Symmetries can be used to integrate scalar Ito equation – or reduce systems of such equations – by the Kozlov substitution, i.e. passing to symmetry adapted coordinates. While the theory is well established for so called deterministic standard symmetries (the class originally studied by Kozlov), some points need clarification for so called random standard symmetries and W-symmetries. This paper is devoted to such clarification; in particular we note that the theory naturally calls, for these classes of symmetries, to also consider generalized Ito equations; and that while Kozlov theory is extended substantially unharmed for random standard symmetries, W-symmetries should be handled with great care, and cannot be used towards integration of stochastic equations, albeit they have different uses.

1.
P. J.
Olver
,
Application of Lie Groups to Differential Equations
(
Springer
,
1986
).
2.
H.
Stephani
,
Differential Equations. Their Solution Using Symmetries
(
Cambridge University Press
,
1989
).
3.
D. V.
Alexseevsky
,
A. M.
Vinogradov
, and
V. V.
Lychagin
,
Basic Ideas and Concepts of Differential Geometry
(
Springer
,
1991
).
4.
P. J.
Olver
,
Equivalence, Invariants and Symmetry
(
Cambridge University Press
,
1995
).
5.
I. S.
Krasil’schik
and
A. M.
Vinogradov
,
Symmetries and Conservation Laws for Differential Equations of Mathematical Physics
(
AMS
,
1999
).
6.
G.
Cicogna
and
G.
Gaeta
,
Symmetry and Perturbation Theory in Nonlinear Dynamics
(
Springer
,
1999
).
7.
N.
Ikeda
and
S.
Watanabe
,
Stochastic Differential Equations and Diffusion Processes
(
North Holland, Amsterdam
,
The Netherlands
,
1981
).
8.
D.
Freedman
,
Brownian Motion and Diffusion
(
Springer
,
1983
).
9.
L.
Arnold
,
Random Dynamical Systems
(
Springer
,
1988
).
10.
N. G.
van Kampen
,
Stochastic Processes in Physics and Chemistry
(
North Holland, Amsterdam
,
The Netherlands
,
1992
), p.
2003
.
11.
D. W.
Stroock
,
Markov Processes from K. Ito’s Perspective
(
Princeton University Press
,
Princeton
,
2003
).
12.
L. C.
Evans
,
An Introduction to Stochastic Differential Equations
(
AMS
,
2013
).
13.
B.
Oksendal
,
Stochastic Differential Equations
, 6th ed. (
Springer
,
2013
).
14.
A.
Amir
,
Thinking Probabilistically
(
Cambridge University Press
,
Cambridge
,
2021
).
15.
R.
Kozlov
, “
Symmetries of systems of stochastic differential equations with diffusion matrices of full rank
,”
J. Phys. A: Math. Theor.
43
,
245201
(
2010
).
16.
R.
Kozlov
, “
The group classification of a scalar stochastic differential equation
,”
J. Phys. A: Math. Theor.
43
,
055202
(
2010
).
17.
R.
Kozlov
, “
On maximal Lie point symmetry groups admitted by scalar stochastic differential equations
,”
J. Phys. A: Math. Theor.
44
,
205202
(
2011
).
18.
R.
Kozlov
, “
Symmetries of Ito stochastic differential equations and their applications
,” in
Nonlinear Systems and Their Remarkable Mathematical Structures
, edited by
N.
Euler
(
CRC Press
,
2018
), pp.
408
436
.
19.
G.
Gaeta
and
F.
Spadaro
, “
Random Lie-point symmetries of stochastic differential equations
,”
J. Math. Phys.
58
,
053503
(
2017
);
Erratum
,
J. Math. Phys.
58
, 129901
129901
(
2017
).
20.
R.
Kozlov
, “
Random Lie symmetries of Ito stochastic differential equations
,”
J. Phys. A: Math. Theor.
51
,
305203
(
2018
).
21.
R.
Kozlov
, “
Lie point symmetries of Stratonovich stochastic differential equations
,”
J. Phys. A: Math. Theor.
51
,
505201
(
2018
).
22.
G.
Gaeta
, “
W-symmetries of Ito stochastic differential equations
,”
J. Math. Phys.
60
,
053501
(
2019
).
23.
F. C.
De Vecchi
,
P.
Morando
, and
S.
Ugolini
, “
Symmetries of stochastic differential equations using Girsanov transformations
,”
J. Phys. A: Math. Theor.
53
,
135204
(
2020
).
24.
F. C.
De Vecchi
,
P.
Morando
, and
S.
Ugolini
, “
Reduction and reconstruction of SDEs via Girsanov and quasi Doob symmetries
,”
J. Phys. A: Math. Theor.
54
,
185203
(
2021
).
25.
N.
