We study the invariance of stochastic differential equations under random diffeomorphisms and establish the determining equations for random Lie-point symmetries of stochastic differential equations, both in Ito and in Stratonovich forms. We also discuss relations with previous results in the literature.

1.
D.
Freedman
,
Brownian Motion and Diffusion
(
Springer
,
1983
).
2.
P. J.
Olver
,
Application of Lie Groups to Differential Equations
(
Springer
,
1986
).
3.
H.
Stephani
,
Differential Equations: Their Solution Using Symmetries
(
Cambridge University Press
,
1989
);
D. V.
Alexseevsky
,
A. M.
Vinogradov
, and
V. V.
Lychagin
,
Basic Ideas and Concepts of Differential Geometry
(
Springer
,
1991
);
G.
Gaeta
,
Nonlinear Symmetries and Nonlinear Equations
(
Kluwer
,
1994
);
P. J.
Olver
,
Equivalence, Invariants and Symmetry
(
Cambridge University Press
,
1995
);
I. S.
Krasil’schik
and
A. M.
Vinogradov
,
Symmetries and Conservation Laws for Differential Equations of Mathematical Physics
(
A.M.S.
,
1999
).
4.
G.
Cicogna
and
G.
Gaeta
,
Symmetry and Perturbation Theory in Nonlinear Dynamics
(
Springer
,
1999
).
5.
H. P.
McKean
,
Stochastic Integrals
(
A.M.S.
,
1969
);
N.
Ikeda
and
S.
Watanabe
,
Stochastic Differential Equations and Diffusion Processes
(
North Holland
,
1981
);
F.
Guerra
, “
Structural aspects of stochastic mechanics and stochastic field theory
,”
Phys. Rep.
77
,
263
312
(
1981
);
N. G.
van Kampen
,
Stochastic Processes in Physics and Chemistry
(
North Holland
,
1992
);
L. C.
Evans
,
An Introduction to Stochastic Differential Equations
(
A.M.S.
,
2013
).
6.
B.
Oksendal
,
Stochastic Differential Equations
, 4th ed. (
Springer
,
1985
).
7.
L.
Arnold
,
Random Dynamical Systems
(
Springer
,
1988
).
8.
D. W.
Stroock
,
Markov Processes from K. Ito’s Perspective
(
Princeton UP
,
2003
).
9.
Y.
Kossmann-Schwarzbach
, “
Les théorèmes de Noether. Invariance et lois de conservation au XXe siècle, editions de l’ecole polytechnique 2004
,” in
Noether Theorems: Invariance and Conservation Laws in the XXth Century
(
Springer
,
2009
).
10.
K.
Yasue
, “
Stochastic calculus of variations
,”
Lett. Math. Phys.
4
,
357
360
(
1980
);
K.
Yasue
,
J. Funct. Anal.
41
,
327
340
(
1981
);
J. C.
Zambrini
, “
Stochastic dynamics: A review of stochastic calculus of variations
,”
Int. J. Theor. Phys.
24
,
277
327
(
1985
);
T.
Misawa
, “
Noether’s theorem in symmetric stochastic calculus of variations
,”
J. Math. Phys.
29
,
2178
2180
(
1988
);
M.
Thieullen
and
J. C.
Zambrini
, “
Probability and quantum symmetries. I. The theorem of Noether in Schrodinger’s Euclidean quantum mechanics
,”
Ann. I.H.P.: Phys. Théor.
67
,
297
338
(
1997
);
S.
Albeverio
,
J.
Rezende
, and
J. C.
Zambrini
, “
Probability and quantum symmetries. II. The theorem of Noether in quantum mechanics
,”
J. Math. Phys.
47
,
062107
(
2006
).
11.
L.
Arnold
and
P.
Imkeller
, “
Normal forms for stochastic differential equations
,”
Prob. Theory Relat Fields
110
,
559
588
(
1998
).
12.
T.
Misawa
, “
New conserved quantities derived from symmetry for stochastic dynamical systems
,”
J. Phys. A
27
,
L777
L782
(
1994
).
13.
T.
Misawa
, “
Conserved quantities and symmetry for stochastic dynamical systems
,”
Phys. Lett. A
195
,
185
189
(
1994
);
T.
Misawa
, “
A method for deriving conserved quantities from the symmetry of stochastic dynamical systems
,”
Nuovo Cimento B
113
,
421
428
(
1998
);
T.
Misawa
, “
Conserved quantities and symmetries related to stochastic dynamical systems
,”
Ann. Inst. Stat. Math.
51
,
779
802
(
1999
).
14.
S.
Albeverio
and
S. M.
Fei
, “
A remark on symmetry of stochastic dynamical systems and their conserved quantities
,”
J. Phys. A
28
,
6363
6371
(
1995
).
15.
Y. N.
Grigoriev
,
N. H.
Ibragimov
,
S. V.
Meleshko
, and
V. F.
Kovalev
,
Symmetries of Integro-Differential Equations with Applications in Mechanics and Plasma Physics
(
Springer
,
2010
).
16.
C.
Wafo Soh
and
F. M.
Mahomed
, “
Integration of stochastic ordinary differential equations from a symmetry standpoint
,”
J. Phys. A
34
,
177
192
(
2001
).
17.
G.
Unal
, “
