The exponential asymptotics of solutions of the Kolmogorov-Feller equation (Q1874771)

The authors consider the Kolmogorov-Feller equation \[ h\partial u/\partial t= L_hu,\tag{1} \] \[ L_h= {h^2\over 2}\langle A(x)\nabla, \nabla\rangle- h\langle b(x),\nabla\rangle+ \int_{\mathbb{R}^n} [\exp\{\langle\beta, h\nabla\rangle\}- 1]\mu (x,d\beta),\tag{2} \] where \(A(x)\) is a symmetric strictly positive \(n\times n\) matrix, \[ b(x)= (b_1(x),\dots, b_n(x)),\quad \nabla= (\partial/\partial x_1,\dots, \partial/\partial x_n), \] \(\mu(x,d\beta)\) is a positive measure compactly supported in \(\mathbb{R}_\beta\) for each \(x\in\mathbb{R}^n\), \(\langle\cdot,\cdot\rangle\) is the inner product in \(\mathbb{R}^n_x\) and \(h\in (0,1]\) is a small parameter. Moreover assume that the entries of \(A(x)\), the vector \(b(x)\) and the measure \(\mu(x,\beta)\) are infinitely differentiable bounded functions of \(x\) for \(x\in\mathbb{R}^n\) and that all of their derivatives are also uniformly bounded; the measure \(\mu(x,\beta)\) is compactly supported with respect to the variable \(\beta\). In this paper, the authors present a method for constructing the exponential asymptotics of the fundamental solution \(\Gamma=\Gamma(x,t,\xi,h)\) of the problem \[ h\partial\Gamma/\partial t= L_h\Gamma,\quad \Gamma|_{t=0}= \delta(x- \xi),\quad\xi\in \mathbb{R}^n, \] where \(L_h\) is the operator given by (2) and \(\delta(x)\) is the Dirac delta function. The authors construct a function \(\Gamma_N= \Gamma_N(x,t,\xi,h)\) such that \[ \max_{x\in\mathbb{R}^n,t\in[0, T)}\Biggl|\int_{\mathbb{R}^n} \exp\{S_1/h\}(\Gamma_N- \Gamma)u(\xi,h) d\xi\Biggr|\leq C_N h^N, \] where \(S_1= S_1(x,t)= \max_{\xi\in\mathbb{R}^n} (S_0(\xi)+ S(x,t,\xi))\), \[ S(x,t,\xi)=-\lim_{h\to+0} h\cdot\ln\Gamma_N(x,t,\xi,h), \] \[ u(x,h)= \varphi(x, h)\exp\{-S_0(x)/h\},\quad S_0\in C^\infty(\mathbb{R}^n), \] \(\varphi(x,h)\in C^\infty_0(\mathbb{R}^n)\) uniformly with respect to \(h\in [0, h_0]\), and \(C_N\) is a positive constant.