Michael Kouritzin - Parabolic equations with random coefficients

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Michael Kouritzin - Parabolic equations with random coefficients

MICHAEL KOURITZIN, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Parabolic equations with random coefficients

Questions related to the asymptotic behavior (as $\epsilon \rightarrow 0$ ) of systems of random ordinary differential equations

$\begin{displaymath}\dot{X}^{\epsilon }(t)=F\Bigl(X^{\epsilon }(t),\frac{t}{\epsilon },\omega \Bigr),\quad \epsilon >0,\ X^{\epsilon }(0)=x_{0}, \end{displaymath}$

where $\{F(x,{s}),s\geq 0\}$ is a random process for each $x\in \Re ^{d}$ , have attracted a multitude of investigations due to applications in such diverse areas as celestial mechanics, oscillation theory, adaptive filtering, recursive identification, and stochastic adaptive control.

A natural question that is important to filtering theory and stochastic control is whether these convergence results continue to hold for the parabolic partial differential equations

$\begin{displaymath}\partial _{t}u^{\epsilon }(x,t)=\sum_{\vert k\vert\leq 2p}A_{... ...psilon }(x,t), \quad \epsilon >0,\ u^{\epsilon}(x,0)=\phi (x). \end{displaymath}$

For second order parabolic equations with various technical and simplifying assumptions, earlier results indicate that laws of large numbers and fluctuation results continue to hold, provided one resorts to spaces of generalized functions for the fluctuation results. In this talk, we will discuss general convergence and rate of convergence results for $u^{\epsilon}$ . In particular, we will only assume that the coefficients themselves satisfy natural convergence or fluctuation results and we will prove our fluctuation results on a natural Hilbert space. Finally, our setting is general enough to allow for long-range dependence and/or heavy-tail distributions within our work on fluctuations.

Next: Reg Kulperger - Empirical Up: Probability Theory / Théorie Previous: Gail Ivanoff - Set-indexed