Symmetries of Ito and Stratonovich dynamical systems and their conserved quantities
,”
Nonlinear Dyn.
32
,
417
426
(
2003
);
B.
Srihirun
,
S. V.
Meleshko
, and
E.
Schulz
, “
On the definition of an admitted Lie group for stochastic differential equations with multi-Brownian motion
,”
J. Phys. A
39
,
13951
13966
(
2006
);
S.
Mei
and
F. X.
Mei
, “
Conserved quantities and symmetries related to stochastic Hamiltonian systems
,”
Chin. Phys.
16
,
3161
3167
(
2007
);
E.
Fredericks
and
F. M.
Mahomed
, “
Symmetries of first-order stochastic ordinary differential equations revisited
,”
Math. Methods Appl. Sci.
30
,
2013
2025
(
2007
);
E.
Fredericks
and
F. M.
Mahomed
, “
A formal approach for handling Lie point symmetries of scalar first-order Ito stochastic ordinary differential equations
,”
J. Nonlinear. Math. Phys.
15-S1
,
44
59
(
2008
);
S. V.
Meleshko
and
E.
Schulz
, “
A new set of admitted transformations for autonomous stochastic ordinary differential equations
,”
J. Nonlinear. Math. Phys.
17
,
179
196
(
2010
);
R.
Kozlov
, “
The group classification of a scalar stochastic differential equation
,”
J. Phys. A
43
,
055202
(
2010
);
R.
Kozlov
, “
On maximal Lie point symmetry groups admitted by scalar stochastic differential equations
,”
J. Phys. A
44
,
205202
(
2011
);
R.
Kozlov
, “
On symmetries of the Fokker-Planck equation
,”
J. Eng. Math.
82
,
39
57
(
2013
).
18.
R.
Kozlov
, “
Symmetry of systems of stochastic differential equations with diffusion matrices of full rank
,”
J. Phys. A
43
,
245201
(
2010
).
19.
G.
Gaeta
and
N.
Rodríguez-Quintero
, “
Lie-point symmetries and stochastic differential equations
,”
J. Phys. A
32
,
8485
8505
(
1999
).
20.
G.
Gaeta
, “
Lie-point symmetries and stochastic differential equations. II
,”
J. Phys. A
33
,
4883
4902
(
2000
).
21.
G.
Gaeta
, “
Symmetry of deterministic versus stochastic non-variational differential equations,
Phys. Rep.
(to be published).
22.
V. I.
Arnold
,
Geometrical Methods in the Theory of Ordinary Differential Equations
(
Springer
,
1983
).
23.
C.
Elphick
,
E.
Tirapegui
,
M. E.
Brachet
,
P.
Coullet
, and
G.
Iooss
, “
A simple global characterization for normal forms of singular vector fields
,”
Phys. D
29
,
95
127
(
1987
);
C.
Elphick
,
E.
Tirapegui
,
M. E.
Brachet
,
P.
Coullet
, and
G.
Iooss
, “
Addendum
,”
Phys. D
32
,
488
(
1988
);
G.
Iooss
and
M.
Adelmeyer
,
Topics in Bifurcation Theory and Applications
(
World Scientific
,
1992
).
24.
G.
Gaeta
, “
Poincaré normal and renormalized forms
,”
Acta Appl. Math.
70
,
113
131
(
2002
);
S.
Walcher
, “
On differential equations in normal form
,”
Math. Ann.
291
,
293
314
(
1991
);
S.
Walcher
, “
On transformation into normal form
,”
J. Math. Anal. Appl.
180
,
617
632
(
1993
).
25.
F.
Finkel
, “
Symmetries of the Fokker-Planck equation with a constant diffusion matrix in 2 + 1 dimensions
,”
J. Phys. A
32
,
2671
2684
(
1999
).
26.
G.
Cicogna
and
D.
Vitali
, “
Generalised symmetries of Fokker-Planck-type equations
,”
J. Phys. A
22
,
L453
L456
(
1989
);
G.
Cicogna
and
D.
Vitali
, “
Classification of the extended symmetries of Fokker-Planck equations
,”
J. Phys. A
23
,
L85
L88
(
1990
).
27.
W. M.
Shtelen
and
V. I.
Stogny
, “
Symmetry properties of one-and two-dimensional Fokker-Planck equations
,”
J. Phys. A
22
,
L539
L543
(
1989
);
S.
Spichak
and
V.
Stognii
, “
Symmetry classification and exact solutions of the one-dimensional Fokker-Planck equation with arbitrary coefficients of drift and diffusion
,”
J. Phys. A
32
,
8341
8353
(
1999
).
28.

As opposed to the “indirect” action due to the modification of the Wiener process induced by the action on the time variable; see below.

29.

Actually, in order to end up with an equation possibly of the same type (not to say about it being exactly the same equation as the original one), the transformed processes w should be only a function of the (transformed) time t, i.e., w=w(t); this amounts to requiring that the transformation of time does not depend on the space coordinates xi. On the other hand, it could depend on the Wiener processes themselves.

You do not currently have access to this content.