| Contents |
| We study differential equations of the form
\begin{equation*}
dX/dt=f(X),
\end{equation*}%
where $f$ may be only measurable. We intend to obtain more regularity and
numerical feasability by the stochastic discretization
\begin{equation*}
\delta x_{t}=f(x_{t}+w_{t})\delta t.
\end{equation*}%
The time-step $\delta t$ is infinitesimal and the stochastic variables $w_{t}
$ are Lebesgue-measurable, independent and identically distributed, but are
not necessarily standard: their variance may be infinitesimal. We give
conditions to ensure that the solutions are almost surely infinitely close
to the more regular system%
\begin{equation*}
\delta y_{t}=Ef(y_{t}+w_{t})\delta t
\end{equation*}
Often the expectations are calculated by convolution. The regularizations
obtained are of numerical interest for the original differential equation
when $f$ presents discontinuites or if there is no unqueness of solutions.
The results are based on joint work with C. Lobry (Nice) and T. Sari
(Montpellier). |
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