Determine the boundary slope of v in the primal Parisi PDE

Determine the precise boundary behavior of the function v(t,y) = 1 / ∂_{y,y}Λ(t,y) as y approaches ±1 for t ∈ [0,1], where Λ is the Fenchel–Legendre conjugate of the Parisi PDE solution Φ for the Sherrington–Kirkpatrick model and v satisfies the PDE ∂_t v = −β^2 v^2 (∂_{y,y} v + 2 F_μ(t)) with terminal condition v(1,y) = 1 − y^2. In particular, establish whether v vanishes linearly at the boundary and determine the exact slope (“angle”) of v as y → ±1.

Background

In the primal formulation of the Parisi theory developed in this paper, Λ(t,y) denotes the Fenchel–Legendre conjugate in the spatial variable of the Parisi PDE solution Φ(t,x). The function v(t,y) = 1 / ∂{y,y}Λ(t,y) arises naturally in the analysis and satisfies a nonlinear parabolic PDE: ∂_t v = −β^2 v^2 (∂{y,y} v + 2 F_μ(t)) with terminal condition v(1,y) = 1 − y^2. Here F_μ is the cumulative distribution function of the Parisi measure.

Understanding the boundary behavior of v at y = ±1 is important for characterizing the regularity and free-boundary aspects of the primal PDE, which, in turn, impacts the analysis of the stochastic dynamics and the algorithmic framework developed in the paper. The authors note that v should vanish at the boundary and conjecture a linear decay but point out that the exact boundary slope (the “angle”) remains undetermined.