Versions of the Feynman-Kac formula (Q1575978)

The paper deals with some versions of the Feynman-Kac formula for Brownian motion. One of the versions is as follows. Let \(\varphi(t)\) be a solution of the system of linear stochastic differential equations \[ d\varphi(t)= A(t, w(t)) \varphi(t)dt+ B(t, w(t)) \varphi(t)dw(t)+ c(t, w(t))dt,\quad \varphi(0)= \varphi_0, \] where \(A(t,x)=\|a_{k,l}(t, x)\|^n_{k,l= 1}\), \(B(t,x)=\|b_{k,l}(t, x)\|^n_{k,l= 1}\), \(c(t,x)=\|c_k(t, x)\|^n_{k=1}\), \(w(t)\) is a Brownian motion and the following conditions hold true \[ |b_{kl}(t, x)|+ \Biggl|{\partial\over\partial x_i} b_{kl}(t, x)\Biggr|+ \Biggl|{\partial\over\partial x_i} {\partial\over\partial x_j} b_{kl}(t, x)\Biggr|\leq K_t,\quad |a_{kl}(t, x)|\leq K_t(1+|x|), \] \[ \Biggl|{\partial\over\partial x_i} a_{kl}(t, x)\Biggr|+ \Biggl|{\partial\over\partial x_i} {\partial\over\partial x_j} a_{kl}(t, x)\Biggr|\leq K_t, \] \[ |c_{kl}(t, x)|+ \Biggl|{\partial\over\partial x_i} c_{kl}(t, x)\Biggr|+ \Biggl|{\partial\over\partial x_i} {\partial\over\partial x_j} c_{kl}(t,x)\Biggr|\leq K_t\exp(C_t|x|), \] with constants \(K_t\), \(C_t\) bounded in \(t\) on any finite interval. Then the function \[ g(t,z)= {\partial\over\partial z} E_x[\varphi(t)\mathbf{1}f_{\{w(t)< z\}}(\omega)] \] is a solution of the problem \[ {\partial\over\partial t} g(t,z)= {1\over 2} {\partial^2\over\partial z^2} g(t, z)- B(t,z) {\partial\over\partial z} g(t,z)+\Biggl(A(t,z)- {\partial\over\partial z} B(t,z)\Biggr) g(t,z)+ c(t,z) {e^{-(z- x)^2/2t}\over \sqrt{2\pi t}}, \] \[ g(0,z)= \varphi_0 \delta_x(z). \] Here \(\delta_x(z)\) is Dirac's delta function.