In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes -- classical, quantum or beyond -- that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to $all$ theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.
 Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach (Springer, New York, 2009).
 M. Liao, Applied Stochastic Processes (Chapman and Hall/CRC, Boca Raton, 2013).
 A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer, Berlin, 1933) [Foundations of the Theory of Probability (Chelsea, New York, 1956)].
 W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971).
 H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007).
 T. Tao, An Introduction to Measure Theory (American Mathematical Society, 2011).
 A. J. Leggett, Rep. Prog. Phys. 71, 022001 (2008).
 C. Emary, N. Lambert, and F. Nori, Rep. Prog. Phys. 77, 016001 (2014).
 N. Gilbert, Agent-Based Models (SAGE Publications Inc., Los Angeles, 2007).
 J. Pearl, Causality: Models, Reasoning and Inference (Cambridge University Press, Cambridge, U.K.; New York, 2009).
 F. Costa and S. Shrapnel, , 063032 (2016).
 O. Oreshkov and C. Giarmatzi, New J. Phys. 18, 093020 (2016).
 N. Wiener, A. Siegel, B. Rankin, and W. T. Martin, eds., Differential Space, Quantum Systems, and Prediction (The MIT Press, Cambridge (MA), 1966).
 Z. Ciesielski, Lectures on Brownian motion, heat conduction and potential theory (Aarhus Universitet, Mathematisk Institutt, 1966).
 R. Bhattacharya and E. C. Waymire, A Basic Course in Probability Theory (Springer, New York, NY, 2017).
 E. B. Davis, Quantum Theory of Open Systems (Academic Press Inc, London; New York, 1976).
 G. Chiribella, G. M. D’Ariano, and P. Perinotti, Phys. Rev. Lett. 101, 180501 (2008a).
 G. M. D'Ariano, G. Chiribella, and P. Perinotti, Quantum Theory from First Principles: An Informational Approach, 1st ed. (Cambridge University Press, Cambridge, United Kingdom ; New York, NY, 2017).
 G. Chiribella and D. Ebler, New J. Phys. 18, 093053 (2016).
 G. Chiribella, G. M. D'Ariano, and P. Perinotti, Phys. Rev. Lett. 101, 060401 (2008c).
 E. Kreyszig, Introductory Functional Analysis with Applications, 1st ed. (Wiley, New York, 1989).
 C. M. Caves, C. A. Fuchs, K. K. Manne, and J. M. Renes, Found. Phys. 34, 193 (2004).
 M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, New J. Phys. 17, 102001 (2015).
 F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Phys. Rev. Lett. 120, 040405 (2018b).
 E. Haapasalo, T. Heinosaari, and Y. Kuramochi, J. Phys. A 49, 33LT01 (2016).
 A Smirne, T Nitsche, D Egloff, S Barkhofen, S De, I Dhand, C Silberhorn, S F Huelga, and M B Plenio, "Experimental control of the degree of non-classicality via quantum coherence", Quantum Science and Technology 5 4, 04LT01 (2020).
 Pedro Figueroa-Romero, Kavan Modi, and Felix A. Pollock, "Equilibration on average in quantum processes with finite temporal resolution", Physical Review E 102 3, 032144 (2020).
 Mathias R. Jørgensen and Felix A. Pollock, "Discrete memory kernel for multitime correlations in non-Markovian quantum processes", Physical Review A 102 5, 052206 (2020).
 Simon Milz, Dario Egloff, Philip Taranto, Thomas Theurer, Martin B. Plenio, Andrea Smirne, and Susana F. Huelga, "When Is a Non-Markovian Quantum Process Classical?", Physical Review X 10 4, 041049 (2020).
 Chu Guo, Kavan Modi, and Dario Poletti, "Tensor-network-based machine learning of non-Markovian quantum processes", Physical Review A 102 6, 062414 (2020).
 G. A. L. White, C. D. Hill, F. A. Pollock, L. C. L. Hollenberg, and K. Modi, "Demonstration of non-Markovian process characterisation and control on a quantum processor", Nature Communications 11 1, 6301 (2020).
 Philipp Strasberg, "Operational approach to quantum stochastic thermodynamics", Physical Review E 100 2, 022127 (2019).
