Slowdown in branching Brownian motion with inhomogeneous variance

Transcription

1 Slowdown in branching Brownian motion with inhomogeneous variance Pascal Maillard Ofer Zeitouni February 2, 24 Abstract We consider the distribution of the maximum M T of branching Brownian motion with time-inhomogeneous variance of the form σ 2 t/t ), where σ ) is a strictly decreasing function. This corresponds to the study of the time-inhomogeneous Fisher Kolmogorov- Petrovskii-Piskunov FKPP) equation F t x, t) = σ 2 t/t )F xx x, t)/2 + gf x, t)), for appropriate nonlinearities g ). Fang and Zeitouni 22) showed that M T v σ T is negative of order T /3, where v σ = σs)ds. In this paper, we show the existence of a function m T, such that M T m T converges in law, as T. Furthermore, m T = v σt w σ T /3 σ) log T + O) with w σ = 2 /3 α σs)/3 σ s) 2/3 ds. Here, α = is the largest zero of the Airy function Ai. The proof uses a mixture of probabilistic and analytic arguments. Introduction The classical branching Brownian motion BBM) model in R can be described probabilistically as follows. Fix a law µ of finite variance on [2, ) Z. At time t =, one particle exists and is located at the origin. This particle starts performing standard Brownian motion on the real line, up to an exponentially distributed random time, with parameter β = 2E µ [L] )) that is, branching occurs at rate β ). At that time, the particle instantaneously splits into a random number L 2 of independent particles, and those start afresh performing Brownian motion until their independent) exponential clocks ring. There is an extensive literature on this model and its discrete analog, the branching random walk, in particular concerning the position of the right-most particle see e.g. [M75, Br78, Br83, DS88, R, A3]). In order to state the main result, introduce the F-KPP travelling wave equation φ : R, ) increasing, 2 φ + φ + β E µ [φ L ] φ) =, φ ) =, φ+ ) =..) One has the following theorem: Faculty of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76, Israel. Partially supported by a grant from the Israel Science Foundation Department of Mathematics, The Weizmann Institute of Science, POB 26, Rehovot 76, Israel and Courant Institute, New-York University. Partially supported by a grant from the Israel Science Foundation and the Henri Taubman professorial chair at the Weizmann Institute

2 Theorem Bramson [Br83]). Let M t denote the position of the right-most particle at time t in branching Brownian motion as defined above. Then there exists a solution φ to.), such that for all x R, PM t t 3 2 log t + x) φx), as t. We discuss in this paper a variant of the BBM model, first introduced in [DS88], where the motion of the particles) is controlled by a time-inhomogeneous variance. More precisely, let σ C 2 [, ]) be a strictly decreasing function with σ) > and inf t [,] σ t) >. We assume that the variance of the Brownian motions at time t [, T ] is given by σ 2 t/t ). Let M t = max u Nt) X u t) denote the location of the rightmost particle at time t. The cumulative distribution function of M T is F, T ), where F x, t) is the solution of the timeinhomogeneous Fisher Kolmogorov-Petrovskii-Piskunov FKPP) equation F t x, t) = σ2 t/t ) 2 F 2 2 x x, t) + β E µ [F x, t) L ] F x, t)), t [, T ], x R F x, ) = x..2) See [M75] for this probabilistic interpretation of the FKPP equation in the time homogeneous case. In [FZ2], the authors prove the following. Theorem Fang, Zeitouni [FZ2]). There exist constants C, C > so that C lim inf T M T v σ T T /3 lim sup T M T v σ T T /3 C <,.3) where v σ = σs)ds. The derivation in [FZ2] is for the case that P L = 2) =, but applies with no changes to the current setup. The linear in T asymptotics, i.e. the speed v σ, can be read off with some effort from the results in [DS88] and [BK4].) Our goal in this paper is to significantly refine Theorem. To state our results, introduce the functions v, w : [, ] R + by and vt) = t σs) ds,.4) t wt) = 2 /3 α σs) /3 σ s) 2/3 ds,.5) where α = is the largest zero of the Airy function of the first kind Aix) = ) t 3 cos π 3 + xt dt,.6) see [AS64, Section.4] for definitions; note that Ai satisfies the Airy differential equation Ai x) xaix) =. Note also that v σ = v). Set Our main result is the following. m T = v)t w)t /3 σ) log T. 2

