Elements of stochastic analysis for the case of Grassmann variables. I. Grassmann stochastic integrals and random processes

The present paper reviews some general aspects of the stochastic analysis performed by the author in the field of statistical physics, particularly concerning the order formation from unstable states. First, a brief review and some new results are given on the generalization of the Itô-type and Stratonovich-type stochastic integrals. Their physical meaning is also discussed form the viewpoint of symmetry. Secondly, Kubo's stochastic Liouville equation is presented from the viewpoint of separation of procedures, to give a simple derivation of the Fokker–Planck equation. Thirdly, the scaling theory of order formation from the unstable point is re-formulated by introducing here a new order parameter to characterize macroscopic order formation and to clarify the synergetic effect of the initial fluctuation, random noise and nonlinearity. Finally, some discussions are given, particularly concerning applications of the Hida calculus based on the Gelfand triplet space.

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Stochastic Analysis: Geometry of Random Processes

Oberwolfach Reports ◽

10.4171/owr/2017/26 ◽

2018 ◽

Vol 14 (2) ◽

pp. 1631-1679

Author(s):

Alice Guionnet ◽

Martin Hairer ◽

Grégory Miermont

Keyword(s):

Stochastic Analysis ◽

Random Processes

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Assouad Dimension of Random Processes

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091518000433 ◽

2018 ◽

Vol 62 (1) ◽

pp. 281-290 ◽

Cited By ~ 2

Author(s):

Douglas C. Howroyd ◽

Han Yu

Keyword(s):

Lévy Processes ◽

Random Processes ◽

Levy Processes ◽

Stochastic Integrals ◽

Assouad Dimension

AbstractIn this paper we study the Assouad dimension of graphs of certain Lévy processes and functions defined by stochastic integrals. We do this by introducing a convenient condition which guarantees a graph to have full Assouad dimension and then show that graphs of our studied processes satisfy this condition.

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Wick Calculus of Generalized Operators and its Applications to Quantum Stochastic Calculus

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025798000247 ◽

1998 ◽

Vol 01 (03) ◽

pp. 455-466 ◽

Cited By ~ 8

Author(s):

Zhiyuan Huang ◽

Shunlong Luo

Keyword(s):

Differential Equations ◽

White Noise ◽

Stochastic Analysis ◽

Stochastic Calculus ◽

Stochastic Integrals ◽

Algebra Structure ◽

Quantum Stochastic Calculus ◽

Fundamental Properties ◽

White Noise Calculus ◽

Wick Calculus

A nonlinear and stochastic analysis of free Bose field is established in the framework of white noise calculus. Wick algebra structure is introduced in the space of generalized operators generated by quantum white noise, some fundamental properties of the calculus based on the Wick algebra are investigated. As applications, quantum stochastic integrals and quantum stochastic differential equations are treated from the viewpoint of Wick calculus.

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