Some results for second order elliptic operators having unbounded coefficients. (English) Zbl 1131.35393

Existence and uniqueness results and Schauder estimates are obtained both for the elliptic problem \(\lambda \varphi - {\mathcal A}\varphi = f\) in \(\mathbb R^ {d}\) and for the parabolic equation \(u_ {t} = {\mathcal A}u + f\) in \(\left ]0,T\right ] \times \mathbb R^ {d}\), \(u(0,\cdot ) = \varphi \), where \({\mathcal A}\psi (x) = \frac 12\operatorname {Tr} [A(x)D^ 2\psi (x)] + \langle F(x),D\psi (x)\rangle \) is a second order elliptic differential operator with smooth coefficients of a polynomial growth. More precisely, it is assumed that \(F\in C^ 3(\mathbb R^ {d};\mathbb R^ {d})\), \(A=GG^ {*}\), \(G\in C^ 3 (\mathbb R^ {d};\mathbb R^ {r})\), and \[ \sup _ {x\in \mathbb R^ {d}}\bigl \{(1+| x| ^ {2m -1-j})^ {-1}| D^ \beta F(x)| + (1+| x| ^ {k-j} )^ {-1}\| D^ \beta G(x)\| \bigr \} <\infty \] for some \(m\geq 2\), \(k\leq m-1\) and any multiindex \(\beta \) with \(| \beta | \leq 3\). The matrix \(G(x)\) is supposed to be invertible for all \(x\in \mathbb R^ {d}\) and \(\sup _ {x} \| G^ {-1}(x)\| <\infty \). Finally, let the following dissipativity hypotheses be satisfied: \(\langle F(x+h)-F(x),h\rangle \leq -a| h| ^ {2m} + b| x| ^ {2m} +c\) for some \(a>0\), \(b,c\in \mathbb R\) and all \(x,h\in \mathbb R^ {d}\), and for any \(p>0\) let there exist \(\omega _ {p}\) such that \(\langle DF(x)y,y\rangle + p\| DG(x)y\| ^ 2\leq \omega _ {p}| y| ^ {2}\). Author’s approach is probabilistic, relying on estimating (by means of the Bismut-Elworthy formula) the derivatives of the transition semigroup \((P_ {t})\) defined by a stochastic differential equation \( dX_ {t} = F(X_ {t})\, dt + G(X_ {t})\, dw_ {t}\), \(w\) denoting an \(r\)-dimensional Brownian motion.
The developed methods are also used to prove (under slightly modified assumptions) that the semigroup \((P_ {t})\) is strongly Feller and there exists a unique invariant measure for it, which is ergodic and strong mixing.


35R60 PDEs with randomness, stochastic partial differential equations
35J15 Second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35K10 Second-order parabolic equations
47D06 One-parameter semigroups and linear evolution equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)