Infinite System of Stochastic Ordinary Differential Equations Coupled by Infinite Graphs - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:49:21Z http://mathoverflow.net/feeds/question/28470 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28470/infinite-system-of-stochastic-ordinary-differential-equations-coupled-by-infinite Infinite System of Stochastic Ordinary Differential Equations Coupled by Infinite Graphs Leandro 2010-06-17T08:10:28Z 2010-06-19T03:57:03Z <p>$\ \ \$ In <i>Time Evolution of Infinite Anharmonic Systems</i>, Lanford,Lebowitz and Lieb, roughly speaking, proved that for some families of functions $F_v$ $(v\in\mathbb Z^d)$ and a large set of initial condition, there exist a unique solution defined on a non-degenerated interval $[0,T]$ for the infinite system <code>$$\left\{ \begin{array}{rcl} \frac{d}{dt}q_v(t)&amp;=&amp;p_v(t)\\ \frac{d}{dt}p_v(t)&amp;=&amp;F_v(X(t))\\ X(0)&amp;=&amp;X_0, \end{array} \right.$$</code> where $X_0$ and $X(t)$ belongs to some Banach space contained in $(\mathbb R^n\times\mathbb R^n)^{\mathbb Z^d}$ and as mentioned before $v\in\mathbb Z^d$. </p> <p>$\ \ \$ Although the authors do not mention, their results can be generalized for any infinite uniformly bounded degree graph. But a more interesting generalization, would be made by considering the stochastic case. Because it seems to be physically relevant to consider that the forces in each vertex of the graph are not completely known, due some random perturbations. </p> <p>$\ \ \$ Could you help me to find a reference where this kind of model is considered ? </p>