= Solution
A second-order <nondivergence-form elliptic operator> on a domain $D\subseteq\mathbb R^d$ has the form
$$
\mathcal Af(x)=\sum_{j,k=1}^d a_{jk}(x)\partial_{jk}f(x)
+\sum_{j=1}^db_j(x)\partial_jf(x)+c(x)f(x).
$$
Only the symmetric part of $a$ affects the <Hessian matrix> term, so one may take $a(x)$ symmetric. The operator is a <uniformly elliptic operator> if there is a constant $\kappa>0$, independent of $x$, such that
$$
\boxed{\sum_{j,k=1}^da_{jk}(x)\xi_j\xi_k\geq\kappa|\xi|^2
\qquad(x\in D,\ \xi\in\mathbb R^d).}
$$
A common bounded-coefficient convention additionally requires an upper bound $\sum a_{jk}(x)\xi_j\xi_k\leq\Lambda|\xi|^2$ with $\Lambda<\infty$. The lower bound says that the principal <quadratic form> cannot degenerate in any direction or at any location.
For an explicit stochastic example, standard $d$-dimensional <Brownian motion> has <infinitesimal generator>
$$
\boxed{\mathcal A=\tfrac12\Delta,\qquad a_{jk}=\tfrac12\delta_{jk},\quad b=0,\quad c=0.}
$$
Its principal <quadratic form> is $|\xi|^2/2$, so it is <uniformly elliptic> with $\kappa=\Lambda=1/2$. More generally, a sufficiently regular <stochastic differential equation> $dX_t=b(X_t)\,dt+\Sigma(X_t)\,dB_t$ has <diffusion generator> $b\cdot\nabla+\tfrac12(\Sigma\Sigma^{\mathsf T}):D^2$; it is <uniformly elliptic> when $\Sigma\Sigma^{\mathsf T}$ has a uniformly positive lower bound. A conservative <diffusion process> has no zeroth-order term, since its <infinitesimal generator> annihilates constants.
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