Preface ........................................................ ix
Notation ....................................................... xi
Introduction .................................................... 1
Chapter 1. Introduction ......................................... 3
1.1. Basic methodology .......................................... 4
1.2. The basic setting for Markov processes ..................... 6
1.3. Related approaches ......................................... 8
1.4. Examples .................................................. 10
1.5. An outline of the major obstacles ......................... 25
Chapter 2. An overview ......................................... 29
2.1. Basic setup ............................................... 30
2.2. Compact state spaces ...................................... 30
2.3. General state spaces ...................................... 33
Part 1. The general theory of large deviations ................. 39
Chapter 3. Large deviations and exponential tightness .......... 41
3.1. Basic definitions and results ............................. 41
3.2. Identifying a rate function ............................... 50
3.3. Rate functions in product spaces .......................... 53
Chapter 4. Large deviations for stochastic processes ........... 57
4.1. Exponential tightness for processes ....................... 57
4.2. Large deviations under changes of time-scale .............. 62
4.3. Compactification .......................................... 64
4.4. Large deviations in the compact uniform topology .......... 65
4.5. Exponential tightness for solutions of martingale
problems .................................................. 67
4.6. Verifying compact containment ............................. 71
4.7. Finite dimensional determination of the process rate
function .................................................. 73
Part 2. Large deviations for Markov processes and semigroup
convergence ............................................ 77
Chapter 5. Large deviations for Markov processes and
nonlinear semigroup convergence ..................... 79
5.1. Convergence of sequences of operator semigroups ........... 79
5.2. Applications to large deviations .......................... 82
Chapter 6. Large deviations and nonlinear semigroup
convergence using viscosity solutions ............... 97
6.1. Viscosity solutions, definition and convergence ........... 98
6.2. Large deviations using viscosity semigroup convergence ... 106
Chapter 7. Extensions of viscosity solution methods ........... 109
7.1. Viscosity solutions, definition and convergence .......... 109
7.2. Large deviation applications ............................. 126
7.3. Convergence using projected operators .................... 130
Chapter 8. The Nisio semigroup and a control representation
of the rate function ............................... 135
8.1. Formulation of the control problem ....................... 135
8.2. The Nisio semigroup ...................................... 141
8.3. Control representation of the rate function .............. 142
8.4. Properties of the control semigroup V .................... 143
8.5. Verification of semigroup representation ................. 151
8.6. Verifying the assumptions ................................ 155
Part 3. Examples of large deviations and the comparison
principle ............................................. 163
Chapter 9. The comparison principle ........................... 165
9.1. General estimates ........................................ 165
9.2. General conditions in Rd ................................. 172
9.3. Bounded smooth domains in Rd with (possibly oblique)
reflection ............................................... 179
9.4. Conditions for infinite dimensional state space .......... 184
Chapter 10.Nearly deterministic processes in Rd ............... 199
10.1.Processes with independent increments .................... 199
10.2.Random walks ............................................. 207
10.3.Markov processes ......................................... 207
10.4.Nearly deterministic Markov chains ....................... 219
10.5.Diffusion processes with reflecting boundaries ........... 221
Chapter 11.Random evolutions .................................. 229
11.1.Discrete time, law of large numbers scaling .............. 230
11.2.Continuous time, law of large numbers scaling ............ 244
11.3.Continuous time, central limit scaling ................... 260
11.4.Discrete time, central limit scaling ..................... 266
11.5.Diffusions with periodic coefficients .................... 269
11.6.Systems with small diffusion and averaging ............... 271
Chapter 12.Occupation measures ................................ 283
12.1.Occupation measures of a Markov process - Discrete
time ..................................................... 284
12.2.Occupation measures of a Markov process - Continuous
time ..................................................... 288
Chapter 13.Stochastic equations in infinite dimensions ......... 293
13.1.Stochastic reaction-diffusion equations on a rescaled
lattice .................................................. 293
13.2.Stochastic Cahn-Hilliard equations on rescaled
lattice .................................................. 305
13.3.Weakly interacting stochastic particles .................. 315
Appendix ...................................................... 343
Appendix A. Operators and convergence in function spaces ...... 345
A.l. Semicontinuity .................................. 345
A.2. General notions of convergence .................. 346
A.3. Dissipativity of operators ...................... 350
Appendix B. Variational constants, rate of growth and
spectral theory for the semigroup of positive
linear operators .................................. 353
B.l. Relationship to the spectral theory of
positive operators .............................. 354
B.2. Relationship to some variational constants ...... 357
Appendix C. Spectral properties for discrete and continuous
Laplacians ........................................ 367
C.l. The case of d = 1 ............................... 368
C.2. The case of d > 1 ............................... 368
C.3. E = L2(0) ∩ {p: ∫ pdx = 0} ...................... 369
C.4. Other useful approximations ..................... 370
Appendix D.Results from mass transport theory ................. 371
D.l. Distributional derivatives ...................... 371
D.2. Convex functions ................................ 375
D.3. The p-Wasserstein metric space .................. 376
D.4. The Monge-Kantorovich problem ................... 378
D.5. Weighted Sobolev spaces Hμ1(Rd) and Hμ-1(Rd) ...... 382
D.6. Fisher information and its properties ........... 386
D.7. Mass transport inequalities ..................... 394
D.8. Miscellaneous ................................... 401
Bibliography .................................................. 403
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