Written proof or calculation (limited scope) Chapter 8 establishes extension of coordinate laws; this is not a continuous-path construction.
Proof outline / conditional derivation The existing route from consistent Gaussian finite-dimensional laws to a continuous version invokes extension and continuity tools.
Written proof or calculation (limited scope) Sections 23.6–23.12 contain complete written proofs of continuous modification, C/C_loc measurability and Wiener existence/uniqueness; continuing mathematical verification, not final review.
Written proof or calculation (limited scope) Linear interpolation and the original fourth-moment calculations are written out, under their stated hypotheses.
Proof outline / conditional derivation Finite-dimensional identification and path tightness still invoke external tools; this is not a self-contained functional CLT.
Written proof or calculation (limited scope) Chapter25 and T-FCLT contain complete written finite-second-moment Donsker, empirical-bridge and finite-variance renewal proofs; continuing verification, not final review.
Written proof or calculation (limited scope) The Brownian/Gaussian-kernel and heat-equation calculations are retained.
Mathematical intuition The connection from random paths to deterministic evolution motivates, but does not replace, general operator theory.
Written proof or calculation (limited scope) Chapter 28 and its three companions now provide positive contraction generation, boundary domains, and conservative locally compact Polish Feller paths and strong Markov. Author review and human mathematical final review remain pending; PDE developments remain in Chapter 38.
Written proof or calculation (limited scope) A written quadratic-variation proof is available for fixed T along dyadic grids, assuming Brownian motion exists.
Mathematical intuition Fixed-time difference-quotient scaling motivates nondifferentiability; it is not a complete almost-sure nowhere-differentiability proof.
Theorem still to be proved Chapter 26 handles path regularity; Chapter 31 develops general variation and integration, and Chapter 32 proves Itô’s formula.
Written proof or calculation (limited scope) Chapter 26 adds a uniform-mesh nowhere-differentiability proof and deterministic dyadic square sums; general brackets and stochastic integration remain for Chapter 31.
Written proof or calculation (limited scope) Chapter31 and T-ITO-BRACKET contain complete working proofs for continuous-local-martingale integration, brackets and ucp square sums; author review and human final review remain open; the legacy body is preserved.
Written proof or calculation (limited scope) Chapter 26 supplies written Brownian strong Markov, measurable measure-preserving reflection and maximum/first-hit proofs; the old source is retained in full.
Theorem still to be proved Human independent mathematical final review remains pending; general Feller strong Markov belongs to Chapter 28.
Mathematical intuition Excursions, a pseudometric and branching diagrams provide geometric intuition; the tree structure of the quotient still needs a proof.
Proof outline / conditional derivation The coding discussion is not a complete real-tree, CRT or conditioned-tree limit theorem.
Theorem still to be proved Chapter 37 develops excursions, Chapters 52–53 discrete coding and conditioned limits, Chapter 54 real trees and CRT, and Chapter 55 convergence of spaces.
Foundations and companion proofs
Notes and resources
Further Reading and Routes through the Proofs
Chapter 4 identified, through random walks, the structure that continuous random motion should have. To turn that structure into a rigorous probability model, we can continue along two connected routes: constructing the process and proving path limits. Each entry below develops a particular mathematical question.
How Do Finite-Dimensional Distributions Give Continuous Random Motion?
Consistent joint distributions, measure extension, and continuous modifications connect finitely many coordinates to an entire random path. On continuous-function space, Wiener measure makes sets of paths into objects that probability theory can study.
Linear interpolation places discrete paths in continuous-function space. Finite-dimensional limits must be accompanied by control of oscillations over small time intervals; path tightness addresses precisely this issue.
The square-root scaling of increments makes accumulated second-order effects impossible to ignore. Path regularity, quadratic variation, and stochastic integration will explain further the source of the additional term in Itô's formula.
How Do We Calculate the Probability of “Having Reached” a Level?
Stopping times and the strong Markov property let us study paths after their first hitting time, while the reflection principle connects path events with Gaussian distributions.
The changing height of an excursion can encode branching and distance. Through pseudometrics, quotient spaces, and real trees, we move from random paths toward continuous random geometry.