Regularization of stochastic problems with respect to variables of different kinds. (English. Russian original) Zbl 1188.47010
Dokl. Math. 79, No. 3, 408-411 (2009); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 426, No. 5, 597-601 (2009).
The author considers the stochastic Cauchy problem
\[ X'(t)= AX(t)+ B\mathbb{W}(t),\quad t\in [0,\tau),\;\tau\leq\infty;\;X(0)= \xi,\tag{1} \]
where \(A\) is the generator of a regularized semigroup \(S= \{S(t): t\in [0,\tau)\}\) in a Hilbert space \(H\); \(\mathbb{W}= \{\mathbb{W}(t): t\geq 0\}\) is an \(H\)-valued white noise, which is informally defined as a process with independent distributions at different values of \(t\) with zero mean and infinite variation, \(B\in{\mathcal L}(H)\).
By regularization of (1), we mean the construction of a new problem (corrected, smoothed) and its solution, including a weak solution of (1). The resulting regularized solutions are not assumed to converge to some exact solution of (1) with respect to a parameter. The author uses regularization (i) with respect to variables of \(H\) and the operator \(A\); (ii) with respect to \(t\); and (iii) with respect to a random variable.
\[ X'(t)= AX(t)+ B\mathbb{W}(t),\quad t\in [0,\tau),\;\tau\leq\infty;\;X(0)= \xi,\tag{1} \]
where \(A\) is the generator of a regularized semigroup \(S= \{S(t): t\in [0,\tau)\}\) in a Hilbert space \(H\); \(\mathbb{W}= \{\mathbb{W}(t): t\geq 0\}\) is an \(H\)-valued white noise, which is informally defined as a process with independent distributions at different values of \(t\) with zero mean and infinite variation, \(B\in{\mathcal L}(H)\).
By regularization of (1), we mean the construction of a new problem (corrected, smoothed) and its solution, including a weak solution of (1). The resulting regularized solutions are not assumed to converge to some exact solution of (1) with respect to a parameter. The author uses regularization (i) with respect to variables of \(H\) and the operator \(A\); (ii) with respect to \(t\); and (iii) with respect to a random variable.
Reviewer: Alexandra Rodkina (Kingston/Jamaica)
MSC:
| 47A52 | Linear operators and ill-posed problems, regularization |
| 60H25 | Random operators and equations (aspects of stochastic analysis) |
| 34F05 | Ordinary differential equations and systems with randomness |
| 34K30 | Functional-differential equations in abstract spaces |
| 93E03 | Stochastic systems in control theory (general) |
| 34G10 | Linear differential equations in abstract spaces |
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