finiteGaussianGirsanovCylinderMeasure_eq_withDensity

Plain-English statement

A finite-dimensional shifted standard Gaussian law is the centered standard Gaussian law tilted by the exponential shift likelihood.

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

Law(Z+h) = stdGaussian.withDensity(exp(<h,z> - norm(h)^2/2)).

Intuition

This is the finite-cylinder Radon-Nikodym identity behind Girsanov. Path-space Girsanov additionally needs filtrations, adaptedness, stochastic integration, and martingale conditions.

Conditions

The coordinate index is finite.
Both laws use unit-variance Gaussian coordinates.
The coordinate-to-EuclideanSpace map is the measurable PiLp equivalence.

Why these conditions cannot be dropped

The finite product proof does not automatically extend to an infinite path space.
Changing covariance changes the likelihood and normalization.

Proof route

Use the product Gaussian shift density identity.
Map both measures through the measurable Euclidean equivalence.
Commute map with withDensity.
Rewrite coordinate sums as inner products and squared norms.

Lean interface notes

Measure.withDensity stores an ENNReal likelihood.
MeasurableEquiv transports both the measure and its density.
The local let-bindings expose the intermediate product law and coordinate density.

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Lean statement

theorem finiteGaussianGirsanovCylinderMeasure_eq_withDensity
    {ι : Type*} [Fintype ι] (h : ι → ℝ) :
    finiteShiftedGaussianPathMeasure h =
      (stdGaussian (EuclideanSpace ℝ ι)).withDensity
        (fun z => ENNReal.ofReal (finiteGaussianGirsanovWeight h z)) := by
  classical
  let μ0 : Measure (ι → ℝ) :=
    Measure.pi (fun _ : ι => gaussianReal 0 (1 : NNReal))
  let coordDensity : (ι → ℝ) → ℝ≥0∞ :=
    fun x => ENNReal.ofReal
      (Real.exp ((∑ i, h i * x i) - (∑ i, (h i) ^ 2) / 2))
  let e : (ι → ℝ) ≃ᵐ EuclideanSpace ℝ ι :=
    MeasurableEquiv.toLp 2 (ι → ℝ)
  have hProduct :
      Measure.pi (fun i : ι => gaussianReal (h i) (1 : NNReal)) =
        μ0.withDensity coordDensity := by
    simpa [μ0, coordDensity] using
      (_root_.AutoSamplingTheory.TechnicalLemmas.Gaussian.pi_gaussianReal_withDensity_exp_shift h).symm
  have hMapDensity :
      (μ0.withDensity coordDensity).map e =
        (μ0.map e).withDensity (fun z => coordDensity (e.symm z)) :=
    AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.measurableEquiv_map_withDensity
      e μ0 (by fun_prop)
  have hStd : μ0.map e = stdGaussian (EuclideanSpace ℝ ι) := by
    simpa [μ0, e, MeasurableEquiv.coe_toLp] using
      ProbabilityTheory.map_pi_eq_stdGaussian (ι := ι)
  have hDensityEq :
      (fun z : EuclideanSpace ℝ ι => coordDensity (e.symm z)) =
        fun z => ENNReal.ofReal (finiteGaussianGirsanovWeight h z) := by
    funext z
    have hInner :
        inner ℝ (WithLp.toLp 2 h : EuclideanSpace ℝ ι) z =
          ∑ i, h i * (WithLp.ofLp z) i := by
      simpa using
        _root_.AutoSamplingTheory.TechnicalLemmas.Gaussian.inner_toLp_toLp_eq_sum_mul
          h (WithLp.ofLp z)
    have hNorm :
        ‖(WithLp.toLp 2 h : EuclideanSpace ℝ ι)‖ ^ 2 =
          ∑ i, (h i) ^ 2 :=
      _root_.AutoSamplingTheory.TechnicalLemmas.Gaussian.norm_sq_toLp_eq_sum_sq h
    simp [coordDensity, finiteGaussianGirsanovWeight, e, hInner, hNorm]
  calc
    finiteShiftedGaussianPathMeasure h
        = (μ0.withDensity coordDensity).map e := by
          simpa [finiteShiftedGaussianPathMeasure, e, μ0,
            MeasurableEquiv.coe_toLp] using congrArg (fun ν => ν.map e) hProduct
    _ = (μ0.map e).withDensity (fun z => coordDensity (e.symm z)) := hMapDensity
    _ = (stdGaussian (EuclideanSpace ℝ ι)).withDensity
          (fun z => ENNReal.ofReal (finiteGaussianGirsanovWeight h z)) := by
          rw [hDensityEq, hStd]

/-- The finite-dimensional Girsanov weight has unit mass under the centered
`stdGaussian` cylinder. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Girsanov.lean:56published source at 7bcd37294df1