3 Theorem.. The family of random variables M T m T ) T is tight. Further, there exists a solution φx) to.) and a function m T with C σ = lim sup T m T m T <, such that for all x R, lim PM T m T + x) = φx/σ)). T Furthermore, for a fixed travelling wave φ, the constant C σ above is uniformly bounded for σ {σ : [, ] R + : σ) + /σ) < c, sup σ t) < c, inf t [,] t [,] σ t) > /c } =: Ξ c. Parallel to our work, and an inspiration to it, was the study [NRR3], by PDE techniques, of a class of time-inhomogeneous FKPP equations that includes.2). Compared with [NRR3], we deal with a slightly restricted class of equations, but are able to obtain finer up to order ) asymptotics and convergence to a travelling wave. We hope that our techniques can be pushed to yield convergence in distribution of the family M T m T ) T instead of M T m T ) T ), in parallel with the recent results in [BDZ3], but this requires significant changes in the approach of [BDZ3] mainly, because unlike in the time-homogeneous case, extremal particles at time T will, with positive probability, be extremal at some random intermediate time between ɛt and ɛ)t ). We therefore leave the adaptation for possible future work. We remark that Mallein [M3] has recently published results similar to ours which are much less precise but hold for a rather general class of not necessarily Gaussian) timeinhomogeneous branching random walks. The core of the proof of Theorem. is based on a constrained first and second moment analysis of the number of particles that reach a target value but remain below a barrier for the duration of their lifetime. Due to the time inhomogenuity of σ ), the choice of barrier is not straight-forward, and in particular it is not a straight line; rectifying it introduces a killing potential. The analysis of the survival of Brownian motion in this potential eventually leads to a time-inhomogeneous Airy-type differential equation which we study by analytic means, exploiting the anti-symmetry of the differential operator. As pointed out to us by Dima Ioffe, a similar phenomenon with related T /3 scaling was already observed in [G89, SF6].) These methods together lead to estimates of the right tail of M T which are sharp up to a multiplicative factor Proposition 3.). By a bootstrapping procedure that may be of independent interest, these estimates are then turned into convergence in law by using a convergence result for the derivative Gibbs measure of time-homogeneous) branching Brownian motion. The structure of the paper is as follows. In the next section, we introduce a barrier γ T ), and show that with high probability, no particle crosses a shifted version of) the barrier, see Lemma 2.. Using the barrier, we then control the distribution of extremal particles at all times large enough Lemma 2.2). In these lemmas, results concerning time-inhomogeneous Airy-type PDE s are needed, and the proof of those is given in Appendix A Section 5). Section 3 combines the results of Section 2 taken at time T T 2/3 ) together with an analysis of the last segment of time of length T 2/3, and provides the first-and-second moment results needed to obtain lower and upper bound on the right tail of M T. The proof of Theorem. is then completed in Section 4, using a result about the convergence of the derivative Gibbs measure of time-homogeneous) branching Brownian motion, which is given in Appendix B Section 6). 3

4 Notation In the rest of this article except in the appendix), the symbols C,C,C,C 2 etc. stand for positive constants, possibly depending on c see Theorem.), whose values may change from line to line. The phrase X holds for large T means that there exists T, possibly depending on c, such that X holds for T T. We further use the Landau symbols O ) and o ), which are always to be interpreted with respect to T, and which may depend on c as well. Finally, the symbols P and E possibly with sub-/superscripts) always stand for the law of a branching Markov process branching Brownian motion with time-varying or constant variance and with or without absorption of particles) and the expectation with respect to this law. On this other hand, the symbols P and E are used for probability and expectation with respect to a single particle i.e. a Markov process, usually a Brownian motion or a three-dimensional Bessel process). The location of the initial particle is denoted by a subscript, e.g. P x, without a subscript the initial particle is implicitly located at the origin. Acknowledgements We thank Lenya Ryzhik for very stimulating conversations concerning the PDE approach to time-inhomogeneous BBMs, and for making [NRR3] available to us before we completed work on this paper. We also thank Bastien Mallein for describing to us his progress on analogous questions for branching random walks. 2 Crossing estimates Fix T. Define the curve γ T : [, T ] R by γ T t) = T vt/t ) T /3 wt/t ). In this section we prove two lemmas. The first lemma bounds, for any fixed K, the probability that there exists a particle that reaches the curve γ T t) + K. The second lemma estimates the expected number of particles that have stayed below the curve up to time t, and reach a given terminal value at time t. Lemma 2.. There exists a constant C = Cc ), such that for large T, for any σ Ξ c every K [, T /3 ], and P t [, T ] : max u Nt) X ut) γ T t) + K) CKe K/σ). Proof. The proof goes by a first moment estimate of the number of particles hitting the curve γ T + K. For an interval I [, T ], let R I be the number of particles hitting the curve γ T + K for the first time during the interval I. Let B t be a Brownian motion with variance σ 2 t/t ) started from the point x under P x see the remarks on notation in the introduction). For a path X t ) t, define H X) = inf{t : X t = }. By the first moment formula for branching Markov processes [INW69, Theorem 4.] also known as Many-to-one lemma ) we then have taking x = K) E[R I ] = E [e H γ T t)+k B t)/2 H γ T t)+k B t) I ] [ ] = E K e H B t+γ T t))/2 H B t+γ T t)) I, Note that due to our choice of the branching rate β, the expected number of particles in the system at the time t is E[Nt)] = e t/2, which is the reason for the exponential term arising in the formula. 4