Privault
and
J. C.
Zambrini
, “
Stochastic deformation of integrable dynamical systems and random time symmetry
,”
J. Math. Phys.
51
,
082104
(
2010
).
26.
R.
Chetrite
and
H.
Touchette
, “
Nonequilibrium Markov processes conditioned on large deviations
,”
Ann. Henri Poincare
16
,
2005
2057
(
2015
).
27.
F. C.
De Vecchi
,
P.
Morando
, and
S.
Ugolini
, “
Symmetries of stochastic differential equations: A geometric approach
,”
J. Math. Phys.
57
,
063504
(
2016
).
28.
F. C.
De Vecchi
,
P.
Morando
, and
S.
Ugolini
, “
Reduction and reconstruction of stochastic differential equations via symmetries
,”
J. Math. Phys.
57
,
123508
(
2016
).
29.
F. C.
De Vecchi
,
P.
Morando
, and
S.
Ugolini
, “
A note on symmetries of diffusions within a martingale problem approach
,”
Stoch. Dyn.
19
,
1950011
(
2019
).
30.
F. C.
De Vecchi
,
P.
Morando
, and
S.
Ugolini
, “
Integration by parts formulas and Lie’s symmetries of SDEs
,” arXiv:2307.05089 (
2023
).
31.
S.
Albeverio
and
F. C.
De Vecchi
, “
Some recent developments on Lie symmetry analysis of stochastic differential equations
,” in
Geometry and Invariance in Stochastic Dynamics
, edited by
S.
Ugolini
et al
(
Springer
,
2021
).
32.
G.
Gaeta
and
N. R.
Quintero
, “
Lie-point symmetries and stochastic differential equations
,”
J. Phys. A: Math. Gen.
32
,
8485
8505
(
1999
).
33.
G.
Gaeta
, “
Lie-point symmetries and stochastic differential equations: II
,”
J. Phys. A: Math. Gen.
33
,
4883
4902
(
2000
).
34.
G.
Gaeta
, “
Symmetry of stochastic non-variational differential equations
,”
Phys. Rep.
686
,
1
62
(
2017
);
Erratum
,
Phys. Rep.
713
, 18
129901
(
2017
).
35.
G.
Gaeta
and
C.
Lunini
, “
On Lie-point symmetries for Ito stochastic differential equations
,”
J. Nonlinear Math. Phys.
24
(
S1
),
90
102
(
2017
).
36.
G.
Gaeta
and
C.
Lunini
, “
Symmetry and integrability for stochastic differential equations
,”
J. Nonlinear Math. Phys.
25
,
262
289
(
2021
).
37.
V. I.
Arnold
,
Ordnary Differential Equations
(
Springer
,
1992
).
38.
V. I.
Arnold
,
Geometrical Methods in the Theory of Ordinary Differential Equations
(
Springer
,
1983
).
39.
G.
Gaeta
and
M. A.
Rodríguez
, “
Symmetry classification of scalar autonomous Ito stochastic differential equations with simple noise
,”
Open Commun. Nonlinear Math. Phys.
2
,
53
101
(
2022
).
40.
G.
Gaeta
and
M. A.
Rodríguez
, “
Integrable Ito equations with multiple noises
,”
Open Commun. Nonlinear Math. Phys.
2
,
122
153
(
2022
).
41.
G.
Gaeta
and
M. A.
Rodríguez
, “
Integrable Ito equations and properties of the associated Fokker-Planck equations
,”
Open Commun. Nonlinear Math. Phys.
3
,
67
90
(
2023
).
42.
G.
Gaeta
,
R.
Kozlov
, and
F.
Spadaro
, “
Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations
,”
Math. Eng.
4
(
5
),
1
52
(
2022
).
43.
T.
Marchand
,
M.
Ozawa
,
G.
Biroli
, and
S.
Mallat
, “
Wavelet conditional renormalization group
,” arXiv:2207.04941.
44.
T.
Arnoulx de Pirey
,
L. F.
Cugliandolo
,
V.
Lecomte
, and
F.
van Wijland
, “
Discretized and covariant path integrals for stochastic processes
,”
Adv. Phys.
(
published online
,
2023
).
45.
K.
Yasue
, “
Stochastic calculus of variations
,”
Lett. Math. Phys.
4
,
357
360
(
1980
);
K.
Yasue
Stochastic calculus of variations
,”
J. Funct. Anal.
41
,
327
340
(
1981
).
46.
J. C.
Zambrini
, “
Stochastic dynamics: A review of stochastic calculus of variations
,”
Int. J. Theor. Phys.
24
,
277
327
(
1985
).
47.
J. C.
Zambrini
, “
Variational processes and stochastic versions of mechanics
,”
J. Math. Phys.
27
,
2307
2330
(
1986
).
48.
B.
Coutinho dos Santos
and
C.
Tsallis
, “
Time evolution towards q-Gaussian stationary states through unified Ito-Stratonovich stochastic equation
,”
Phys. Rev. E
82
,
061119
(
2010
).
49.
Z. G.
Arenas
,
D. G.
Barci
, and
C.
Tsallis
, “
Nonlinear inhomogeneous Fokker-Planck equation within a generalized Stratonovich prescription
,”
Phys. Rev. E
90
,
032118
(
2014
).
50.

