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Session 34. SPDE: stochastic analysis and dynamics
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\(L^p\)-parabolic regularity and non-degenerate Ornstein-
Uhlenbeck type operators
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Enrico Priola, University of Torino, Italy
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We prove
\(L^p\)-parabolic a-priori estimates for \(\partial_t u + \sum_{i,j=1}^d
c_{ij}(t)\partial_{x_i x_j}^2 u = f \) on \(R^{d+1}\) when the coefficients
\(c_{ij}\) are locally bounded functions on \(R\) and \(p \in (1, \infty)\).
We slightly generalize
the usual parabolicity assumption and show that still \(L^p\)-estimates hold for the
second spatial derivatives of \(u\). We also investigate the dependence of
the constant appearing in such estimates from the parabolicity constant.
When \(p \not = 2\) the proof requires the use of the stochastic integral.
Finally we
extend our estimates to parabolic equations involving non-degenerate
Ornstein-Uhlenbeck type operators.
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