5 where the second equality follows from the fact that the law of K B t under P is equal to the law of B t under P K by symmetry. Applying Girsanov s theorem we get that E[R I ] = E K [exp H γ T t) σ 2 t/t ) db t + H = e Kγ T )/σ2 )+o) E K [exp T H 2 H γ T t))2 ) ] 2σ 2 t/t ) dt H I ) q T t/t )B t + T /3 w t) σt/t ) ) ] dt H I, where the last equation follows by integration by parts and the function q T, k T : [, ] R is defined by q T t) = σ t) σ 2 t) + T 2/3 w /σ 2 ) t). For large T, this yields by.5) and the assumptions on σ and K, [ E[R I ] = e K/σ)+o) E K exp T H { q T t/t )B t + T /3 α q T t) 2/3 ) /3 } ) ] 2 σ2 t/t ) dt H I 2.) Set Jt) = t 2 σs)2 ds. 2.2) Define s := J 4T /3 )T, so that CT 2/3 s C T 2/3 for large T. We will bound separately E[R [,s ]] and E[R [s,t ]]. For the first term, 2.) immediately gives I E[R [,s ]] Ce K/σ), 2.3) because under P K, B t is positive until the time H and the deterministic term in the integral in 2.) is bounded by a constant C. In order to bound E[R [s,t ]], we note that the expectation on the right side of 2.) equals dgk, y; t) 2 σ2 t/t ) dt, 2.4) dy y= where Gx, y; t) is the fundamental solution to the PDE 5.4), with Qt) = q T Jt/T )) see [G85, Sections 5.2. and 5.2.8] for an elementary, but somewhat non-rigorous presentation, and [D84, Sections I.XI.7 and 2.IX.3] for the formal definition of parabolic measure and its relation to hitting time distributions for Brownian motion). Now, by Proposition 5.2 and 5.5), there exists a positive function Q t) Qt), such that for every t [s, T ], dgk, y; t) dy T Cψ Q t)k ) e C n 2/3 ψ y= n) T CK, where the last inequality follows from Lemma 5.. Together with 2.) and 2.4), this yields for large T, E[R [s,t ]] CKe K/σ). 2.5) The lemma now follows from 2.3) and 2.5) and Markov s inequality. n= 5

6 We next control the expected number of particles that stay below the curve γ T ) + K up to time t T and reach a prescribed value at time t. In what follows, for measures µ, ν we use the notation µ dy) ν dy), y, to mean that for any interval I R +, µ I) ν I). Lemma 2.2. Let t [J 4T /3 )T, T ]. Then there exist constants C, C > depending on c only), such that for large T and for all K [, T /3 ] and y >, E[#{u Nt) : γ T t) + K X u t) dy and X u s) γ T s) + K, s t}] [CKe y/σt/t ) K/σ) T 2/3 e C n 2/3 ψ q t/t ) n T /3 y) ] dy n= where q t) q T t) and q T t) q t) 2T /3 sup t [,] q T t). Proof. By a similar argument as the one leading to 2.), the expectation in the statement of the lemma equals e yγ T t)/σ2 t/t ) Kγ T )/σ2 )+o) GK, y; t) dy, where Gx, y; t) is the same as in the proof of Lemma 2.. By the assumption on K, we have Kγ T )/σ2 ) = K/σ) + o) and by definition of γ T, we have γ T t) σt/t ). The claim now follows from the analytical Proposition 5.2 and 5.5) in Appendix A Section 5). 3 Tail estimates We derive in this section tail estimates on the distribution of M T summarized in the following proposition. Proposition 3.. There exists a constant C = Cc ) and T R, such that for any σ Ξ c, T T and K [, T /3 ], C Ke K/σ) PM T mt ) + K) CKe K/σ). The proof of Proposition 3. goes by a suitably truncated first-second moment method, inspired by analogous results in the time-homogeneous case [Br78, A3, R, BDZ3]. The key ingredients are estimates on a single Brownian particle with time-inhomogeneous variance staying below a curve and reaching a certain point at a given time t. These results, which have already been used in the previous section, are obtained in the appendix by analytic methods. However, as in the time-homogeneous case, the first-second moment method applied directly to the particles staying under the curve γ T would not yield the O) precision on the maximum at time T that we are aiming at, but would rather induce an error of magnitude Olog log T ). This can be rectified in our case by slightly changing the curve in the time interval [T T 2/3, T ] in a way similar to the time-homogeneous case namely, by having it end at the point γ T T ) σ) log T. Luckily, for the upper bound it is possible to shortcut this approach, as Slepian s inequality allows us here to directly use existing results in the time-homogeneous case for the system during the time interval [T T 2/3, T ] see Section 3. for details). 6

