Recent results on branching Brownian motion on the positive real axis Pascal Maillard (Université Paris-Sud (soon Paris-Saclay)) CMAP, Ecole Polytechnique, May 18 2017 Pascal Maillard Branching Brownian motion with selection 1 / 27
Outline Introduction 1 BBM with absorption 2 BBM with absorption, near-critical drift 3 BBM with absorption, critical drift 4 Pascal Maillard Branching Brownian motion with selection 2 / 27
Branching Brownian motion (BBM) Picture by Matt Roberts Definition A particle performs standard Brownian motion started at a point x ∈ R . With rate 1 / 2 , it branches into 2 offspring (can be generalized) Each offspring repeats this process independently of the others. → A Brownian motion indexed by a tree. − Pascal Maillard Branching Brownian motion with selection 3 / 27
Why BBM ? Discrete counterpart: branching random walk, has lots of applications in diverse domains Generalisation of age-dependent branching processes ( Crump–Mode–Jagers process ), model for asexual population undergoing mutation (position = fitness) Toy model for log-correlated field , e.g. 2-dimensional Gaussian free field appearing notably in Liouville quantum gravity theory . Used to study random walk in random environment on trees Hu–Shi et al. , growth-fragmentation processes Bertoin–Budd–Curien–Kortchemski , loop O ( n ) model on random quadrangulations Chen–Curien–M. , . . . Intimate relation with (F-)KPP equation With diffusion constant depending on time : also known as Derrida ’s CREM spin glass model Pascal Maillard Branching Brownian motion with selection 4 / 27
Maximum : LLN M t = maximum at time t . LLN ( Biggins ’77 ) Almost surely, as t → ∞ . M t / t → 1 , Picture by Éric Brunet Pascal Maillard Branching Brownian motion with selection 5 / 27
A family of martingales For every θ ∈ R , 1 1 2 ( 1 − θ 2 ) t . 2 t P ( B t ≈ θ t ) ≈ e E [# { u ∈ N t : X u ( t ) ≈ θ t } ] = e Martingales: e θ X u ( t ) − 1 W ( θ ) 2 ( 1 + θ 2 ) t � = t u ∈N t Theorem ( Biggins 78 ) The martingale ( W ( θ ) ) t ≥ 0 is uniformly integrable if and only if | θ | < 1 . In this t case, for every a , b ∈ R , a < b , # { u ∈ N t : X u ( t ) ∈ θ t + [ a , b ] } E [# { u ∈ N t : X u ( t ) ∈ θ t + [ a , b ] } ] → W ( θ ) := W ( θ ) a.s. as t → ∞ . ∞ , Pascal Maillard Branching Brownian motion with selection 6 / 27
Derivative martingale For θ = 1 , W ( 1 ) → 0 , almost surely as t → ∞ . Derivative martingale: t D t = − d � d θ W ( θ ) � ( t − X u ( t )) e X u ( t ) − t . θ = 1 = � t � u ∈N t Theorem ( Lalley–Sellke 87 ) Almost surely, D t converges as t → ∞ to a non-degenerate r.v. D . Theorem ( Bramson 83 + Lalley–Sellke 87, Aïdekon 11 ) Let M t = maximum at time t . Then, conditioned on D , for some constant C > 0 , M t − ( t − 3 2 log t ) ⇒ log CD + G , where G is a standard Gumbel-distributed random variable. Pascal Maillard Branching Brownian motion with selection 7 / 27
Outline Introduction 1 BBM with absorption 2 BBM with absorption, near-critical drift 3 BBM with absorption, critical drift 4 Pascal Maillard Branching Brownian motion with selection 8 / 27
Absorption at the origin Start with one particle at x ≥ 0 . Add drift − µ , µ ∈ R to motion of particles. Kill particles upon hitting the origin. Theorem (Kesten 78) P ( survival ) > 0 ⇐ ⇒ µ < 1 . Why should we do this? Useful for the study of BBM without absorption (e.g., convergence of derivative martingale) Biological interpretation: natural selection Appears in other mathematical models, e.g. infinite bin models Aldous , Mallein–Ramassany Pascal Maillard Branching Brownian motion with selection 9 / 27
Absorption at the origin, µ ≥ 1 Start with one particle at 0 , absorb particles at − x . N x = number of particles absorbed at − x . Set µ 2 − 1 . � θ ± = µ ± Theorem ( Neveu 87, Chauvin 88 ) ( N x ) x ≥ 0 is a continuous-time Galton–Watson process. Moreover, almost surely as x → ∞ , If µ > 1 , e − θ − x N x → W ( θ − ) . If µ = 1 , xe − x N x → D . Theorem As x → ∞ , µ > 1 : P ( W ( θ − ) > x ) ∼ C ( µ ) x − θ + /θ − Guivarc’h 90, Liu 00 µ = 1 : P ( D > x ) ∼ 1 / x Buraczewski 09, Berestycki–Berestycki–Schweinsberg 10, M. 12 Pascal Maillard Branching Brownian motion with selection 10 / 27
Absorption at the origin, µ ≥ 1 (contd.) µ 2 − 1 . � θ ± = µ ± Theorem As x → ∞ , µ > 1 : P ( W ( θ − ) > x ) ∼ C ( µ ) x − θ + /θ − Guivarc’h 90, Liu 00 µ = 1 : P ( D > x ) ∼ 1 / x Buraczewski 09, Berestycki–Berestycki–Schweinsberg 10, M. 12 Theorem ( M. 10, Aïdekon–Hu–Zindy 12 ) As n → ∞ , µ > 1 : P ( N x > n ) ∼ C ( e θ + x − e θ − x ) / n − θ + /θ − . µ = 1 : P ( N x > n ) ∼ xe x / ( n ( log n ) 2 ) . Pascal Maillard Branching Brownian motion with selection 11 / 27
