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Dawson-Watanabe Superprocesses and Measure-Valued Diffusions by Edwin Perkins1 University of British Columbia 1 Research Partial

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Dawson-Watanabe Superprocesses and Measure-Valued Diffusions by Edwin Perkins1 University of British Columbia 1 Research Partial Dawson-Watanabe Superprocesses and Measure-valued Diffusions by Edwin Perkins1 University of British Columbia 1 Research partially supported by an NSERC Research Grant. Superprocesses 127 Table of Contents Page Glossary of Notation 129 I. Introduction 132 II. Branching Particle Systems and Dawson-Watanabe Superprocesses II.1. Feller’s Theorem 135 II.2. Spatial Motion 137 II.3. Branching Particle Systems 143 II.4. Tightness 147 II.5. The Martingale Problem 159 II.6. Proof of Theorem II.5.11 172 II.7. Canonical Measures 178 II.8. Historical Processes 187 III. Sample Path Properties of Superprocesses III.1. Historical Paths and a Modulus of Continuity 193 III.2. Support Propagation and Super-L´evyProcesses 200 III.3. Hausdorff Measure Properties of the Supports 207 III.4. One-dimensional Super-Brownian Motion and Stochastic PDE’s 215 III.5. Polar Sets 223 III.6. Disconnectedness of the Support 240 III.7. The Support Process 244 IV. Interactive Drifts IV.1. Dawson’s Girsanov Theorem 247 IV.2. A Singular Competing Species Model-Dimension One 257 IV.3. Collision Local Time 262 IV.4. A Singular Competing Species Model-Higher Dimensions 276 V. Spatial Interactions V.1. A Strong Equation 281 V.2. Historical Brownian Motion 283 V.3. Stochastic Integration on Brownian Trees 292 V.4. Pathwise Existence and Uniqueness 300 V.5. Martingale Problems and Interactions 310 V.6. Appendix 314 References 318 128 Superprocesses Outline of Lectures at St. Flour 1. Introduction 2. Particle Sytems and Tightness (II.1-II.4) 3. The Martingale Problem and Non-linear Equation (II.4-II.8) 4. Path Properties of the Support of Super-Brownian Motion (III.1-III.3) 5. Polar Sets (III.5-III.6) 6. Interactive Drifts (IV) 7. Spatial Interactions 1. Stochastic Integration on Trees and a Strong Equation (V.1-V.3) 8. Spatial Interactions 2. Pathwise Existence & Uniqueness, and the Historical Martingale Problem (V.4-V.5) 9. Interacting Particle Systems 1. The Voter Model 10. Interacting Particle Systems 2. The Contact Process Note. A working document with Ted Cox and Rick Durrett was distributed to provide background material for lectures 9 and 10. Superprocesses 129 Glossary of Notation α ∼ t |α|/N ≤ t < (|α| + 1)/N = ζα, i.e., α labels a particle alive at time t |A| Lebesgue measure of A Aδ the set of points less than a distance δ from the set A Agφ Aφ + gφ Aˆ weak generator of path-valued Brownian motion–see Lemma V.2.1 A¯τ,m see Proposition V.2.6 A~ generator of space-time process–see prior to Proposition II.5.8 bp → bounded pointwise convergence bE the set of bounded E-measurable functions BSMP a cadlag strong Markov process with x 7→ P x(A) Borel measurable for each measurable A in path space B(E) the Borel σ-field on E C C(Rd) C(E) continuous E-valued functions on R+ with the topology of uniform convergence on compacts Cb(E) bounded continuous E-valued functions on R+ with the supnorm topology k d d Cb (R ) functions in Cb(R ) with bounded continuous partials of order k or less ∞ d d Cb (R ) functions in Cb(R ) with bounded continuous partials of any order CK (E) continuous functions with compact support on E with the supnorm topology C`(E) continuous functions on a locally compact E with a finite limit at ∞ C(g)(A) the g-capacity of a set A–see prior to Theorem III.5.2 C Borel σ-field for C = C(Rd) Ct sub-σ-field of C generated by coordinate maps ys, s ≤ t D ≡ equal in distribution D(E) the space of cadlag paths from R+ to E with the Skorokhod J1 topology Ds the set of paths in D(E) which are constant after time s Dfd smooth functions of finitely many coordinates on R+ × C –see Example V.2.8 D(n, d) space of Rn×d-valued integrands–see after Proposition V.3.1 ∆ cemetary state added to E as a discrete point D(A) domain of the weak generator A–see II.2 and Proposition II.2.2 D(A~)T domain of weak space-time generator–see prior to Proposition