feat: define IsKolmogorovProcess (#27202)

A stochastic process `X : T → Ω → E` on an index space `T` and a measurable space `Ω` with measure `P` is said to satisfy the Kolmogorov condition with exponents `p, q` and constant `M` if for all `s, t : T`, the pair `(X s, X t)` is measurable for the Borel sigma-algebra on `E × E` and the following condition holds:
`∫⁻ ω, edist (X s ω) (X t ω) ^ p ∂P ≤ M * edist s t ^ q`.

This is the main assumption of the Kolmogorov-Chentsov theorem, which gives the existence of a continuous modification of the process.

From the Brownian motion project.

Author: RemyDegenne

Mathlib.lean

@@ -5100,6 +5100,7 @@ import Mathlib.Probability.ProbabilityMassFunction.Monad
 import Mathlib.Probability.Process.Adapted
 import Mathlib.Probability.Process.Filtration
 import Mathlib.Probability.Process.HittingTime
+import Mathlib.Probability.Process.Kolmogorov
 import Mathlib.Probability.Process.PartitionFiltration
 import Mathlib.Probability.Process.Stopping
 import Mathlib.Probability.ProductMeasure

Mathlib/Probability/Process/Kolmogorov.lean

@@ -0,0 +1,183 @@
+/-
+Copyright (c) 2025 Rémy Degenne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rémy Degenne
+-/
+import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
+import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
+import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
+
+/-!
+# Stochastic processes satisfying the Kolmogorov condition
+
+A stochastic process `X : T → Ω → E` on an index space `T` and a measurable space `Ω`
+with measure `P` is said to satisfy the Kolmogorov condition with exponents `p, q` and constant `M`
+if for all `s, t : T`, the pair `(X s, X t)` is measurable for the Borel sigma-algebra on `E × E`
+and the following condition holds:
+`∫⁻ ω, edist (X s ω) (X t ω) ^ p ∂P ≤ M * edist s t ^ q`.
+
+This condition is the main assumption of the Kolmogorov-Chentsov theorem, which gives the existence
+of a continuous modification of the process.
+
+The measurability condition on pairs ensures that the distance `edist (X s ω) (X t ω)` is
+measurable in `ω` for fixed `s, t`. In a space with second-countable topology, the measurability
+of pairs can be obtained from measurability of each `X t`.
+
+## Main definitions
+
+* `IsKolmogorovProcess`: property of being a stochastic process that satisfies
+  the Kolmogorov condition.
+* `IsAEKolmogorovProcess`: a stochastic process satisfies `IsAEKolmogorovProcess` if it is
+  a modification of a process satisfying the Kolmogorov condition.
+
+## Main statements
+
+* `IsKolmogorovProcess.mk_of_secondCountableTopology`: in a space with second-countable topology,
+  a process is a Kolmogorov process if each `X t` is measurable and the Kolmogorov condition holds.
+
+-/
+
+open MeasureTheory
+open scoped ENNReal NNReal
+
+namespace ProbabilityTheory
+
+variable {T Ω E : Type*} [PseudoEMetricSpace T] {mΩ : MeasurableSpace Ω} [PseudoEMetricSpace E]
+  {p q : ℝ} {M : ℝ≥0} {P : Measure Ω} {X : T → Ω → E}
+
