2021/03/18 by Pascal Maillard, Maillard, Pascal, Michel Pain +1
Economics, Econometrics and Finance · Mathematics · #35K57 #60F17 #60J80 #82B44 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Financial Risk and Volatility Modeling #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2103.10412
openalex publication_date 2021/03/18 · openalex created_date 2021/03/29 · openalex updated_date 2026/08/02
Let μt denote the critical derivative Gibbs measure of branching Brownian motion at time t. It has been proved by Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470--502) and Maillard and Zeitouni (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 3, 1144--1160) that μt converges weakly to the random measure Z_∞ √(2/π) x2 e-x2/2 \boldsymbol 1x >0 d x, where Z_∞ is the limit of the derivative martingale. In this paper, we are interested in the fluctuations that occur in this convergence and prove for a large class of functions F that √(t) ( ∫\mathbb R F d μt - Z_∞ ∫0^∞ F(x) √\frac2π x2 e-x2/2 d x - (c(F) log t)/(√(t)) Z_∞ ) → S(F), in law, as t→∞, where c(F) is a constant depending on F and, given Z_∞, S(F) has an explicit 1-stable distribution. Moreover, we extend this result to a functional convergence, and we identify precisely the particles responsible for the fluctuations. In particular, this proves the following result for the critical additive martingale (Wt)t≥ 0: √(t) ( √(t) Wt - √\frac2π Z_∞ ) \xrightarrow[t→∞] C Z_∞, in law, where here C is a Cauchy variable independent of Z_∞, confirming a conjecture by Mueller and Munier (Phys. Rev. E 90 (2014), 042143) in the physics literature.