sq_fellerOperator_apply_le

Plain-English statement

A Markov transition operator cannot make the square of an average larger than the average of the square.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

For every bounded continuous f, time t, and state x, (P_t f(x))^2 <= P_t(f^2)(x).

Intuition

At a fixed starting state, P_t is expectation under a probability law, so this is Jensen's inequality for the convex square function.

Conditions

a Feller family of probability transition kernels
a bounded continuous real observable

Why these conditions cannot be dropped

probability normalization is the normalization needed by Jensen
bounded continuity gives measurability and integrability of f and f squared

Proof route

view K_t(x,dy) as a probability measure
prove integrability of f and f squared from the sup-norm bound
apply ConvexOn.map_integral_le to the even convex power r maps to r squared
identify the two integrals with the Feller operators

Lean interface notes

IsMarkovKernel installs IsProbabilityMeasure for K_t x
the theorem derives equation (1.2.11) from kernels instead of assuming it

Lean learning studio · mathematics → formal proof

Read the mathematics first, then descend into Lean

You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.

BeginnerWhy this theorem exists → intuition → statement → one hand calculation. Hide proof-engineering detail.

RigorousExpose assumptions, hidden measure/limit/domain issues, proof route, and rigorous references.

Lean learnerOpen the exact declaration, proof tree/network, syntax glossary, and line-by-line explanation.

Proof architecture

Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.

Loading source-derived dependency evidence…

Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.

How to read the exact Lean declaration

Read a Lean theorem left-to-right exactly as you would unpack a mathematical sentence: name → ambient types → automatically inferred structures → explicit hypotheses → conclusion → proof. Then read the proof top-to-bottom as transformations of the current goal.

NameWhat reusable mathematical fact is being created?
ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
PropositionAfter the colon, translate the Lean expression back into a paper statement.
Proof actionsAfter by, ask what each tactic does to the mathematical goal—not only what syntax it uses.

Syntax used on this page

This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.

Source voice and ASTIS voice stay separate

When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.

Lean statement

theorem sq_fellerOperator_apply_le {K : ℝ≥0 → Kernel E E}
    (hK : FellerTransitionKernelContract K) (t : ℝ≥0)
    (f : E →ᵇ ℝ) (x : E) :
    (fellerOperator hK t f x) ^ 2 ≤
      fellerOperator hK t (f * f) x := by
  let : IsMarkovKernel (K t) := hK.toTransitionKernelContract.isMarkov t
  have hsquare : Integrable ((fun r : ℝ => r ^ 2) ∘ fun y => f y) (K t x) := by
    simpa [Function.comp_def, pow_two] using
      (integrable_boundedContinuousFunction (f * f) (K t x))
  have hJensen :=
    (even_two.convexOn_pow : ConvexOn ℝ Set.univ fun r : ℝ => r ^ 2).map_integral_le
      (continuousOn_pow 2) isClosed_univ
      (ae_of_all _ fun y => Set.mem_univ (f y))
      (integrable_boundedContinuousFunction f (K t x))
      hsquare
  simpa [Function.comp_def, pow_two] using hJensen

/-- The zero-time Feller operator is the identity continuous linear map. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:143published source at 7bcd37294df1