Polynomial slowdown in an angle-dependent 2d branching Brownian motion

Technical Summary: Polynomial Slowdown in an Angle-Dependent 2d Branching Brownian Motion

Problem Statement
The paper investigates the asymptotic behavior of the maximal displacement, $M_t$, in a spatially inhomogeneous branching Brownian motion (BBM) in $\mathbb{R}^2$. Unlike standard BBM where particles branch at a constant rate, this model features a branching rate $b(\theta)$ that depends solely on the angular coordinate $\theta$ of the particle. The function $b$ is assumed to be $2\pi$-periodic, reaching its unique maximum at $\theta = 0$ (the preferred direction), and satisfying the local expansion $b(\theta) = 1 - \beta|\theta|^\alpha + O(\theta^2)$ as $\theta \to 0$, with $\alpha \in (2/3, 2)$ and $\beta > 0$.

The central question is how this angular inhomogeneity affects the speed and fluctuations of the leading particles compared to the homogeneous case. In standard 1d BBM, the maximum grows as $\sqrt{2t} - \frac{3}{2\sqrt{2}}\log t$. In 2d homogeneous BBM, the logarithmic correction changes to $-\frac{1}{\sqrt{2}}\log t$. The authors aim to determine the precise asymptotic expansion of $M_t$ when the branching rate is maximal only in a specific direction and decays polynomially with the angle.

Methodology
The proof strategy combines probabilistic techniques with partial differential equation (PDE) analysis, specifically leveraging the Feynman-Kac formula to relate the expectation of the branching process to the solution of a heat equation with a potential.

1. Many-to-One and Many-to-Few Lemmas: The authors reduce expectations over the branching population to expectations over a single (or two) Brownian motion(s) weighted by the integral of the branching rate along the path.
   $$\mathbb{E}\left[\sum_{u \in N(t)} f(\text{path}_u)\right] = \mathbb{E}\left[f(\text{path}) \exp\left(\int_0^t b(\theta_s) ds\right)\right]$$
2. PDE Analysis via Sturm-Liouville Theory: A critical component is the analysis of the kernel $G(s, x; t, y)$ associated with the weighted Brownian motion. By rescaling time and space, the problem is mapped to a time-inhomogeneous PDE on $[0, 1) \times \mathbb{R}$:
   $$\partial_r u = \varrho \left( \partial_{xx} u - (1-r)^{-\alpha}|x|^\alpha u \right)$$
   The authors utilize Sturm-Liouville theory to analyze the associated operator $L_q f = -f'' + q|x|^\alpha f$. They establish precise estimates for the fundamental solution of this PDE, showing that the decay is governed by the principal eigenvalue $\lambda_0$ of the operator $L = -f'' + |x|^\alpha f$.
3. Path Localization: The proof relies on localizing the trajectories of extremal particles. It is shown that particles achieving the maximum displacement must stay within a narrow angular cone around $\theta=0$. Specifically, the $Y$-coordinate (transverse to the preferred direction) is constrained to scale as $t^{\kappa/2}$ where $\kappa = \frac{2\alpha}{2+\alpha}$. This localization allows the approximation $\theta_s \approx Y_s / (\sqrt{2}s)$, transforming the angular dependence into a transverse coordinate dependence.
4. Upper and Lower Bounds:
   • Upper Bound: Established using a first-moment calculation (many-to-one lemma) combined with a "pseudo-derivative martingale" and barrier arguments. The authors compare the process to a standard 1d BBM with a curved barrier to control the logarithmic corrections.
   • Lower Bound: Established using a second-moment calculation (many-to-two lemma) on a specific set of particles $\Gamma_{s_0}(t)$ that satisfy strict trajectory constraints (staying within the angular cone and near the predicted maximum). The second moment is shown to be bounded by the square of the first moment, ensuring the existence of such particles with high probability.

Key Contributions and Results

• Polynomial Slowdown: The primary result (Theorem 1.1) is that the maximal displacement $M_t$ exhibits a polynomial slowdown relative to the standard $\sqrt{2t}$ speed. The asymptotic expansion is given by:
  $$M_t = \sqrt{2t} - \vartheta_1 \sqrt{2} t^{(2-\alpha)/(2+\alpha)} - \left( \frac{3}{2\sqrt{2}} - \frac{\alpha}{2\sqrt{2}(2+\alpha)} \right) \log t + O_P(1)$$
  Here, $\vartheta_1$ is an explicit constant depending on $\beta$ and the principal eigenvalue $\lambda_0$ of the operator $-f'' + |x|^\alpha f$.
• Optimality of Exponents: The paper proves that the exponent of the polynomial term, $\frac{2-\alpha}{2+\alpha}$, is optimal. Furthermore, Proposition 1.2 establishes that particles at distance $m(t)$ have a $Y$-coordinate of order $t^{\alpha/(2+\alpha)}$ and an angle of order $t^{-2/(2+\alpha)}$.
• Tightness: The authors prove that the sequence $(M_t - m(t))_{t \ge 1}$ is tight (bounded in probability), though they do not prove convergence in distribution. They note that the limiting random variable would likely be the limit of an approximate martingale rather than the derivative martingale found in homogeneous cases.
• Phase Transition: The results are specific to the regime $\alpha \in (2/3, 2)$. The authors discuss that for $\alpha \ge 2$, the polynomial correction should vanish (leaving only a logarithmic correction), and for $\alpha \le 2/3$, the behavior becomes more complex with potentially additional polynomial terms.

Significance and Claims
The paper claims to uncover a "richer variety of behaviors" in branching processes by making the branching rate dependent on the angle rather than a linear coordinate (which would resemble a strip model). The significance lies in:

1. Universal Polynomial Correction: The model provides a new occurrence of polynomial slowdown for maximal displacement, distinct from the $T^{1/3}$ correction found in time-inhomogeneous BBM. Crucially, this model allows for a continuous range of polynomial exponents in $(0, 1/2)$ by varying $\alpha$, whereas time-inhomogeneous models typically yield a fixed exponent.
2. Interplay of Geometry and Branching: The work demonstrates how the geometry of the preferred direction (the angle) interacts with the diffusion and branching to create a specific scaling regime for the transverse coordinate ($t^{\kappa/2}$).
3. Technical Novelty: The derivation of precise heat kernel estimates for a time-inhomogeneous PDE with a singular potential $(1-t)^{-\alpha}|x|^\alpha$ using Sturm-Liouville theory is a significant technical contribution, extending methods used in previous works on time-inhomogeneous BBM (e.g., Maillard and Zeitouni).

The authors remain modest regarding the convergence in distribution, acknowledging that the lack of an exact martingale structure makes proving the limit distribution an open question. They also leave the cases $\alpha \le 2/3$ and $\alpha \ge 2$ as open problems for future research.