7 3. Proof of Proposition 3.: Upper bound Set t = T T 2/3 and let K 2. Let F t ) t be the natural filtration of the BBM. A union bound gives, PM T mt ) + K F t ) P Xut ),t )M T mt ) + K), u Nt ) where P x,t) denotes the law of BBM with variance σ 2 /T ) starting with one particle at the point x at time t. We will estimate the summands on the right-hand side by comparison with a BBM with constant variance. Set σc 2 = T /3 T σ 2 t) dt. By the assumption on σ, we 2/3 have mt ) γ T t ) T T /3 σt) dt σ) log T C σ c T 2/3 3 2 log T 2/3) C, for some constant C that we fix for the remainder of this proof. Now, let X u T 2/3 )) u and X c ut 2/3 )) u be the positions of the particles at time T 2/3 in branching Brownian motion with branching rate β and variance σ 2 +t )/T ) and σ 2 c, respectively. Conditioned on the genealogy, we have E[X u T 2/3 ) 2 ] = E[X c ut 2/3 ) 2 ] and E[X u T 2/3 )X v T 2/3 )] E[X c ut 2/3 )X c vt 2/3 )] for every u and v, by the definition of σ 2 c and the fact that σ 2 is decreasing. Hence, setting M c = max u X c ut 2/3 ), we have by Slepian s inequality [S62] for every x, P γ T t )+K C x,t ) M T mt ) + K) PM c σ c T 2/3 3 2 log T 2/3) + x) The tail estimates for the maximum of time-homogeneous BBM are available e.g. in [Br83], and we obtain that P γ T t )+K C x,t ) M T mt ) + K) Cxe x/σc Cxe x/σt /T ), 3.) for large T, uniformly in x. Let A denote the event that no particle reaches the curve γ T t) + K C until time t. Integrating 2 the upper bound in Lemma 2.2 taken at time t = t ) with respect to the distribution in 3.) now yields for K 2C + ) and large T, ) P{M T mt ) + K} A) CKe K/σ) ψ, x + T /3 e C n 2/3 ψ n, x. The upper bound in the statement of Proposition 3. now follows from this inequality, together with the fact that PA c ) CKe K/σ) for large T by Lemma Proof of Proposition 3.: Lower bound As discussed above, the proof involves a second moment Many-to-two ) argument. In order to carry it out, we need to modify the curve γ T ) at the last interval [T T 2/3, T ]. Toward this end, fix K > and let φ T t) be an increasing, twice differentiable function 3 2 We can integrate term by term because n=2 e C n 2/3 ψ n, x converges by point 4 of Lemma The construction of such a function is possible for large enough T, for example by gluing together a parabola on [T T 2/3, T T 2/3 /2] and a line on [T T 2/3 /2, T ]. n=2 7

8 such that φ T t) on [, T T 2/3 ], φ T T ) = σ) log T, φ T t) 2σ) log T/T 2/3 and φ T t) 4σ) log T/T 4/3. Define the curve For s, t [, T ], let ζ T t) = γ T t) + K φ T t). G ζ x, y; s, t) dy = E K x,s) [#{u Nt) : X u r) ζ T r) s r t} ζt t) X ut) dy] denote the expected number of descendants at time t of a particle present at time s at location K x, so that the path of the descendant stayed below the curve ζ T ) until time t, and reached, at time t, an infinitesimal neighborhood of the value ζ T t) y. Similarly to the proof of 2.), we have [ t G ζ x, y; s, t) dy = E x,s) exp = exp s ζ t) σ 2 t/t ) y ζ T r) σ 2 r/t ) db r + t s 2 t s ζ T r))2 ) 2σ 2 r/t dr ) ζ s) σ 2 s/t ) x + φ T t) φ T s) + o) σ) Bt dy ] Gx, y; s, t) dy, where under P x,s), B t, t) t s is the law of space-time Brownian motion with time-inhomogeneous variance σ 2 /T ) started at the space-time point x, s) and Gx, y; s, t) is the fundamental solution to 5.4), with Qt) = σ Jt/T )) /σ 2 Jt/T )) + Olog T/T 2/3 ) The o) term in the last display comes from the time-inhomogeneity in the quadratic term of Girsanov s theorem.) In particular, if N T denotes the number of particles, at time T, whose trajectory stayed under the curve ζ T ) and reached the interval [ζ T T ) 2, ζ T T ) ] at time T, then, for large T, E[N T ] = 2 2 G ζ, y;, T ) dy CT e K/σ) GK, y; T ) dy CKe K/σ), 3.2) where the last inequality follows from Proposition 5.2 and 5.5). As for the second moment, the second moment formula 4 Many-to-two lemma ) for branching Markov processes [INW69, Theorem 4.5] yields for large T, T E[NT 2 ] = E[N T ] + β E µ [L 2 L] T E[N T ] + Ce K/σ) T dt 2 dt dy G ζ K, y;, t) 2 dy GK, y;, t) ) 2 G ζ y, z; t, T ) dz ) 2 Gy, z; t, T ) dz e c y+ φ T T ) φ T t) σ), for C = Q /2, a constant that we fix for the rest of the proof. We split the integral into three parts, according to intervals of time [, T 2/3 ], [T 2/3, T T 2/3 ] and [T T 2/3, T ] and denote the three parts by I, I 2 and I 3. In order to estimate the first and third part, we bound the Green kernel Gx, y; s, t) for t s T 2/3 by the Green kernel of Brownian motion killed at the origin. Namely, writing V t) = t σ2 s/t ) ds, we have for t s T 2/3 and x, y, Gx, y; s, t) C t s exp x y) 2 2V t) V s)) 4 It can be derived by conditioning on the splitting time of pairs of particles. ) xy t s ) 3.3) 3.4) 8

