Peano phenomenon and large deviations (Q2741010)

Let \(X^\varepsilon\) be the solution of the stochastic differential equation NEWLINE\[NEWLINEdX^\varepsilon= \varepsilon dB_t+ b(X^\varepsilon_t) dt,\;t\in [0,T];\qquad X^\varepsilon_0= 0,\tag{1.}\(\varepsilon\) NEWLINE\]NEWLINE where \(B\) is a one-dimensional Brownian motion and \(b\in C(R)\) is an increasing odd function, continuously differentiable on \(R\setminus\{0\}\), such that \(b^{-1}\) is integrable in a small neighborhood of \(0\) and, for some \(\gamma\in (0,1)\), \(C>0\), \(b'(x)\) behaves like \(C\gamma|x|^{\gamma- 1}\), as \(x\to 0\). For \(\varepsilon= 0\), the equation possesses an infinite number of solutions (Peano's phenomenon). \textit{R. Bafico} and \textit{P. Baldi} [Stochastics 6, 279-292 (1982; Zbl 0487.60050)] have shown that the law of \(X^\varepsilon\) converges weakly towards \({1\over 2}(\delta_{\rho_1}+ \delta_{\rho_2})\), where \(\rho_1\), \(\rho_2\) are the extremal solutions of equation (1.0).NEWLINENEWLINENEWLINEThe author studies the rate of convergence of the density \(p^\varepsilon_t(x)\) of \(X^\varepsilon_t\) where, according to the position of \((t,x)\) with respect to the reciprocal to \(K(u)= \int^u_0 dy/b(y)\), he gets two kinds of rate: If \(|x|> K^{-1}(t)\), \(\lim_{\varepsilon\to 0} \varepsilon^2\log p^\varepsilon_t(x)= k_t(|x|)\), for some strictly positive function \(k_t\), while for \(|x|< K^{-1}(t)\), \(\lim_{\varepsilon\to 0} \varepsilon^{2(1- \gamma)/(1+ \gamma)}\log p^\varepsilon_t(x)= \lambda_1(K(|x|)- t)\), where \(\lambda_1\) is the first positive eigenvalue of some Schrödinger operator. Moreover, the author deduces a large deviation principle. The paper generalizes an earlier work of the author with \textit{M. Gradinaru} and \textit{B. Roynette} [Ann. Inst. Henri Poincaré, Probab. Stat. 37, No. 5, 555-580 (2001)] which was devoted to the case \(b(x)= \text{sign}(x)|x|^\gamma\).