Symmetry of the isotropic Ornstein-Uhlenbeck process in a force field

Abstract

We classify simple symmetries for an Ornstein-Uhlenbeck process, describing a particle in an external force field $f(x)$. It turns out that for sufficiently regular (in a sense to be defined) forces there are nontrivial symmetries only if $f(x)$ is at most linear. We fully discuss the isotropic case, while for the non-isotropic we only deal with a generic situation (defined in detail in the text).

Symmetry classification of scalar autonomous Ito stochastic differential equations with simple noise

“…This point is discussed in detail e.g. in [29] (see Remark 12 therein), to which we refer for further detail. In the case of deterministic standard symmetries we are guaranteed that the new random variable y(t) obtained applying the Kozlov transformation obeys an Ito equation, so that this is a substantial reason to study this standard symmetry case in more detail.…”

“…The general theory has been developed in a number of publications; see e.g. [17][18][19][20][21][22][23][24][25][26][27][28][29][30] or the review paper [31]; we will recall some basic facts in Section 2 below, referring to the literature (see e.g. the papers quoted above) for details.…”

Integrable Ito equations with multiple noises

“…Moreover, the vector field ( 27) is just Y = ϕ(x, t; w, z) ∂ x . When tackling the system ( 29), ( 30), (31), our general approach will consist in first solving the equations which are independent of f , i.e. ( 30) and (31); this determines a general class of functions ϕ.…”

“…When tackling the system ( 29), ( 30), (31), our general approach will consist in first solving the equations which are independent of f , i.e. ( 30) and (31); this determines a general class of functions ϕ. We will then turn our attention to (29), where we have to determine both f and the exact function ϕ within the class identified before; in general, it will be convenient to consider differential consequences of (29) providing some separation of variables (see below for details).…”

“…In this sense we speak of reduction, or also partial integrability, as in the case of deterministic equations [10][11][12][13][14][15]. 2 It turns out that, similarly to what happens in the deterministic case [10][11][12][13][14][15], integrability of Ito equations is strongly related to their symmetry properties [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. In fact, once a symmetry of (1) of a suitable form (see below for details) is known, there is a standard way of constructing the integrating change of variables, known as the Kozlov substitution (so named after Roman Kozlov, who first devised it [19][20][21][22][23][24]).…”