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To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the quantum random walk.

  For a physics perspective, see Quantum random walks - an introductory overview, but you might prefer the more math-oriented exposition of Martin boundary theory of some quantum random walks and On algebraic and quantum random walks.

We give a concise prescription of the concept of a quantum random walk (QRW), using the example of QRW on integers as paradigm. It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and the entropy increase without majorization effect of its probability distributions. We conclude with a group theoretical scheme of classification of various known QRW's.

Concerning a mathematical description of the physical phenomenon of Brownian motion: We are then concerned with the effect of an environment having a large (infinite) number of degrees of freedom on the dynamics of a particle with a few degrees of freedom. So we are seeking a quantum theory of friction, diffusion, and thermalization. The seminal paper here is the path integral theory of Caldeira and Leggett. The literature is very extensive, an older but still relevant review is Quantum Brownian Motion: The Functional Integral Approach.

The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.

A. To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the quantum random walk. For a physics perspective, see Quantum random walks - an introductory overview, but you might prefer the more math-oriented exposition of Martin boundary theory of some quantum random walks and On algebraic and quantum random walks.

We give a concise prescription of the concept of a quantum random walk (QRW), using the example of QRW on integers as paradigm. It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and the entropy increase without majorization effect of its probability distributions. We conclude with a group theoretical scheme of classification of various known QRW's.

B. Concerning the relation between Wiener processes and quantum Brownian motion: A quantum version of the wavelet expansion of a Wiener process has been developed in A Levy-Cielsielski expansion for quantum Brownian motion and the construction of quantum Brownian bridges.

Classical Brownian motion has a delightful wavelet expansion obtained by combining the Schauder system with a sequence of i.i.d. standard normals. Our main technical result is to obtain a quantum version of this expansion and so construct quantum Brownian motion in Fock space. Consequently, only the discrete skeleton provided by a "quantum random walk" is required to generate the continuous time process. Our result seems easier to establish than the classical one of Lévy-Cielsielski as we don’t require logarithmic growth estimates on the squares of i.i.d. Gaussians, thanks to the nice action of annihilation operators on exponential vectors.

C. Concerning a mathematical description of the physical phenomenon of Brownian motion: We are then concerned with the effect of an environment having a large (infinite) number of degrees of freedom on the dynamics of a particle with a few degrees of freedom. So we are seeking a quantum theory of friction, diffusion, and thermalization. The seminal paper here is the path integral theory of Caldeira and Leggett. The literature is very extensive, an older but still relevant review is Quantum Brownian Motion: The Functional Integral Approach.

The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.

To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the quantum random walk.

For a physics perspective, see Quantum random walks - an introductory overview, but you might prefer the more math-oriented exposition of Martin boundary theory of some quantum random walks and On algebraic and quantum random walks.

We give a concise prescription of the concept of a quantum random walk (QRW), using the example of QRW on integers as paradigm. It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and the entropy increase without majorization effect of its probability distributions. We conclude with a group theoretical scheme of classification of various known QRW's.

Concerning a mathematical description of the physical phenomenon of Brownian motion: We are then concerned with the effect of an environment having a large (infinite) number of degrees of freedom on the dynamics of a particle with a few degrees of freedom. So we are seeking a quantum theory of friction, diffusion, and thermalization. The seminal paper here is the path integral theory of Caldeira and Leggett. The literature is very extensive, an older but still relevant review is Quantum Brownian Motion: The Functional Integral Approach.

The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.

To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the quantum random walk.

For a physics perspective, see Quantum random walks - an introductory overview, but you might prefer the more math-oriented exposition of Martin boundary theory of some quantum random walks and On algebraic and quantum random walks.

We give a concise prescription of the concept of a quantum random walk (QRW), using the example of QRW on integers as paradigm. It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and the entropy increase without majorization effect of its probability distributions. We conclude with a group theoretical scheme of classification of various known QRW's.

Concerning a mathematical description of the physical phenomenon of Brownian motion: We are then concerned with the effect of an environment having a large (infinite) number of degrees of freedom on the dynamics of a particle with a few degrees of freedom. So we are seeking a quantum theory of friction, diffusion, and thermalization. The seminal paper here is the path integral theory of Caldeira and Leggett. The literature is very extensive, an older but still relevant review is Quantum Brownian Motion: The Functional Integral Approach.

The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.

To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the quantum random walk.

For a physics perspective, see Quantum random walks - an introductory overview, but you might prefer the more math-oriented exposition of Martin boundary theory of some quantum random walks and On algebraic and quantum random walks.

We give a concise prescription of the concept of a quantum random walk (QRW), using the example of QRW on integers as paradigm. It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and the entropy increase without majorization effect of its probability distributions. We conclude with a group theoretical scheme of classification of various known QRW's.

I understand the question is more oriented towards the physical phenomenon of Brownian motion, which I would describe as the effect of an environment with a large (infinite) number of degrees of freedom on the dynamics of a particle with a few degrees of freedom. We would then be seeking a quantum theory of friction, diffusion, and thermalization. The seminal paper here is the path integral theory of Caldeira and Leggett. The literature is very extensive, an older but still relevant review is Quantum Brownian Motion: The Functional Integral Approach.

The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.

To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the quantum random walk.

For a physics perspective, see Quantum random walks - an introductory overview, but you might prefer the more math-oriented exposition of Martin boundary theory of some quantum random walks and On algebraic and quantum random walks.

We give a concise prescription of the concept of a quantum random walk (QRW), using the example of QRW on integers as paradigm. It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and the entropy increase without majorization effect of its probability distributions. We conclude with a group theoretical scheme of classification of various known QRW's.

Concerning a mathematical description of the physical phenomenon of Brownian motion: We are then concerned with the effect of an environment having a large (infinite) number of degrees of freedom on the dynamics of a particle with a few degrees of freedom. So we are seeking a quantum theory of friction, diffusion, and thermalization. The seminal paper here is the path integral theory of Caldeira and Leggett. The literature is very extensive, an older but still relevant review is Quantum Brownian Motion: The Functional Integral Approach.

The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.