Plain-English statement
A generator satisfies the positive Bakry-Emery curvature condition when Gamma_2 dominates alpha times Gamma everywhere.
Mathematical statement
CD(alpha,infinity) means alpha > 0 and alpha Gamma(f,f)(x) <= Gamma_2(f,f)(x) for every f and x.
Intuition
The inequality is the generator analogue of a uniform lower curvature bound. For Langevin diffusion it is later connected to strong convexity of the potential.
Conditions
- alpha is strictly positive
- the pointwise inequality holds for every observable and state
Why these conditions cannot be dropped
- strict positivity is part of the source definition and determines a nontrivial convergence rate
- the abstract predicate alone does not prove Poincare or log-Sobolev inequalities
Proof route
- package positivity and the universal Gamma_2 inequality as one predicate
- leave the functional-inequality implication and Langevin verification to their own source routes
Lean interface notes
- the universal quantifiers are explicit
- no typeclass or axiom supplies the inequality automatically
Read the mathematics first, then descend into Lean
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Lean statement
def SatisfiesBakryEmery
(generator : (X → ℝ) →ₗ[ℝ] (X → ℝ))
(alpha : ℝ) : Prop :=
0 < alpha ∧ ∀ (f : X → ℝ) (x : X),
alpha * carreDuChamp generator f f x ≤
iteratedCarreDuChamp generator f f x
/-- Chewi Lemma 1.2.13: the Markov-semigroup Jensen inequality implies
nonnegativity of the carre du champ after taking the right-generator limit.
The theorem is pointwise. `hf` and `hf2` are the actual right difference-
quotient limits for `f` and `f²`; `hcontinuous` is strong/right continuity of
the orbit at the selected state. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:74published source at 7bcd37294df1