 Philip Taranto, Felix A. Pollock, Simon Milz, Marco Tomamichel, and Kavan Modi, "Quantum Markov Order", Physical Review Letters 122 14, 140401 (2019).
 Simon Milz, M. S. Kim, Felix A. Pollock, and Kavan Modi, "Completely Positive Divisibility Does Not Mean Markovianity", Physical Review Letters 123 4, 040401 (2019).
 Philipp Strasberg and Andreas Winter, "Stochastic thermodynamics with arbitrary interventions", Physical Review E 100 2, 022135 (2019).
 Mathias R. Jørgensen and Felix A. Pollock, "Exploiting the Causal Tensor Network Structure of Quantum Processes to Efficiently Simulate Non-Markovian Path Integrals", Physical Review Letters 123 24, 240602 (2019).
 Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Quantum Causal Models", arXiv:1906.10726.
 Jacques Pienaar, "Quantum causal models via QBism", arXiv:1806.00895.
 Philipp Strasberg and María García Díaz, "Classical quantum stochastic processes", Physical Review A 100 2, 022120 (2019).
 Philip Taranto, Simon Milz, Felix A. Pollock, and Kavan Modi, "Structure of quantum stochastic processes with finite Markov order", Physical Review A 99 4, 042108 (2019).
 Joshua Morris, Felix A. Pollock, and Kavan Modi, "Non-Markovian memory in IBMQX4", arXiv:1902.07980.
 A. Smirne, D. Egloff, M. G. Díaz, M. B. Plenio, and S. F. Huelga, "Coherence and non-classicality of quantum Markov processes", Quantum Science and Technology 4 1, 01LT01 (2019).
 Simon Milz, Felix A. Pollock, and Kavan Modi, "Reconstructing non-Markovian quantum dynamics with limited control", Physical Review A 98 1, 012108 (2018).
 Philipp Strasberg, "Repeated Interactions and Quantum Stochastic Thermodynamics at Strong Coupling", Physical Review Letters 123 18, 180604 (2019).
 Hong-Bin Chen, Ping-Yuan Lo, Clemens Gneiting, Joonwoo Bae, Yueh-Nan Chen, and Franco Nori, "Quantifying the nonclassicality of pure dephasing", Nature Communications 10, 3794 (2019).
 Philip Taranto, Felix A. Pollock, and Kavan Modi, "Memory Strength and Recoverability of Non-Markovian Quantum Stochastic Processes", arXiv:1907.12583.
 Graeme D. Berk, Andrew J. P. Garner, Benjamin Yadin, Kavan Modi, and Felix A. Pollock, "Resource theories of multi-time processes: A window into quantum non-Markovianity", arXiv:1907.07003.
 Fattah Sakuldee, Simon Milz, Felix A. Pollock, and Kavan Modi, "Non-Markovian quantum control as coherent stochastic trajectories", Journal of Physics A Mathematical General 51 41, 414014 (2018).
 Simon Milz, Felix A. Pollock, and Kavan Modi, "Reconstructing open quantum system dynamics with limited control", arXiv:1610.02152.
 Pedro Figueroa-Romero, Felix A. Pollock, and Kavan Modi, "Markovianization by design", arXiv:2004.07620.
 Philip Taranto, "Memory effects in quantum processes", International Journal of Quantum Information 18 2, 1941002-574 (2020).
 Jacques Pienaar, "Quantum causal models via quantum Bayesianism", Physical Review A 101 1, 012104 (2020).
 Philipp Strasberg, "Thermodynamics of Quantum Causal Models: An Inclusive, Hamiltonian Approach", Quantum 4, 240 (2020).
 Matheus Capela, Lucas C. Céleri, Kavan Modi, and Rafael Chaves, "Monogamy of temporal correlations: Witnessing non-Markovianity beyond data processing", Physical Review Research 2 1, 013350 (2020).
 Kavan Modi, "George Sudarshan and Quantum Dynamics", Open Systems and Information Dynamics 26 3, 1950013 (2019).
 Pedro Figueroa-Romero, Kavan Modi, and Felix A. Pollock, "Almost Markovian processes from closed dynamics", arXiv:1802.10344.
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