## Journals

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CPAA

We study evolution by horizontal mean curvature flow in sub-
Riemannian geometries by using stochastic approach to prove the existence of
a generalized evolution in these spaces. In particular we show that the value
function of suitable family of stochastic control problems solves in the viscosity
sense the level set equation for the evolution by horizontal mean curvature flow.

DCDS

Some quantum fluid models are written as the Lagrangian flow of mass distributions
and their geometric properties are explored. The first model includes magnetic effects
and leads, via the Madelung transform,
to the electromagnetic Schrödinger equation in the Madelung representation.
It is shown that the Madelung transform is a symplectic map between Hamiltonian
systems. The second model is obtained from the Euler-Lagrange equations with
friction induced from a quadratic dissipative potential. This model corresponds to the
quantum Navier-Stokes equations with density-dependent viscosity. The fact that
this model possesses two different energy-dissipation identities is explained
by the definition of the Noether currents.

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