On the Boolean model of Wiener sausages
Consider a \(d\)-dimensional Brownian path \((W_t)_{t\in[0,T]}\) started at \(0\) and its corresponding Wiener sausage of radius \(r>0\) \[ S_{r,T}=\{y\in \mathbb{R}^d: \|y-W_t\|\leq r\text{ for some }t\in[0,T]\}. \] The Boolean model is the random set \(\Xi\subset\mathbb{R}^d\) obtained as the union of Wiener sausages when each point of a homogeneous Poisson point process serves as the starting point of one Brownian path. By symmetry, the two-point function \[ C(h)=\mathbf{P}[\{0,x\}\subset\Xi]\qquad\text{for \(\|x\|=h\)} \] only depends on the norm of the vector \(x\). It can be computed using the one-point function \(p=\mathbf{P}[0\in\Xi]\), the expected volume of \(S_{r,T}\) (for both quantities there are more or less explicit formulas) and the expected volume of \(S_{r,T}\cap(x+S_{r,T})\). The authors use a Monte Carlo scheme to estimate this latter volume. To this end the Brownian paths are approxiated by random walks with normally distributed increments. They present the estimated curves \(h\mapsto C(h)\) for different values of \(T\) rather than providing numerical values with errors bounds. As an alternative to computing the expected volume of \(S_{r,T}\cap (x+S_{r,T})\), it is enough to compute the expected volume of \(S_{r,T}\cup (x+S_{r,t})\). This, however, has a representation as the integral \[ \int_{\mathbb{R}^d}u(T,y)\,dy, \] where \(u(t,y)\) is the solution of the heat equation with boundary values \(u(t,y)=1\) for \(\|y\|\leq r\) or \(\|y-x\|\leq r\) and \(u(0,y)=0\) for all other \(y\). The authors propagate a finite element scheme to compute \(u\) which is more efficient than the Monte Carlo scheme considered above.
scientific article; zbMATH DE number 5314358
scientific article