Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces
Siva Athreya, Richard F. Bass, Edwin Perkins · Transactions of the American Mathematical Society · 2005
We introduce a new method for proving the estimate \[ ‖ ∂ 2 u ∂ x i ∂ x j ‖ C α ≤ c ‖ f ‖ C α , \left \Vert \frac {\partial ^2 u}{\partial x_i \partial x_j} \right \Vert _{C^\alpha }\leq c\|f\|_{C^\alpha }, \] where u u solves the equation Δ u − λ u = f \Delta u-\lambda u=f . The method can be applied to the Laplacian on R ∞ \mathbb {R}^\infty . It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.