## 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.

### MSC:

 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)