II.5.7 D(Aˆ) domain of the weak generator for path-valued Brownian motion –see Lemma V.2.1 D(∆/2) domain of the weak generator of Brownian motion–see Example II.2.4 D the Borel σ-field on the Skorokhod space D(E) (Dt)t≥0 the canonical right-continuous filtration on D(E) −W (φ) eφ(W ) e E the Borel σ-field on E E+ the non-negative E-measurable functions Eˆ {(t, y(· ∧ t)) : y ∈ D(E), t ≥ 0} β −1 fβ(r) r if β > 0, (log 1/r) if β = 0 Fˆ F × B(C(Rd)) Fˆt Ft × Ct ˆ∗ ˆ Ft the universal completion of Ft 130 Superprocesses FX the Borel σ-field on ΩX −β + gβ(r) r if β > 0, 1 + (log 1/r) , if β = 0 and 1, if β < 0 Gεφ see (IV.3.4) t R G(f, t) sup Psf(x)ds 0 x G(X) ∪δ>0cl{(t, x): t ≥ δ, x ∈ S(Xt)}, the graph of X h − m the Hausdorff h-measure–see Section III.3 h(r) L´evy’s modulus function (r log(1/r))1/2 2 + + 2 + + + + hd(r) r log log 1/r if d ≥ 3, r (log 1/r)(log log log 1/r) if d = 2 s,y Ht the Ht measure of {w : w = y on [0, s]}, s ≤ t, y(·) = y(· ∧ s) bp H the bounded pointwise closure of H H+ the set of non-negative functions in H ∞ S {0,...,n} I N = {(α0, . , αn): αi ∈ N, n ∈ Z+} n=0 IBSMP time inhomogeneous Borel strong Markov process–see after Lemma II.8.1 I(f, t) stochastic integral of f on a Brownian tree–see Proposition V.3.2 K the compact subsets of Rd Lip1 Lipschitz continuous functions with Lipschitz constant and supnorm ≤ 1 (LE) Laplace functional equation–see prior to Theorem II.5.11 (LMP )ν local martingale problem for Dawson-Watanabe superprocess with initial law ν–see prior to Theorem II.5.1 1 2 Lt(X) the collision local time of X = (X ,X )–see prior to Remarks IV.3.1 log+(x) (log x) ∨ ee −W (φ) LW (φ) the Laplace functional of the random measure W , i.e., E(e ) 2 Lloc see after Lemma II.5.2 M1(E) space of probabilities on E with the topology of weak convergence MF (E) the space of finite measures on E with the topology of weak convergence t MF (D) the set of finite measures on D(E) supported by paths which are constant after time t MF the Borel σ-field on MF (E) Mloc the space of continuous (Ft)-local martingales starting at 0 (ME) mild form of the nonlinear equation–see prior to Theorem II.5.11 (MP )X0 martingale problem for Dawson-Watanabe superprocess with initial state X0 –see Proposition II.4.2 ΩΩˆ × C(Rd) n t o ΩH [τ, ∞) H· ∈ C [τ, ∞),MF D(E) : Ht ∈ MF (D) ∀t ≥ τ ΩH ΩH [0, ∞) ΩX the space of continuous MF (E)-valued paths ΩD the space of cadlag MF (E)-valued paths pt(x) standard Brownian density x pt (y) pt(x − y) P the σ-field of (Ft)-predictable subsets of R+ × Ω t g x nR o Pt φ(x) E φ(Yt) exp g(Ys) ds 0 PX0 the law of the DW superprocess on (ΩX , FX ) with initial state X0 –see Theorem II.5.1 Pν the law of the DW superprocess with initial law ν Superprocesses 131 ˆ PT the normalized Campbell measure associated with KT , i.e., ˆ PT (A × B) = P(1AKT (B))/m(1) (PC) x 7→ P x is continuous (QLC) quasi-left continuity, i.e., Y is a Hunt process–see Section II.2 Qτ,m the law of the historical process starting at time τ in state m–see Section II.8 S R(I) S(Xt), the range of X on I t∈I R(I) R(I) is the closed range of X on I R S R[δ, ∞) is the range of X. δ>0 S(µ) the closed support of a measure µ St S(Xt) S simple P × E-measurable integrands–see after Lemma II.5.2 (SE) strong form of nonlinear equation–see prior to Theorem II.5.11 t [Nt]/N Tb bounded (Ft)t≥τ -stopping times ucb→ convergence on E which is uniform on compacts and bounded on E Uλ the λ resolvent of a Markov process ⇒w weak convergence of finite (usually probability) measures Wt the coordinate maps on D(Eˆ) y/s/w the path equaling y up to s and w(t − s) thereafter yt(·) y(t ∧ ·) th ζα the lifetime of the α branch–see after Remark II.3.2 132 Superprocesses I. Introduction Over the years I have heard a number of complaints about the impenetrable literature on measure-valued branching processes or Dawson-Watanabe superpro- cesses. These concerns have in part been addressed by some recent publications including Don Dawson’s St. Flour notes (Dawson (1993)), Eugene Dynkin’s mono- graph (Dynkin (1994)) and Jean-Francois Le Gall’s ETH Lecture Notes (Le Gall (1999)). Nonetheless, one still hears that several topics are only accessible to ex- perts.
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