# How we define the Lie symmetry of a stochastic differential equation?

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)?

• 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. Feb 18 at 1:45

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 $$dx = \hat{f}( t) dt + \hat{\sigma}( t) dw$$.