9 For t T 2/3, we use Proposition 5.2 and 5.5) in order to bound GK, y;, t) and Gy, z; t, T ) for the latter, we consider the time-reversal of 5.4)). Note that ψnx) q qx for every n, q, x. This yields GK, y;, t) CT Ky and Gy, z; T t, T ) CT y for every t T 2/3 and z [, 2]. For the first part, we now get by exchanging integrals, ) T 2/3 I T 2 T y) 2 e C y GK, y;, t) dt dy C y 2 e Cy + Ky) dy CK, for K and large T. Here, we used the fact that by 3.4), T 2/3 For the second part, we have GK, y;, t) dt C I 2 CT 2 T T 2/3 T 2/3 dt Ky dt + C dt C + Ky). t t3/2 T 3 Ky 3 e C y dy CK, For the third part, we note that by 3.4) and the assumptions on φ T, we have for every y, for large T, T 2/3 2 ) 2 Gy, z; T t, T ) dz e φ T T ) φ T T t) σ) dt Cy 2 T 2/3 / log T ) 3 t 3 dt + T 2/3 T 2/3 T Cy 2. log T Furthermore, for t, we have for every y. This gives, 2 Gy, z; T t, T ) dz ) 2 expφt T ) φ T T t))/σ)) C In total, we have I 3 Ky 3 e C y dy + CK dt CK. This now yields, E[N 2 T ] E[N T ] + Ce K/σ) I + I 2 + I 3 ) CE[N T ]. PN T ) E[N T ] 2 E[N 2 T ] /C, which finishes the proof of the lower bound in Proposition Proof of Theorem. Armed with the tail estimates provided by Proposition 3., the proof of Theorem. follows by considering the descendants of the particles living at a large but fixed) time t. Here are the details: 9

10 We assume without loss of generality that σ) = otherwise we can rescale space). Write P T and E T in place of P and E, similarly, we write P T x,t) in place of P x,t) see Section 3.). Furthermore, we will denote by P hom and E hom the law of time-homogeneous) branching Brownian motion with variance and branching rate β, starting with one particle at the origin. In what follows, we fix y R and let t large enough, such that y < log t 2. We will later let first T, then t go to infinity, i.e. we will choose t as a function of T, such that tt ) goes to infinity slowly enough as T. As in Section 3., let F t ) t be the natural filtration of the BBM. Define the F t - measurable random variable W t,t by W t,t = P T M T mt ) + y F t ) = ) P T X M ut),t) T mt ) + y). Furthermore, define D t = u Nt) u Nt) t X u t))e Xut) t. By Proposition 3. applied with the function σt ) = σt T t)+t)/t ), there exists a constant C and for each large T a function g t,t : R + [C, C], such that for each x [ t, t log t], P T x,t) M T mt ) + y) = exp g t,t y x + t)/ t)y x + t)e y x+t)). 4.) By the continuity of P T x,t) in x, the functions g t,t are actually continuous, in particular, they are Lebesgue-measurable. As in Section 6 note that if B t ) t is a Brownian motion started at the origin, then t B t ) t is a Brownian motion with drift + started at the origin), define the derivative Gibbs measure µ t = t X u t))e t Xut)) δ t Xut))/ t. u N t) Then, on the event A t = { u Nt) : t X u t) t log t}, we get by 4.) W t,t At = exp e y g t,t y/ ) t + x)µ t dx) At 4.2) and P hom A t ) as t goes to infinity [Br83]. Now, note that as T, the law of the process until time t converges to its law under P hom, because conditioned on the genealogical structure and the branching times, the particle motion until time t on each of the finitely many branches of the genealogical tree converges to Brownian motion with variance. Moreover, thanks to the continuity and positivity of the Gaussian density, we can construct a probability space with probability measure P which supports random variables µ T ) T and µ, such that, under P, µ T follows the law of µ t under P T, µ follows the law of µ t under P hom and µ T = µ on an event G T with P G T ) as T. In particular, g t,t y/ t + x) µ T dx) = g t,t y/ t + x) µdx) on G T, for every y. 4.3) By a diagonalization argument, we can now choose t = tt ) growing slowly with T, so that 4.3) continues to hold with this choice of tt ). By Theorem 6., we have that for every

11 bounded continuous function f, E hom [f g tt ),T y/ )] tt ) + x)µ tt ) dx) E hom [f D g tt ),T y/ )] tt ) + x)ρdx) T, where ρ is the law of a BES3) process at time, started at, and the variable D is the derivative martingale limit from Section 6. Using the above coupling we conclude that E [f T g tt ),T y/ )] tt ) + x)µ tt ) dx) E hom [f D g tt ),T y/ )] tt ) + x)ρdx). 4.4) On the other hand, since ρ has a continuous density with respect to Lebesgue measure, we have, lim sup g tt ),T y/ tt ) + x)ρdx) g tt ),T x)ρdx) = 4.5) T Setting C T = g tt ),T x)ρdx), we get by 4.2), 4.4), 4.5) and dominated convergence, lim T PT M T mt ) + y log C T ) = E hom [e e y D ] = φx), where φ is a solution to.). This yields Theorem.. Remark: While a-priori, the constant C T depends on the particular choice of sequence tt ), it is clear that the conclusion of Theorem. implies that a-posteriori, it is independent of this choice. 5 Appendix A: An Airy-type PDE with time-varying parameters We are interested in the following parabolic PDE: w t = ε { w xx qt)xw }, wt, ) = t, 5.) for q C [, ], q >. We want to study its behaviour as ε. Before solving this equation, we recall some facts about the Airy differential operator Lψ = ψ xψ. Let L 2, ) be the space of square-integrable functions on, ) and let, be the associated scalar product with norm 2. Recall the definition.6) of the Airy function of the first kind Aix). We denote by α > α 2 > its discrete set of zeros, with α = The functions ψ n defined by ψ n x) = Aix α n) Ai α n ) 2, n =, 2,... then form an ONB of L 2, ) and ψ n is an eigenfunction of L with eigenvalue α n [VS4, Section 4.4]. The following lemma collects some other facts about the functions ψ n x), which are probably well-known, although we could not find a reference to some of them.

