## Conserved quantities and symmetries related to

stochastic dynamical systems

### Tetsuya Misawa

*Faculty of Economics, Nagoya City University, Mizuho-ku, Nagoya 467-8501, Japan*
(Received May 26, 1997; revised March 30, 1998)

**Abstract.**
The present article focuses on the
three topics related to the notions of "conserved
quantities" and "symmetries" in stochastic dynamical systems
described by stochastic differential equations of
Stratonovich type. The first topic is concerned with the
relation between conserved quantities and symmetries in
stochastic Hamilton dynamical systems, which is established
in a way analogous to that in the deterministic Hamilton
dynamical theory. In contrast with this, the second topic is
devoted to investigate the procedures to derive conserved
quantities from symmetries of stochastic dynamical systems
without using either the Lagrangian or Hamiltonian structure.
The results in these topics indicate that the notion of
symmetries is useful for finding conserved quantities in
various stochastic dynamical systems. As a further important
application of symmetries, the third topic treats the
similarity method to stochastic dynamical systems. That is,
it is shown that the order of a stochastic system can be
reduced, if the system admits symmetries. In each topic, some
illustrative examples for stochastic dynamical systems and
their conserved quantities and symmetries are given.

*Key words and phrases*:
Stochastic dynamical
systems, conserved quantities, symmetries, similarity method.

**Source**
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