+/-- A stochastic process `X : T → Ω → E` on an index space `T` and a measurable space `Ω`
+with measure `P` is said to satisfy the Kolmogorov condition with exponents `p, q` and constant `M`
+if for all `s, t : T`, the pair `(X s, X t)` is measurable for the Borel sigma-algebra on `E × E`
+and the following condition holds: `∫⁻ ω, edist (X s ω) (X t ω) ^ p ∂P ≤ M * edist s t ^ q`. -/
+structure IsKolmogorovProcess (X : T → Ω → E) (P : Measure Ω) (p q : ℝ) (M : ℝ≥0) : Prop where
+  measurablePair : ∀ s t : T, Measurable[_, borel (E × E)] fun ω ↦ (X s ω, X t ω)
+  kolmogorovCondition : ∀ s t : T, ∫⁻ ω, edist (X s ω) (X t ω) ^ p ∂P ≤ M * edist s t ^ q
+  p_pos : 0 < p
+  q_pos : 0 < q
+
+/-- Property of being a modification of a stochastic process that satisfies the Kolmogorov
+condition (`IsKolmogorovProcess`). -/
+def IsAEKolmogorovProcess (X : T → Ω → E) (P : Measure Ω) (p q : ℝ) (M : ℝ≥0) : Prop :=
+  ∃ Y, IsKolmogorovProcess Y P p q M ∧ ∀ t, X t =ᵐ[P] Y t
+
+lemma IsKolmogorovProcess.IsAEKolmogorovProcess (hX : IsKolmogorovProcess X P p q M) :
+    IsAEKolmogorovProcess X P p q M := ⟨X, hX, by simp⟩
+
+namespace IsAEKolmogorovProcess
+
+/-- A process with the property `IsKolmogorovProcess` such that `∀ t, X t =ᵐ[P] h.mk X t`. -/
+protected noncomputable
+def mk (X : T → Ω → E) (h : IsAEKolmogorovProcess X P p q M) : T → Ω → E :=
+  Classical.choose h
+
+lemma IsKolmogorovProcess_mk (h : IsAEKolmogorovProcess X P p q M) :
+    IsKolmogorovProcess (h.mk X) P p q M := (Classical.choose_spec h).1
+
+lemma ae_eq_mk (h : IsAEKolmogorovProcess X P p q M) : ∀ t, X t =ᵐ[P] h.mk X t :=
+  (Classical.choose_spec h).2
+
+lemma kolmogorovCondition (hX : IsAEKolmogorovProcess X P p q M) (s t : T) :
+    ∫⁻ ω, edist (X s ω) (X t ω) ^ p ∂P ≤ M * edist s t ^ q := by
+  convert hX.IsKolmogorovProcess_mk.kolmogorovCondition s t using 1
+  refine lintegral_congr_ae ?_
+  filter_upwards [hX.ae_eq_mk s, hX.ae_eq_mk t] with ω hω₁ hω₂
+  simp_rw [hω₁, hω₂]
+
+lemma p_pos (hX : IsAEKolmogorovProcess X P p q M) : 0 < p := hX.IsKolmogorovProcess_mk.p_pos
+
+lemma q_pos (hX : IsAEKolmogorovProcess X P p q M) : 0 < q := hX.IsKolmogorovProcess_mk.q_pos
+
+lemma congr {Y : T → Ω → E} (hX : IsAEKolmogorovProcess X P p q M)
+    (h : ∀ t, X t =ᵐ[P] Y t) :
+    IsAEKolmogorovProcess Y P p q M := by
+  refine ⟨hX.mk X, hX.IsKolmogorovProcess_mk, fun t ↦ ?_⟩
+  filter_upwards [hX.ae_eq_mk t, h t] with ω hX hY using hY.symm.trans hX
+
+end IsAEKolmogorovProcess
+
+section Measurability
+
+lemma IsKolmogorovProcess.stronglyMeasurable_edist
+    (hX : IsKolmogorovProcess X P p q M) {s t : T} :
+    StronglyMeasurable (fun ω ↦ edist (X s ω) (X t ω)) := by
+  borelize (E × E)
+  exact continuous_edist.stronglyMeasurable.comp_measurable (hX.measurablePair s t)
+
+lemma IsAEKolmogorovProcess.aestronglyMeasurable_edist
+    (hX : IsAEKolmogorovProcess X P p q M) {s t : T} :
+    AEStronglyMeasurable (fun ω ↦ edist (X s ω) (X t ω)) P := by
+  refine ⟨(fun ω ↦ edist (hX.mk X s ω) (hX.mk X t ω)),
+    hX.IsKolmogorovProcess_mk.stronglyMeasurable_edist, ?_⟩
