Asymptotics of the solution to the Cauchy problem for linear parabolic equations of second order with small diffusion (Q5959467)

Let \(u ^\varepsilon(t,x)\), \((t,x)\in \mathbb{R} ^+\times \mathbb{R} ^n\), be the solution of the equation: \[ u ^\varepsilon_t+(a,\nabla u ^\varepsilon)+ \frac{\varepsilon ^2}{2} (\sigma \sigma ^{\text{T}}\nabla, \nabla u ^\varepsilon)+\frac{d}{\varepsilon ^2} u ^\varepsilon =0,\tag{1} \] complemented with the initial condition \(u ^\varepsilon|_{t=T}=\varphi(x) \exp(-\frac{\Phi(x)}{\varepsilon ^2}).\) Here, \(a:\mathbb{R} ^+\times\mathbb{R } ^n \to \mathbb{R} ^n\) is a vector function, \(\sigma(t,x)\) is an \((n\times n)\)-matrix, \(d\) [resp. \(\varphi,\Phi\)], is a real function defined on \(\mathbb{R ^+}\times \mathbb{R} ^n\) [resp. \(\mathbb{R} ^n\)]; \(\varepsilon >0\) is a small parameter, and \(\nabla\) denotes the gradient in the direction of \(x\). Under sufficient conditions which are only expressed in terms of the coefficients of \((1)\) and of the initial data, the author proves the existence of unique real functions \(v,g,\) and \(\psi\) such that: \[ \lim_{\varepsilon\to 0} u ^\varepsilon(t,x)\exp \Biggl(\frac{v(t,x)}{\varepsilon ^2}\Biggr)g(t,x)=\psi(t,x)\tag{2} \] for all \(t\in \mathbb{R ^+}\) and almost all \(x\in \mathbb{R} ^n .\) The main step in the proof is a representation of \(u ^\varepsilon \) of the form: \[ u ^\varepsilon (t,x)= \psi ^\varepsilon (t,x)\exp\Biggl(-\frac{v ^\varepsilon(t,x)}{\varepsilon ^2}\Biggr),\tag{3} \] where \(v ^\varepsilon \) and \(\psi ^\varepsilon\) are the unique solutions of second order parabolic equations; relation (3) is established thanks to the Itô stochastic equation and the Girsanov theorem. Then, a-priori estimates and a passage to the limit in relation (3) lead to formula (2).