On uniqueness problems related to elliptic equations for measures

We consider equations of the form L * μ  = 0 for bounded measures on , where L is a second order elliptic operator, for example, Lu  = Δ u  + ( b ,∇ u ), and the equation is understood as the identity for all compactly supported smooth functions u . Stationary Kolmogorov equations for invariant meas...

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2011-08, Vol.176 (6), p.759-773
Main Authors: Bogachev, V. I., Röckner, M., Shaposhnikov, S. V.
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:We consider equations of the form L * μ  = 0 for bounded measures on , where L is a second order elliptic operator, for example, Lu  = Δ u  + ( b ,∇ u ), and the equation is understood as the identity for all compactly supported smooth functions u . Stationary Kolmogorov equations for invariant measures of diffusion processes belong to this type. Solutions are considered in the class of probability measures and in the class of signed measures with integrable densities. We discuss the following problems: When is a probability solution to this equation unique? When does a given probability solution have the property that any integrable solution is a multiple of it? Which dimension can have the simplex of probability solutions? We present some recent positive results, give counterexamples and formulate open problems. Bibliography: 19 titles.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-011-0434-3