Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter \(H < 1/2\) (Q428140)

The authors consider the stochastic heat equation on \(\mathbb{R}^{d}\) defined for \(t\geq0\), \(x\in\mathbb{R}^{d}\) by NEWLINE\[NEWLINE \frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+u\,\frac{\partial W}{\partial t}(t,x) , NEWLINE\]NEWLINE where the initial condition \(u(0,x)=u_{0}(x)\) is a bounded measurable function and \(W=\{W(t,x), t\geq 0,x\in\mathbb{R}^{d}\}\) is a fractional Brownian motion of Hurst parameter \(H\in ( \frac{1}{4},\frac{1}{2} ) \) in time and it has a spatial covariance \(Q(x,y)\), which is locally \(\gamma\)-Hölder continuous, with \(\gamma>2-4H\). They obtain a generalization of the well-known Feyman-Kac formula to the case of a random potential of the form \(\frac{\partial W}{\partial t}(t,x)\). More precisely, they show that the solution to above equation is given by NEWLINE\[NEWLINE u(t,x)=\operatorname{E}^{B} \biggl[ u_{0}(B_{t}^{x})\exp\int _{0}^{t}W(ds,B_{t-s}^{x}) \biggr] , NEWLINE\]NEWLINE where \(B=\{B_{t}^{x}=B_{t}+x,t\geq0,x\in\mathbb{R}^{d}\}\) is a \(d\)-dimensional Brownian motion starting at \(x\in\mathbb{R}^{d}\), independent of \(W\).