# Chernoff's theorem for evolution families

GFM seminar

CIUL, B1-01

2007-09-28 14:30
2007-09-28 15:30
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.

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.