Affiliation:
1. Institute of Natural Sciences and Mathematics , Ural Federal University , Ekaterinburg 620075 , Russia
Abstract
Abstract
The paper is devoted to the regularization of ill-posed stochastic Cauchy problems in Hilbert spaces:
(0.1)
d
u
(
t
)
=
A
u
(
t
)
d
t
+
B
d
W
(
t
)
,
t
>
0
,
u
(
0
)
=
ξ
.
du(t)=Au(t)dt+BdW(t),\quad t>0,\qquad u(0)=\xi.
The need for regularization is connected with the fact that in the general case the operator A is not supposed to generate a strongly continuous semigroup and with the divergence of the series defining the infinite-dimensional Wiener process
{
W
(
t
)
:
t
≥
0
}
{\{W(t):t\geq 0\}}
.
The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q-Wiener processes.
Funder
Russian Science Foundation