12 Lemma 5... Ai α n ) 2 = Ai α n ) for all n. In particular, ψ n) = for all n. 2. α n n 2/3 3π/2 as n. 3. ψ n x) x for all n and x. 4. For some numerical constant C, ψ n, x Cn 4/3 for all n. Proof. The first and second points are [VS4, 4.52) and 2.52)], respectively. For the third point, we first note that since Ai x) = xaix), the local extrema of Ai on R are exactly the zeros of Ai and the origin. Furthermore, by the first point of the lemma, Ai α n ) is increasing in n and by [VS4, 3.5)], Ai ) < Ai α ). This yields Ai x) Ai α n ) for all x α n, from which the third point of the lemma follows. For the fourth point, we first recall that for some x, Aix) exp 2/3)x 3/2 ) for x x [AS64,.4.59]. Together with the first point of the lemma and the definition of ψ n, it follows that ψ n x αn, x ψ, x + α n α ) Cα n, for some numerical constant C. Furthermore, by the third point of the lemma, we have ψ n x αn, x α 2 n/2 for all n. Application of the second point of the lemma now finishes the proof of the fourth point. We get back to the equation 5.). Define for a constant q the operator L q u = u xx qxu. One easily checks that the function ψ q nx) = q /6 ψ n q /3 x) is an eigenfunction of L q with eigenvalue α n q 2/3 and the functions ψ q n form an ONB of L 2, ). We further denote by gx, y; t) the fundamental solution of 5.). Proposition 5.2. Set Q = inf t [,] qt) 2/3 and Q 2 = sup t [,] log q) t). Suppose Q >. Then there exist constants C, C 2 > depending on Q and Q 2, such that uniformly for all x [, ], t [4ε, ] and δ [ε, ε] with δ/ ε as ε there exist sequences c n ) n 2 and c n) n 2, with c n c n C exp C 2 t δ)ε n 2/3 ) and ψ q t) + ε n=2 c nψ q t) n ) gx, ; t) t ) ε ψ q) x) exp α qs) 2/3 ds ψ q t) + ε n=2 c n ψ q t) n where denotes an inequality up to a multiplicative factor depending on Q and Q 2 ) tending to as ε and q t) qt) q t) with q t) q t) 2t δ) 2 ε sup t [,] q t). Before providing the proof of Proposition 5.2, we derive some a-priori estimates on solutions of 5.). Lemma 5.3. Define Q and Q 2 as in Proposition 5.2 and assume Q >. Let wt, x) be the solution to 5.) with initial condition satisfying w, ) 2. Define for each t the function W t x) = exp t ε α qs) 2/3 ds)wt, x). Then there exist numerical constants C, C, such that for all t [, ],. W t 2, 2. W t, ψ qt) W, ψ q) C Q n 2 W t, ψ qt) n 2 ) /2 C Q 2 + Q ε and Q ε Q2 + Q ) ε + W, ψ q) + exp C ε Q t). ), 2

13 Proof. After decomposing the solution of 5.) in the eigen-basis determined by the Airy functions, the proof proceeds by analyzing a coupled system of linear, time inhomogeneous, ordinary differential equations. Throughout the proof, C, C and C 2 are some numerical constants which may change from line to line. Define the vector ct) = c t), c 2 t),...), where c n t) = W t, ψn qt). From 5.), one gets whence ċ n t) = ε α n α )qt) 2/3 c n t) + k c k t)q t) ψ qt) d k, dq ψq n t), q =qt) ċt) = Dt) + At))ct), Dt) = ε qt) 2/3 diagα i α ) i, At) = log q) t)a. 5.2) Here, A is the antisymmetric matrix A = 6 I + 2xψ i, ψ j )) i,j = 6 xψ i, ψ j ) xψ j, ψ x )) i,j, where the equality is easily verified by integration by parts 5. Since Dt)+At) and Dt )+At ) do not commute unless qt) 2/3 log q) t ) = qt ) 2/3 log q) t), there is no obvious explicit expression for the solution to 5.2). However, since D is diagonal and A antisymmetric, we have d dt ct) 2 2 = d dt ct t)ct) = c T t)d T t) + A T t) + Dt) + At))ct) = 2c T t)dt)ct), by the positivity of qt). This implies the first claim. In particular, c t) for all t. Setting ct) =, c 2 t), c 3 t),...), the previous equation yields, d dt ct t) ct) = 2 c T t)dt) ct) 2c t) A j t)c j t) j=2 2ε qt) 2/3 α 2 α ) c T t) ct) + 2 c t) A j t)) j 2 2 ct) 2, by the Cauchy Schwarz inequality. By Parseval s formula, A j ) j 2 2 xψ 2/3 <. This yields d dt ct) 2 C ε Q ct) 2 + C 2 Q 2 c t). Note that the general solution to the equation f t) = aft) + b is ft) = b/a) + Ce at. Since c), Grönwall s inequality now yields that ct) 2 CQ 2 /Q )ε sup c s) + exp C ε Q t), 5.3) s [,t] In order to show the second claim, we note that by 5.2), for every t [, ], t t c t) c ) A j t)c j t) dt C ct) 2 dt, j=2 where the last inequality follows from the Cauchy Schwarz inequality as above. Together with 5.3) and the fact that sup t [,] c t), this implies the second claim. The third claim follows from this, together with 5.3). 5 In fact, A ij = 2 ) i+j α i α j) 3 for i j, which can be easily verified by the equation [VS4, 3.54)], however, we will not use this fact. 3

