Does anyone know how to produce a stochastic flow that preserves the gaussian measure on R^d? I haven't found any literature about this. Of course, for the lebesgue measure we can say something with restriction on the divergence of driving vector fields.
You may find necessary and sufficient conditions (not only for Gaussian measure) in
1) Kunita H. Stochastic flows and stochastic differential equations. Theorem 4.3.2.
or
2) T.E. Harris: Brownian motions on the diffeomorphisms of the
plane, Ann. Prob. 9 (1981), 232–254.
Hi.
There are such papers. Some of them are technically demanding.
1) Da Prato, Debussche: Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196 (2002)
2) Debussche, A. The 2D-Navier-Stokes equations perturbed by a delta correlated noise. Probabilistic methods in fluids, 115–129, World Sci. Publ., River Edge, NJ, 2003
3) Albeverio, S.; Ferrario, B. Uniqueness results for the generators of the two-dimensional Euler and Navier-Stokes flows. The case of Gaussian invariant measures. J. Funct. Anal. 193 (2002)
Cheers, Ben
You may find necessary and sufficient conditions (not only for Gaussian measure) in
1) Kunita H. Stochastic flows and stochastic differential equations. Theorem 4.3.2.
or
2) T.E. Harris: Brownian motions on the diffeomorphisms of the
plane, Ann. Prob. 9 (1981), 232–254.
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