Advances in Superprocesses and Nonlinear PDEs

Cover......Page 1
Advances in Superprocesses 
and Nonlinear PDEs......Page 4
Super-Brownian Motion: Lp-Convergence of Martingales Through the Pathwise Spine Decomposition......Page 6
Contents......Page 8
1 Dual Markov Processes and Kuznetsov Measures......Page 12
2 Markov Property and the Existence of a Transition Function......Page 13
3 Existence and Uniqueness of Decomposition of Excessive Functions into Extremes......Page 14
4 Superdiffusions and Semilinear PDEs......Page 15
4.2 Positive Solutions and Their Boundary Traces......Page 16
References......Page 17
1 Introduction......Page 21
2 Extreme Points of Pμ and Strong Solutions......Page 24
3 Representation Theory and the Existence of Strong Solutions......Page 30
4 Freezing......Page 34
5 Groups Indexed by the Lattice......Page 37
6 Automorphisms of the Torus......Page 40
References......Page 42
1 Introduction......Page 43
2 The ODE Model......Page 44
3 The Spatial Stochastic Model......Page 45
3.2 The Space Is the Tree Td......Page 46
References......Page 47
1 Introduction......Page 50
2 Asymptotic Results on Near Critical Branching Processes......Page 51
3 Catalyst-Reactant Branching Processes with Controlled Immigration......Page 59
3.1 Diffusion Limit under Comparable Timescales......Page 61
3.2 Asymptotic Behavior of the Catalyst Population......Page 64
3.3.1 Stochastic Averaging in a Diffusion Setting......Page 65
3.3.2 Stochastic Averaging for Scaled Branching Processes......Page 66
References......Page 67
1 Overview......Page 69
2 The Git et al.\'s Almost-Sure Result......Page 71
2.1 The Almost-Sure Upper Bound......Page 73
3 Large Deviations Heuristics......Page 74
4 The Spine Approach, Martingales and Measures......Page 78
5 Proving the Large-Deviations Lower Bound......Page 84
6 Proving the Large-Deviations Upper Bounds......Page 86
7 A Spine Proof of the Martingale Upper-Bound......Page 88
7.1 Factoring Out the Spine Term......Page 89
7.2 A First Application of Varadhan\'s Lemma......Page 90
7.3 Dealing with the Sum Term......Page 93
7.4 A Second Application of Varadhan\'s Lemma......Page 95
7.5 Concluding the Upper-Bound for Zt(τ)......Page 96
8 An Alternative Approach to the Lower Bound......Page 97
References......Page 98
1.1 Finite Rate Symbiotic Branching Processes......Page 100
1.2 Infinite Rate Symbiotic Branching Processes......Page 103
2 The Longtime Behavior for Negative Correlations......Page 106
2.1 Some Known Results......Page 107
2.3 An Open Problem: Longtime Behavior for Positive Correlations......Page 109
3.1 Some Properties of SBMγ......Page 110
3.2 Proof of Theorem 1.2......Page 112
3.3.1 Proof of Theorem 2.3, γ<∞......Page 114
3.3.2 Proof of Theorem 2.3, γ=∞......Page 115
References......Page 117
1 Introduction......Page 119
2 Spine Decomposition......Page 121
3 Proof of Theorem 1.1......Page 123
References......Page 126
Index......Page 127