Random walks associated with non-divergence from elliptic equations (Q1582129)

Consider a nondivergence form elliptic operator \(L =\sum_{i,j=1}^d a_{ij}(x) \partial_i \partial_j\), where \(d \geq 2\), \(\partial_i= \partial / \partial x_i\), and the symmetric matrix \(A(x) = ( a_{ij}(x))\) satisfies the condition \[ \lambda I \leq A(x) \leq \Lambda I, \qquad x \in {\mathbb R}^d, \tag \(*\) \] for some constants \(\lambda, \Lambda\) with \(0< \lambda < \Lambda < \infty\). Motivated by the estimation problem of the solution to parabolic equation with \(L\) naturally arising in finance, the authors introduce a new technique to study the \(L\)-diffusion \(X_L(t)\). The basis of the technique consists in estimation of the escape probability of \(X_L(t)\) from \((d-1)\)-dimensional manifolds, and also in comparison of the behaviors of random walks induced by the diffusion to that of random walks induced by Brownian motion. Moreover, another peculiar feature of this paper is that the comparison is implemented in terms of only \(\lambda\), \(\Lambda\) in \((*)\), implying that it is free from dependence on the degree of oscillation of \(A(x)\). In other words, the authors study the diffusion \(X_L(t)\) by using only the knowledge that the local variance of the process is bounded above and below. More precisely, let \(\chi_{U(r)}\) be the characteristic function of \(r\)-neighborhood \(U(r)\) of a \((d-1)\)-dimensional \(C^{1,\alpha}\)-manifold with radius \(r < R\), and \(w_r(x)\) be the solution of the Dirichlet problem \(-Lw= \chi_{U(r)}\) for \(|x |< R\) and \(w = 0\) on \(|x |= R\). Then \(w_r(x)\) is the expected time that the diffusion process \(X_L(t)\) started at \(x\) spends in \(U(r)\) before exiting the disc \(|x |\leq R\). The first theorem says that there is a constant \(C\) depending only on \(d\) and the uniform ellipticity constants \(\lambda, \Lambda\) of \((*)\) such that \[ \|w \|_{\infty}\leq C R r.\tag \(**\) \] It is interesting to note that \((**)\) is much better than the estimate \[ \|w \|_{\infty}\leq C R^{2-1/d} r^{1/d} \tag \(***\) \] given by the Alexandroff-Bakelman-Pucci inequality [cf. \textit{D. Gilbarg} and \textit{N. S. Trudinger}, ``Elliptic partial differential equations of second order'' (1983; Zbl 0562.35001)], and also that for the case of Lipschitz manifolds \((***)\) remains valid while \((**)\) fails to hold.