Asymptotic behavior of the density in a parabolic SPDE (Q5939309)

Consider the density of \(X(t,x)\) at fixed \(t>0\) and \(x\in[0,1]\), where \(X\) is given as the solution of the stochastic heat equation \[ {\partial X \over \partial t} = {\partial^2 X \over \partial x^2} +\varepsilon\sigma(X)\dot{W}+b(X) \] subject to Neumann or Dirichlet boundary conditions with smooth nonlinearities with uniformly bounded derivatives. Suppose \(\sigma>C>0\). To define a solution use the corresponding variation of constants formula involving Green kernel. The main result of this paper describes the asymptotic of the density as the noise vanishes (i.e., \(\varepsilon\to 0\)). The authors obtain a Taylor series expansion in powers of \(\varepsilon\) and explicitly determine the coefficients and the residue of the expansion. This result generalizes a previous one by \textit{D. Márquez-Carreras} and \textit{M. Sanz-Solé} [Collect. Math. 49, No. 2-3, 399-415 (1998; Zbl 0939.60067)]. For the proof they use exponential estimates of tail probabilities of the difference between the approximation and the limiting process. Moreover, they develop an iterative local integration by parts formula.