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# Chernoff's theorem for evolution families

GFM seminar
CIUL, B1-01
2007-09-28 14:30 .. 15:30

by Evelina Shamarova (Post-doc at GFM)

A generalized version of Chernoff's theorem has been obtained. Namely,
the version of Chernoff's theorem for semigroups obtained in a paper
by Smolyanov, Weizsaecker, and Wittich is generalized for a
time-inhomogeneous case. The main theorem obtained in the current
paper, Chernoff's theorem for evolution families, deals with a family
of time-dependent generators of semigroups $A_t$ on a Banach space, a
two-parameter family of operators $Q_{t,t+\Delta t}$ satisfying the
relation: $\frac{\partial}{\partial \Delta t}Q_{t,t+\Delta t}|_{\Delta t = 0}=A_t$, whose products $Q_{t_i,t_{i+1}}... Q_{t_{k-1},t_k}$ are
uniformly bounded for all subpartitions $s = t_0 < t_1 < >... < t_n = t$. The theorem states that $Q_{t_0,t_1}... Q_{t_{n-1},t_n}$ converges
to an evolution family $U(s,t)$ solving a non-autonomous Cauchy
problem. Furthermore, the theorem is formulated for a particular case
when the generators $A_t$ are time dependent second order differential
operators. Finally, an example of application of this theorem to a
construction of time-inhomogeneous diffusions on a compact Riemannian
manifold is given.
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