## Invariant sets for systems of stochastic differential equations without aftereffect.(English. Ukrainian original)Zbl 0990.60060

Theory Probab. Math. Stat. 63, 123-130 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 112-118 (2000).
The authors study the invariant sets for the system of stochastic differential equations $d\xi(t)=a(t,\xi(t))dt+\sum\limits_{k=1}^{n}b_{k}(t,\xi(t))dw_{k}(t)+ \int_{R^{n}}c(t,\xi(t),u)\nu(dt,du),\quad \xi(t_0)=x_0,$ where $$a(t,x)=(a_{i}(t,x)$$, $$i=1,\ldots,n)$$, $$b_{k}(t,x)=(b_{ik}(t,x),i=1,\ldots,n)$$, $$c(t,x,u)=(c_{i}(t,x,u)$$, $$i=1,\ldots,n)$$ are real non-random functions; $$x\in R^{n}$$, $$u\in R^{n}$$; $$w_{k}(t)$$ are independent one-dimensional Wiener processes; $$\nu([0,t),A)$$ is a Poisson measure independent on $$w_{k}(t)$$ with parameter $$t\Pi(A)$$; $$A$$ is a Borel set from $$R^{n}$$; $$\Pi(R^{n})<\infty$$. Let $$Q=[0,\infty)\times D$$, where $$D$$ is an open set in $$R^{n}$$ such that $$\Pi\{u:x+c(t,x,u)\not\in D\}=0$$ for all $$(t,x)\in Q$$. Let us denote $$\tau_{Q}(t_0,x_0)=\inf\{t\geq t_0:\xi(t)\not\in D\}$$, $$(t_0,x_0)\in Q$$, and $$\Gamma_{Q}(G)=\{(t,x):G(t,x)=C\}\subset Q$$, where sufficiently smooth function $$G(t,x)$$ is defined in $$Q$$ and there exist constants $$l>0, \delta>0$$, such that in $$Q$$ we have $\int_{R^{n}}|G(t,x+c(t,x,u))-G(t,x)|^{2+\delta}\Pi(du)\leq l.$ The set $$\Gamma_{Q}(G)$$ is called invariant in the domain $$Q$$ of the considered stochastic differential equation if for all $$(t_0,x_0)\in \Gamma_{Q}(G)$$ and for all $$t\geq t_0$$ we have $$[G(t,\xi(t))-G(t_0,x_0)]\varphi(t)=0$$ a.s., where $$\varphi(t)=1$$ as $$t_0\leq t<\tau_{Q}(t_0,x_0)$$ and $$\varphi(t)=0$$ as $$t\geq \tau_{Q}(t_0,x_0)$$. The authors obtain necessary and sufficient conditions for invariance of the set $$\Gamma_{Q}(G)$$. For some class of second order system the explicit form of invariant curves is presented.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)