Formulae for the derivatives of degenerate diffusion semigroups
The author considers the diffusion semigroup \(P_t\) associated with the possibly degenerate elliptic operator \[ \textstyle{{1\over 2}} \text{Tr}(QD^2\psi(x))+ \langle Ax, D\psi(x)\rangle+ \langle F(x) D\psi(x)\rangle, \] \[ \psi\in C^\infty_0(\mathbb{R}^n),\quad x\in\mathbb{R}^n, \] where \(Q\) and \(A\) are \(n\times n\) real matrices, \(Q\) is symmetric and nonnegative definite, Tr\((.)\) denotes the trace and \(F: \mathbb{R}^n\to\mathbb{R}^n\) is a possibly unbounded regular function. Under the hypothesis \(F\in C^3(\mathbb{R}^n, \mathbb{R}^n)\), \(F(x)\in \text{Im}(Q)\) and with adequate rank constraints the author proves the following main result: for any \(t> 0\), \(f\in C_b(\mathbb{R}^n)\), the map \(x\to P_t(f(x))\) admits all classical bounded derivatives up to the third-order on \(\mathbb{R}^n\). The proof is probabilistic, the spatial derivatives of \(P_t f\) are given in terms of expected values of random variables. As an application it is shown that for any \(f\in C_b(\mathbb{R}^n)\) the function \(u(t, x)= P_t(f(x))\) is the unique classical bounded solution of the associated Cauchy problem.
scientific article; zbMATH DE number 5134734