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In the literature of symmetry groups, Sophus Lie define the symmetry of a pde or an ode by a vector field defined in the tangent space of the submanifold (defined by solutions of the pde or ode) and search when this pde or ode is invariant under the action of the one parameter group of transformations generated by the vector field of the submanifold .

In a stochastic differential equations we have a random term in it, so how we can define a Lie symmetry of this kind of equations (if possible of course)?

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  • $\begingroup$ I don't know if it is helpful, but there is a generalization of the concept of flow for SDE on manifolds (see here) So I'm expecting the symmetries on the case of SDE (if they can be defined) to be based of second order operators, more than vector fields. $\endgroup$
    – Marco
    Feb 18 at 1:45

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In "On Lie-point symmetries for Ito stochastic differential equations", they explain some of the difficulties of doing that and some approaches eg. by Kozlov in "Symmetry of systems of stochastic differential equations with diffusion matrices of full rank".

For example, Itô equations don't satisfy chain rule and so we lose the invariance in coordinates, so one might prefer using the Stratonovich formulation that satisfies chain rule.

Kozlov studied $dx = f(x, t) dt + \sigma(x, t) dw$ that admit a simple Lie-point symmetry with generator $X = \xi(x, t) \partial_{X}$, in order to reduce them to linear equations $dx = \hat{f}( t) dt + \hat{\sigma}( t) dw$.

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