14 Proof of Proposition 5.2. Fix t [4ε, ] and δ [ε, t 3ε]. We can construct q, q C [, t]) such that the following holds: Q q q q q q q on [2ε, t ε δ], q and q are constant on [, ε] [t δ, t], sup s [,t] max{ log q ) s), log q ) s) } Q 2 and q q 2t δ) 2 ε sup t [,] q t). Now let x [, ]. Let w and w denote the solutions to 5.) with q replaced by q or q, respectively, and with initial condition w, ) = w, ) = δ x). By the parabolic maximum principle [E98, Theorem 7..9], we then have w t, y) gx, y; t ) w t, y) for all y and t [, t]. Write Wt y) = w t, y) expε t α q s) 2/3 ds) for all t, y. For every n, we have by the first point of Lemma 5. and the fact that ψ x) > for all x >, W, ψ q ) n = ψ q ) n x) Cψ q) x), for some constant C depending on Q. By 5.2), we then have Wε, ψ q ) n C exp α n α )q ) 2/3 )ψ q) x), for every n, since the off-diagonal terms cancel by the fact that q is constant on [, ε]. Together with the second point of Lemma 5., this yields W ε 2 C for some constant C as ε is small enough. Furthermore, W ε, ψ q ) = W, ψ q ) = + o))ψ q) x), where o) is a term depending on Q and Q 2 which vanishes as ε. Applying Lemma 5.3 with initial condition w, ) = Wε /C, we get that Wt, ψ q t) = + o))ψ q) x) and ) n 2 W t δ, t) /2 ψq n 2 C2 εψ q) x) for small ε, where C 2 depends on Q and Q 2. As above, this now implies that for every n 2, for small ε, Wt, ψ q ) n exp CQ t δ)ε n 2/3 )C 2 εψ q) x). Together with the previous estimates, this finally yields the existence of a sequence of constants c n) n 2 with c n exp CQ t δ)ε n 2/3 )C 2, such that as ε, t ) w t, ) + o)) exp ε α q s) 2/3 ds ψ q) x) ψ q t) + ε n=2 c nψ q t) n An analogous formula holds for w. The statement now follows from the fact that q s) 2/3 q s) 2/3 ds = Ot δ) 2 ) by construction. Fix T >. In the application in Section 2, we need to consider PDE s on the time interval [, T ]. The results obtained in the current section can be easily transported to the following PDE on [, T ] R + : u t t, x) = 2 σ2 t/t )u xx t, x) + { T Qt)x + T 2/3 α Qt) 2/3 2 σ2 t/t ) ) /3 }ut, x), 5.4) 4 ).

15 with Dirichlet boundary condition at and where Q C [, T ]) with Qt) > for all t [, T ]. Setting Jt) = t 2 σs)2 ds as in 2.2), defining qt) by qjt)/j)) = 2Qt)/σ 2 t), and changing variables by ) Jt/T )/J) ut, x) = wjt/t )/J), T /3 x) exp J)T /3 α qs) 2/3 ds, we see that the function wt, x) solves 5.) on [, ] R + with ε = J)T /3 and the qt) defined here. In particular, if Gx, y; t) and gx, y; t) denote the fundamental solutions of 5.4) and 5.), respectively, then we have the relation ) Jt/T )/J) Gx, y; t) = J)T /3 gt /3 x, T /3 y; Jt/T )/J)) exp J)T /3 α qs) 2/3 ds. 5.5) 6 Appendix B: Convergence of the derivative Gibbs measure of time-homogeneous) branching Brownian motion In this section, we consider branching Brownian motion with time-homogeneous) variance σ 2 =, drift + and reproduction law and branching rate as before. In particular, the leftmost particle drifts off to + with zero speed, i.e. if M t = min u N t) X u t), then almost surely, as t, M t /t and M t + [Br78]. Define the derivative Gibbs measure at time t: µ t = X u t)e Xut) δ Xut)/ t u N t) The quantity D t = dµ t is then known as the derivative martingale, and it is known [LS87, N88, YR] that D t converges almost surely as t to a strictly) positive limit D whose Laplace transform is given by E[exp e x D )] = φx), where φ is a solution to.). Let ρ denote the law of a BES3) process at time, started at, i.e. 2 ρdx) = π x2 e x2 /2 x dx. Theorem 6.. In probability, µ t converges weakly to D ρ. Moreover, for every family f t ) t of uniformly bounded measurable functions i.e. sup t,x f t x) < ), we have f t dµ t D f t dρ, in probability. Remark 6.2. Convergence in probability of the Gibbs measure µ t = t e Xut) δ Xut)/ t, u N t) has recently been shown by Madaule [M] for general branching random walks. While Theorem 6. at least the first statement) could be in principle recovered from the results in [M] see in particular Proposition 3.4 of that paper), we present below for completeness a fairly simple proof. 5