Outline Introduction 1 BBM with absorption 2 BBM with absorption, near-critical drift 3 BBM with absorption, critical drift 4 Pascal Maillard Branching Brownian motion with selection 12 / 27
Absorption at the origin, µ = 1 − ε Few works on µ < 1 ( Berestycki–Brunet–Harris–Milo´ s, Corre ). But near-critical case µ = 1 − ε , 0 < ε ≪ 1 well understood. Parametrize ε by ε = π 2 ( ε → 0 ⇐ ⇒ L → ∞ ) . 2 L 2 Theorem ( Brunet–Derrida 06, Gantert–Hu–Shi 08 ) P 1 ( survival ) = exp ( − ( 1 + o ( 1 )) L ) , L → ∞ . Pascal Maillard Branching Brownian motion with selection 13 / 27
Absorption at the origin, µ = 1 − ε Few works on µ < 1 ( Berestycki–Brunet–Harris–Milo´ s, Corre ). But near-critical case µ = 1 − ε , 0 < ε ≪ 1 well understood. Parametrize ε by ε = π 2 ( ε → 0 ⇐ ⇒ L → ∞ ) . 2 L 2 Theorem ( Brunet–Derrida 06, Gantert–Hu–Shi 08 ) P 1 ( survival ) = exp ( − ( 1 + o ( 1 )) L ) , L → ∞ . Theorem ( BBS 10 ) There exists C > 0 , such that, as L → ∞ , φ ( x ) := E [ exp ( − CDe x )] . P L + x ( survival ) → 1 − φ ( x ) , and if x = x ( L ) such that L − x → ∞ , P x ( survival ) ∼ C ( L /π ) sin ( π x / L ) e x − L . Pascal Maillard Branching Brownian motion with selection 13 / 27
BBS 10 proof Define � Z L L sin ( π X u ( t ) / L ) e x − L . t = u ∈N t Then ( Z L t ) t ≥ 0 is (almost) a martingale for BBM with absorption at 0 and at L . Theorem ( BBS 10 ) Suppose the initial configurations are such that Z L 0 ⇒ z 0 as L → ∞ , and L − max u X u ( 0 ) → ∞ . Then ( Z L L 3 t ) t ≥ 0 converges as L → ∞ (wrt fidis) to a continuous-state branching process started at z 0 . Moreover, P ( BBM survives forever ) → P ( CSBP started from z 0 goes to ∞ ) . The CSBP in the above theorem is Neveu ’s CSBP and has branching mechanism � ∞ ( e − ux − 1 + ux 1 x ≤ 1 ) dx ψ ( u ) = au + π 2 u log u = a ′ u + π 2 x 2 , 0 for some (implicit) constants a , a ′ ∈ R . In particular, it is supercritical (with ∞ mean). Pascal Maillard Branching Brownian motion with selection 14 / 27
BBS 10 proof (2) Theorem ( BBS 10 ) If x = x ( L ) such that L − x → ∞ , P x ( survival ) ∼ CL π sin ( π x / L ) e x − L . Proof: Set w ( x ) := L sin ( π x / L ) e x − L . Start BBM with 1 / w ( x ) particles at x at time 0 . Then P ( survival ) → P ( CSBP started at 1 goes to ∞ ) ∈ ( 0 , 1 ) . Also, by independence, � − P x ( survival ) � 1 − P ( survival ) = ( 1 − P x ( survival )) 1 / w ( x ) ∼ exp , w ( x ) and so P x ( survival ) ∼ Cw ( x ) . Pascal Maillard Branching Brownian motion with selection 15 / 27
BBS 10 proof (3) Theorem ( BBS 10 ) There exists C > 0 , such that, as L → ∞ , φ ( x ) = E [ exp ( − CDe x )] . P L + x ( survival ) → 1 − φ ( x ) , Proof: Wait a long time T (independent of L ), so that L − max u X u ( T ) ≫ 1 . Then using L sin ( π x / L ) ∼ π ( L − x ) for L − x ≪ L , we get Z L T ≈ π e x D T , with ( D t ) t ≥ 0 the derivative martingale of usual BBM. Let first L → ∞ then T → ∞ to get P L + x ( survival ) = 1 − E [ P L + x ( extinction | F T )] ≈ 1 − E [ P ( CSBP started from π e x D T goes to 0 )] ≈ 1 − E [ exp ( − CDe x )] = 1 − φ ( x ) . Pascal Maillard Branching Brownian motion with selection 16 / 27
BBS 10 convergence to CSBP Basic idea Decompose process into bulk + fluctuations by putting an additional absorbing barrier at L . bulk: Particles that don’t hit L . fluctuations: Particles from the moment they hit L . Then, Z L , bulk stays bounded over time scale L 3 . t Z L , fluctuations increases from the contributions of the particles hitting L , t an increase being roughly distributed as π D , with D derivative martingale limit. Particles hit L with rate O ( L − 3 ) . Pascal Maillard Branching Brownian motion with selection 17 / 27
BBS 10 convergence to CSBP Basic idea Decompose process into bulk + fluctuations by putting an additional absorbing barrier at L − A , where A is a large constant. bulk: Particles that don’t hit L − A . fluctuations: Particles from the moment they hit L − A . Then, Z L , bulk decreases almost deterministically as exp ( − At / L 3 ) . t Z L , fluctuations increases from the contributions of the particles hitting L , t an increase being roughly distributed as π e − A D , with D derivative martingale limit. Particles hit L − A with rate O ( e A / L 3 ) . Pascal Maillard Branching Brownian motion with selection 17 / 27
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