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Cover -- Title page -- Acknowledgement -- Chapter 1. Introduction -- Chapter 2. Background -- 2.1. (Riemannian) laminations and Laplacians -- 2.2. Covering laminations -- 2.3. Heat kernels, (weakly) harmonic measures and Standing Hypotheses -- 2.4. Brownian motion and Wiener measures without holonomy -- 2.5. Wiener measures with holonomy -- Chapter 3. Statement of the main results -- 3.1. Multiplicative cocycles -- 3.2. First Main Theorem and applications -- 3.3. Second Main Theorem and applications -- 3.4. Plan of the proof -- Chapter 4. Preparatory results -- 4.1. Measurability issue -- 4.2. Markov property of Brownian motion -- Chapter 5. Leafwise Lyapunov exponents -- Chapter 6. Splitting subbundles -- Chapter 7. Lyapunov forward filtrations -- 7.1. Oseledec type theorems -- 7.2. Fibered laminations and totally invariant sets -- 7.3. Cylinder laminations and end of the proof -- Chapter 8. Lyapunov backward filtrations -- 8.1. Extended sample-path spaces -- 8.2. Leafwise Lyapunov backward exponents and Oseledec backward type theorem -- Chapter 9. Proof of the main results -- 9.1. Canonical cocycles and specializations -- 9.2.-weakly harmonic measures and splitting invariant bundles -- 9.3. First Main Theorem and Ledrappier type characterization of Lyapunov spectrum -- 9.4. Second Main Theorem and its corollaries -- APPENDICES -- Appendix A. Measure theory for sample-path spaces -- A.1. Multifunctions and measurable selections -- A.2.-algebras: approximations and measurability -- A.3.-algebra \Ac on a leaf -- A.4. Holonomy maps -- A.5. Metrizability and separability of sample-path spaces -- A.6. The leafwise diagonal is Borel measurable -- A.7.-algebra \Ac on a lamination -- A.8. The cylinder laminations are Riemannian continuous-like -- A.9. The extended sample-path space is of full outer measure