16 Proof. Note that we can and will) assume w.l.o.g. that f t for each t. For every s t, define the measure µ s t = X u t)e Xut) Xur) s r t)δ Xut)/ t u N t) Since min u N t) X u t) + almost surely [M75], there exists a random time S, such that we have µ s t = µ t for all S s t. Since moreover D s D almost surely, as s, it is enough to show that almost surely, for any family of nonnegative functions f t ) t as in the statement of the theorem, lim s lim sup t E[e f t dµ s t F s ] e Ds ft dρ =, a.s. 6.) Let s t. Define f s,t x) = f t x t s)/t). By the branching property and Jensen s inequality, E[e f tdµ s t F s ] = E Xus)[e f s,t dµ t s ] exp [ t s] f s,t dµ, u N s) u N s) E Xus) 6.2) We now have for every x, by the first moment formula for branching Markov processes [INW69, Theorem 4.] and Girsanov s theorem, for every bounded measurable function f, [ ] E x fdµ t = e t/2 W x [B t + t)e Bt+t) fb t + t)/ t) Br r t)] = e x W x [B t fb t / t) Br r t)] = xe x E x [fr t / t)] = xe x E x/ t [fr )], where under P x, R t ) t is a three-dimensional Bessel process started at x [RY99, Section XI.]. The law of R under P x has a continuous density with respect to Lebesgue measure for every x which converges uniformly to the density of ρ as x. It follows easily from this that for every x, [ ] E x f s,t dµ t s xe x f t dρ, as t. 6.3) Equations 6.2) and 6.3) now yield the inequality in 6.). In order to obtain the other inequality, we have by Lemma 6.3 below, for some constants C, C, ) ] 2 E x [ f s,t dµ t s C E x[d 2 t s] Ce x, 6.4) where the superscript in E x indicates that the particles are killed upon hitting the origin. By the branching property and the inequalities e x x + x 2 e x+x2 for x, we then get by 6.4), E[e f tdµ s t F s ] exp [ ] [ ) ] 2 E Xus) f s,t dµ t s + E Xus) f s,t dµ t s u N s) exp D s f t dρ + CW s + E s,t ), 6.5) 6

17 where W s = u N s) e Xus) and E s,t = u N s) E Xus) [ f s,t dµ t s ] + D s f t dρ is an F s -measurable term. By 6.4) and the fact that ρ has a continuous density with respect to Lebesgue s measure, E s,t tends to zero almost surely, as t, for each fixed s. Since W s almost surely, as s see e.g. [LS87, N88]), the inequality 6.5) yields the inequality in 6.). This finishes the proof of the theorem. Lemma 6.3. Let E x be the law of BBM as in the beginning of this section but where in addition particles are killed upon hitting the origin. For some constant C, E x[d 2 t ] Ce x for every x and t. Proof. We first note that D t ) t is a martingale as well under E x. In particular, E x[d t ] = xe x for every x and t. By the second moment formula for branching Markov processes [INW69, Theorem 4.5], this gives for some constant C, E x[d 2 t ] = E x [ u N t) X u t) 2 e 2Xut)] + CE x [ t u N s) ] X u s) 2 e 2Xus) ds. By the first moment formula for branching Markov processes and Girsanov s theorem we get as in the proof of Theorem 6., [ E x[dt 2 ] = e x E x [Bt 2 t T e Bt Bs s t] + CE x Bs 2 e ds] ) Bs, where T is the first hitting time of the origin. The term in the first expectation is bounded by a constant. As for the second expectation, by the inequality x 2 e x C e x/2 and Ito s formula, we have E x [ t T This yields the lemma. References ] Bs 2 e Bs ds 4C E x [e B t T /2 e x/2 ] 4C. [A3] [AS64] [BK4] [Br78] [Br83] E. Aïdékon, Convergence in law of the minimum of a branching random walk, Ann. Probab. 4, 3A 23), pp M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series ). A. Bovier and I. Kurkova, Derrida s generalized random energy models. II. Models with continuous hierarchies, Ann. Inst. H. Poincaré Probab. Statist. 4 24), pp M. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math ), pp M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc ), #285. 7

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