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).
Highlights
Symmetry analysis is since a long time one of the key tools to attack deterministic nonlinear differential equations, both ordinary and partial [1–7]; what is nowadays known as Lie theory was created by Sophus Lie precisely to study nonlinear differential equations
If we determine a symmetry of the general form (7) for the Ito equation (6), we can seek a change of coordinates (x, t; w) → (y, t; z) mapping this symmetry into (20); as symmetries are preserved under diffeomorphisms [34], in the new coordinates the equation will be in the form (21), and promptly integrated
For F (x) = ax + b, with a = 0, the symmetry algebra G of the equation (70) has the structure G = X ⊕ Y, where X is the Abelian subalgebra spanned by the two real simple symmetries given by (90), and Y is the one-dimensional algebra of W-symmetries generated by the pseudo-scaling vector field (91); the subalgebra X is an Abelian ideal in G
Summary
Symmetry analysis is since a long time one of the key tools to attack deterministic nonlinear differential equations, both ordinary and partial [1–7]; what is nowadays known as Lie theory was created by Sophus Lie precisely to study nonlinear (ordinary) differential equations. From the point of view of Physics, there is a class of Dynamical Systems which has a special status, i.e. that corresponding to Newtonian Mechanics of point particles in a force field, possibly with dissipation: xi = vi vi = (1/m) F i(x, v, t) In this case one sets the equations as a second order system, xi = (1/m) F i(x, x , t) ,. The ideal system (1) should be replaced by the system of Ito stochastic differential equations [12–18] depending on independent Wiener processes wi(t), dxi = vi dt , dvi = (1/m) F i(x) − λ vi dt + σ dwi This is known as the Ornstein-Uhlenbeck process for a point particle of mass m in the force field F (x) [19, 20]. We will only work in the Ito framework, but a completely equivalent (up to the subtle points mentioned above) Stratonovich formulation would be possible
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