On limiting values of stochastic differential equations with small noise intensity tending to zero (Q1017648)

The authors consider the ordinary differential equation \[ x'(t)=f(x(t)),\quad t\geq 0,\quad x(0)=x\tag{1} \] and stochastic differential equation obtained by adding to the right-hand side of (1) a noise with small intensity: \[ dX_\varepsilon(t)=f(X_\varepsilon(t))dt+ \varepsilon dW_t,\;t\geq 0,\;X(0)=x,\tag{2} \] where \(\varepsilon>0\) is small, \((W_t:t\geq 0)\) denotes \(d\)-dimensional standard Brownian motion. Let \[ F_f(x)=\cap_{\lambda(N)=0} \cap_{\delta>0}co\,f((x+\delta B)\setminus N) \] with Lebesgue measure \(\lambda\) and \(B\) closed unit ball. An absolutely continuous solution \(t\in[0,\infty) \mapsto x(t)\in \mathbb{R}^d\) is a Filippov solution of (1) if and only if it is a solution of the following differential inclusion \[ x'(t)\in F_f(x(t)),\;t\geq 0,\;x(0)=x. \] The authors prove the theorem: Theorem. Suppose that \(f:\mathbb{R}^d\to \mathbb{R}^d\) is Lebesgue measurable and satisfies \[ \|f(x)\|\leq M(1+|x|),\quad\forall x\in \mathbb{R}^d. \] For any \(\varepsilon>0\), let \(X\) be the solution to (2). Then, there exists \(\varepsilon_n\to 0\) such that \(X_{\varepsilon_n}\) converges in law, as \(\varepsilon_n\to 0\), to some \(X\) which belongs almost surely to the set of Filippov's solutions to (1). Furthermore, any cluster point of \(X_\varepsilon\) is also almost surely in the set of Filippov's solutions.