| import OAI.Geometry.RCD.Heat.CanonicalCompactLogLower |
| import OAI.Geometry.RCD.Heat.CanonicalCompactLogUpper |
|
|
| namespace OAI |
|
|
| /-! # Uniform logarithmic small-noise limit of the actual heat kernel |
|
|
| On any fixed bounded ball, the kernel at time ε*T/2 has scaled logarithm |
| converging uniformly to -dist²/(2*T). Both estimates are proved for the |
| same canonical RCD kernel; no heat-kernel asymptotic is assumed. |
| -/ |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped NNReal ENNReal Topology |
| noncomputable section |
| namespace RegularPointCoordinates.NoncollapsedRCD |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [ProperSpace X] {K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X) |
| (hn : 2 ≤ n) (p : X) [SeparableSpace X] |
| [IsLocallyFiniteMeasure (hausdorff n X)] [(hausdorff n X).Regular] |
|
|
| theorem canonicalHeatKernel_compact_log_limit (hK : K ≤ 0) |
| (ε : ℕ → ℝ) (hε : ∀ k, 0 < ε k) (hε1 : ∀ k, ε k ≤ 1) |
| (hε0 : Tendsto ε atTop (𝓝 0)) {R T : ℝ} (hR : 0 ≤ R) (hT : 0 < T) (hT1 : T < 1) : |
| TendstoUniformly (fun k (q : closedBall p R × closedBall p R) => |
| ε k*Real.log (hX.canonicalHeatKernel hn p (ε k*T/2) q.1 q.2)) |
| (fun q => -(dist (q.1 : X) (q.2 : X))^2/(2*T)) atTop := by |
| apply Metric.tendstoUniformly_iff.mpr |
| intro η hη |
| have hl := hX.canonicalHeatKernel_compact_log_lower hn p hK ε hε hε1 hε0 |
| (isCompact_closedBall p R) (isCompact_closedBall p R) hT hT1 (half_pos hη) |
| have hu := hX.canonicalHeatKernel_bounded_log_upper hn p ε hε hε1 hε0 hR hT hT1 (half_pos hη) |
| filter_upwards [hl,hu] with k hkl hku |
| intro q |
| have hl' := hkl q.1 q.1.property q.2 q.2.property |
| have hu' := hku q.1 q.1.property q.2 q.2.property |
| rw [Real.dist_eq] |
| apply abs_lt.mpr |
| constructor <;> linarith |
|
|
| end RegularPointCoordinates.NoncollapsedRCD |
|
|
| end |
|
|
| end OAI |
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|
