A bounded property for gradients of diffusion semigroups on Euclidean spaces
The author studies stochastic differential equations of the form \[ dX^i(t,x) = \sum_{j}\sigma_{ij}(X(t,x))\,dB^j(t) + b_i(X(t,x))\,dt, \quad X(0,x)=x \] \((1\leq i,j\leq d)\) driven by a \(d\)-dimensional Brownian motion. The coefficients \(\sigma:=(\sigma_{ij})_{ij}\) are assumed to be \(C^\infty\)-smooth and bounded (with all derivatives) and the matrix \(\sigma^T \sigma\) is uniformly elliptic. The drift coefficients \(b=(b_1,\ldots,b_d)\) are smooth, and satisfy the following estimates \[ \sum_{i,j}\xi_i\xi_j\partial_ib_j(x) \leq \text{const.}\sum_i \xi_i^2 \] and for some \(\gamma_2 \geq \gamma_1 > 1\), \[ \langle x,b(x)\rangle \leq c - c'|x|^{\gamma_1+1} \quad\text{ and }\quad |b(x)| + |(\partial_ib_j)_{ij}| \leq c''(1+|x|^{\gamma_2}). \] Under these assumptions it is shown that the solution to the SDE satisfies for all \(\alpha,\beta\geq 0\) the following estimate: \[ |\nabla_x Ef(X(t,x))| \leq (1+|x|^2)^{\frac 12(\gamma_2-\gamma_1+1+\beta+\alpha)} c_{\alpha,\beta} t^{-d_\beta} \|(1+|\cdot|^2)^{-\alpha/2}f\|_\infty \] for all \(0<t\leq 1\) and \(x\in\mathbb R^d\). The proof uses ideas from complex interpolation theory in the scale of spaces with norms shown on the right-hand side of the inequality above.
scientific article; zbMATH DE number 2119544