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On continuity equations in infinite dimensions with non-Gaussian reference measure. (English) Zbl 1317.60083
The authors study infinite-dimensional equations of the form \(\dot{\mu}+ \text{div}(\mu\cdot b) = 0\), \(\mu_0=\zeta\), where \(\mu=\mu_t(dx)\) is a curve of probability measures on \(\mathbb R^{\infty}\), equipped with the product \(\sigma\)-algebra induced by the Borel \(\sigma\)-algebra on \(\mathbb R\) and \(b\) is the function from \(\mathbb R^{\infty}\) to \(\mathbb R^{\infty}\). The approach to solve this equation is to choose a reference measure \(\nu\) and search for solutions which are of the form \(\mu_t(dx)=\rho(t,x)\cdot\nu(dx)\), where \(\rho\) is the solution of the equation \(\dot{\rho} +\text{div}_{\nu}(\rho\cdot b) = 0\), \(\rho(0, x) = \rho_0\), with \(\zeta=\rho_0\cdot \nu\). The choice of reference measure is at author’s disposal and depends on \(b\). One usually takes a Gaussian measure as reference measure, since they are best studied. However, in many cases this is not the best choice. The key point is to choose the reference measure in such a way that it will be possible to formulate existence and uniqueness conditions. It is possible due to the fact that many probability measures on \(\mathbb R^d\) are images of Gaussian measures under so-called triangular mappings which turn out to have sufficient regularity. Therefore, one can reduce existence and uniqueness questions to the case of a Gaussian reference measure. The uniqueness of the solution is a more difficult problem compared to the existence. An existence result is established for the case of reference measures \(\nu\) on \(\mathbb R^{\infty}\) with logarithmic derivatives integrable in any power. Uniqueness is proved for a wide class of product measures, including log-concave ones. Another uniqueness result is proved for a class of uniformly log-concave Gibbs measures.
The main approach relies on the mass transportation method. More precisely, triangular mass transportation is chosen. Sobolev estimates for triangular mappings are established and the key technical relations between transport equations and mass transfer are established as an auxiliary result.

MSC:
60H30 Applications of stochastic analysis (to PDEs, etc.)
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
35F10 Initial value problems for linear first-order PDEs
35B45 A priori estimates in context of PDEs
35L45 Initial value problems for first-order hyperbolic systems
46E27 Spaces of measures
58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps
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