One could consider time reparametrizations 32–34; but this will not change substantially the situation, and produce some complication in formulas. We note in passing that a rescaling of time might be relevant when considering W-symmetries, which produce a rescaling of the noise term.

51.

Actually,one could argue that it makes no sense speaking of symmetries if these are not preserved under a change of variables. So this question is essential to the very existence of symmetries for Ito stochastic equations.

52.

The Ito-Stratonovich correspondence does actually present some subtle points; for these the reader is referred to 11.

53.

This could be circumvented passing to consider the Stratonovich equation associated to the given Ito one. One should again remember that the correspondence between Ito and Stratonovich equations involve some subtleties 11, and this is one of the reasons why we prefer to deal with the Ito equation itself. We also recall that correspondence between (determining equations, and hence) symmetries of an Ito equation and the associated Stratonovich one has been proved in full generality for standard symmetries in 35. That proof does not apply to W-symmetries (which had not been introduced in the literature at the time); see in this respect the discussion below in Sec. VII.

54.

This is not immediately obvious, given that vector fields transform under the chain rule and equations transform under the Ito rule; but it can be proven, see 35 and 36. Alternatively, it becomes obvious considering the Stratonovich representation of the SDE; but one should then prove that symmetries of Ito and Stratonovich equations coincide.

55.

Actually, (5.7) defines Ψ up to a constant of integration, i.e. an arbitrary function of w and t; we are taking this to be zero.

56.

Since now the (x, t) and the w variables transform independently of each other under the action of X. Split W-symmetries were already introduced and considered in 22; see there for some of their properties. A discussion of the use of (split) W-symmetries in the integration of proper Ito equations is also given in  Appendix B to 40.

57.

It is conceivable that different way of integrating SDEs exist, in the same way as for deterministic equations (think of the Arnold-Liouville versus the Lax approach). In this sense, it can be hoped that (generalized) Ito equations which are invariant under a scale transformation can be handled more effectively than generic equations, e.g. using renormalization group techniques to look at the equilibrium distribution (see e.g. 43), or a path integral approach (see e.g. 44).

58.

The formulas above are slightly simplified by taking a(t) = b(t) = 1; this simplified setting does not change the relevant facts.

59.

One cannot exclude they can be used for integration of the equation via a different mechanism, see also the footnote to Remark 9. I am not aware of any result in this direction.

60.

We note in passing that this is also, usually, a preliminary step in the proof of the straightening (or “flow-box”) theorem 37.

61.

We are not using the notation σdw because now we are dealing with real numbers, not with stochastic differential equations.

62.

It is a trivial remark – but it may be worth recalling, to avoid any possible confusion – that whenever we change variables, the Ito Laplacian Δ is defined using the noise coefficients for the equations of the new variables.

You do not currently have access to this content.