chewi_display_1_1_10 · Samplinglib

Plain-English statement

Local square integrability asks for finite accumulated squared energy on almost every path, not finite expected energy.

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 P-almost every omega, the integral from 0 to T of eta(t,omega)^2 dt is finite.

Intuition

Stopping paths before their energy becomes large converts this weaker condition into the global L2 condition used by the elementary integral.

Conditions

finite terminal time
a probability-space representative of the process

Why these conditions cannot be dropped

the condition is pathwise and therefore uses an almost-everywhere representative
ENNReal keeps divergence visible

Proof route

define IsLocallySquareIntegrableOn by a pathwise lintegral
expose the definition as the source equivalence

Lean interface notes

this is weaker than processL2Energy < infinity
no stopping time or Ito integral is constructed by this declaration

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Syntax used on this page

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Lean statement

theorem chewi_display_1_1_10
    {Omega : Type*} [MeasurableSpace Omega]
    (eta : ℝ≥0 → Omega → ℝ) (mu : Measure Omega) (T : ℝ≥0) :
    IsLocallySquareIntegrableOn eta mu T ↔
      ∀ᵐ omega ∂mu,
        (∫⁻ t, ENNReal.ofReal ((eta t omega) ^ 2) ∂(TimeMeasure.upTo T)) < ∞ :=
  Iff.rfl

end ElementaryItoIntegral
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:115published source at 7bcd37294df1