+  filter_upwards [hX.ae_eq_mk s, hX.ae_eq_mk t] with ω hω₁ hω₂ using by simp [hω₁, hω₂]
+
+lemma IsKolmogorovProcess.measurable_edist (hX : IsKolmogorovProcess X P p q M) {s t : T} :
+    Measurable (fun ω ↦ edist (X s ω) (X t ω)) := hX.stronglyMeasurable_edist.measurable
+
+lemma IsAEKolmogorovProcess.aemeasurable_edist (hX : IsAEKolmogorovProcess X P p q M) {s t : T} :
+    AEMeasurable (fun ω ↦ edist (X s ω) (X t ω)) P := hX.aestronglyMeasurable_edist.aemeasurable
+
+variable [MeasurableSpace E] [BorelSpace E]
+
+lemma IsKolmogorovProcess.measurable (hX : IsKolmogorovProcess X P p q M) (s : T) :
+    Measurable (X s) :=
+  (measurable_fst.mono prod_le_borel_prod le_rfl).comp (hX.measurablePair s s)
+
+lemma IsAEKolmogorovProcess.aemeasurable (hX : IsAEKolmogorovProcess X P p q M) (s : T) :
+    AEMeasurable (X s) P := by
+  refine ⟨hX.mk X s, hX.IsKolmogorovProcess_mk.measurable s, ?_⟩
+  filter_upwards [hX.ae_eq_mk s] with ω hω using hω
+
+lemma IsKolmogorovProcess.mk_of_secondCountableTopology [SecondCountableTopology E]
+    (h_meas : ∀ s, Measurable (X s))
+    (h_kol : ∀ s t : T, ∫⁻ ω, (edist (X s ω) (X t ω)) ^ p ∂P ≤ M * edist s t ^ q)
+    (hp : 0 < p) (hq : 0 < q) :
+    IsKolmogorovProcess X P p q M where
+  measurablePair s t := by
+    suffices Measurable (fun ω ↦ (X s ω, X t ω)) by
+      rwa [Prod.borelSpace.measurable_eq] at this
+    fun_prop
+  kolmogorovCondition := h_kol
+  p_pos := hp
+  q_pos := hq
+
+end Measurability
+
+section ZeroDist
+
+lemma IsAEKolmogorovProcess.edist_eq_zero (hX : IsAEKolmogorovProcess X P p q M)
+    {s t : T} (h : edist s t = 0) :
+    ∀ᵐ ω ∂P, edist (X s ω) (X t ω) = 0 := by
+  suffices ∀ᵐ ω ∂P, edist (X s ω) (X t ω) ^ p = 0 by
+    filter_upwards [this] with ω hω
+    simpa [hX.p_pos, not_lt_of_gt hX.p_pos] using hω
+  refine (lintegral_eq_zero_iff' (hX.aemeasurable_edist.pow_const p)).mp ?_
+  refine le_antisymm ?_ zero_le'
+  calc ∫⁻ ω, edist (X s ω) (X t ω) ^ p ∂P
+  _ ≤ M * edist s t ^ q := hX.kolmogorovCondition s t
+  _ = 0 := by simp [h, hX.q_pos]
+
+lemma IsKolmogorovProcess.edist_eq_zero (hX : IsKolmogorovProcess X P p q M)
+    {s t : T} (h : edist s t = 0) :
+    ∀ᵐ ω ∂P, edist (X s ω) (X t ω) = 0 :=
+  hX.IsAEKolmogorovProcess.edist_eq_zero h
+
+lemma IsAEKolmogorovProcess.edist_eq_zero_of_const_eq_zero (hX : IsAEKolmogorovProcess X P p q 0)
+    (s t : T) :
+    ∀ᵐ ω ∂P, edist (X s ω) (X t ω) = 0 := by
+  suffices ∀ᵐ ω ∂P, edist (X s ω) (X t ω) ^ p = 0 by
+    filter_upwards [this] with ω hω
+    simpa [hX.p_pos, not_lt_of_gt hX.p_pos] using hω
+  refine (lintegral_eq_zero_iff' (hX.aemeasurable_edist.pow_const p)).mp ?_
+  refine le_antisymm ?_ zero_le'
+  calc ∫⁻ ω, edist (X s ω) (X t ω) ^ p ∂P
+  _ ≤ 0 * edist s t ^ q := hX.kolmogorovCondition s t
+  _ = 0 := by simp
+
+lemma IsKolmogorovProcess.edist_eq_zero_of_const_eq_zero (hX : IsKolmogorovProcess X P p q 0)
+    (s t : T) :
+    ∀ᵐ ω ∂P, edist (X s ω) (X t ω) = 0 :=
+  hX.IsAEKolmogorovProcess.edist_eq_zero_of_const_eq_zero s t
+
+end ZeroDist
+
+end ProbabilityTheory