 Reflection Principle 39
8 Blumenthal?s Zero-One Law 53
9 Properties of Brownian Motion in One Dimension 57
10 Dirichlet Problem and Brownian Motion 63
11 Stochastic Integration 71
12 Change of Variable Formula 91
13 Extension to Vector-Valued Ito Processes 95 ?
14 Brownian Motion as a Gaussian Process 101
15 Equivalent For of Ito Process 105 ?
16 Ito?s Formula ? 117
17 Solution of Poisson?s Equations 129
18 The Feynman-Kac Formula 133
19 An Application of the Feynman-Kac Formula.... 139
20 Brownian Motion with Drift 147
21 Integral Equations 155
22 Large Deviations 161
23 Stochastic Integral for a Wider Class of Functions 187
24 Explosions 195
25 Construction of a Diffusion Process 201
26 Uniqueness of Diffusion Process 211
27 On Lipschitz Square Roots 223
28 Random Time Changes 229
29 Cameron - Martin - Girsanov Formula 237
30 Behaviour of Diffusions for Large Times 245
31 Invariant Probability Distributions 253
32 Ergodic Theorem 275
33 Application of Stochastic Integral 281
Appendix 284
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