Consider the poisson equation on an unbounded domain. Suppose that the solution is known to exist. $$ \Delta u=f $$
I would like to estimate the solution of the this equation at a given point $x_0$. This is easily doable by numerical approximation, when in $\mathbb{R}^2$ or $\mathbb{R}^3$dimensional. However, I am interesting in $\mathbb{R}^d$ where $d > 1000$ (at least). This clearly, render numerical approximation infeasible.
In the case, of a dirichlet poisson problem,
$$ \Delta u(x)= f(x), x \in \Omega $$ $$ u(x) = g(x), x \in \partial\Omega$$
One could use the Feynmann-Kac theorem, to express the solution $u$ as
$$ u(x) = \mathbb{E}_x[\int_{0}^{\tau_{\partial\Omega}} \frac{g(X_{\tau_{\partial\Omega}})}{\tau_{\partial\Omega}} - f(X_{t}) dt]$$ Where $X_{t}$ is a $d$ dimensional brownian motion with $X_0 = x$
In this case Monte-carlo estimation becomes straighforward.
- Is there a similar approach to the unbounded poisson equation?
- Is there any other estimate the solution of a very high dimensional poisson equation on an unbounded domain at given point?