A generalized parametrix method, smoothness of random fields and applications to parabolic stochastic partial differential equations (Q2760867)

The goal of the thesis is to investigate the asymptotic behaviour as \(\varepsilon\searrow 0\) of solutions to a random parabolic equation NEWLINE\[NEWLINE\partial_t u= Lu+ \varepsilon^{-1}\Phi(\varepsilon^{-2} t,x)\nabla u\tag{1}NEWLINE\]NEWLINE on \([0,T]\times \mathbb{R}^d\), \(u(0,\cdot)= u_0\) on \(\mathbb{R}^d\), where NEWLINE\[NEWLINEL= \sum^d_{i,j=1} a_{ij}(t, x)\partial^2_{x_i x_j}+ \sum^d_{i=1} b_i(t, x)\partial_{x_i}+ c(t,x)NEWLINE\]NEWLINE is a uniformly parabolic second-order differential operator, \(\Phi\) is a centered homogeneous Gaussian random field, and \(u_0\) is a continuous function. The problem (1) is treated pathwise, by means of the variation of constants formula. Since the random field \(\Phi\) usually does not have bounded trajectories, the author shows in the first chapter that the standard parametrix method may be used to yield existence of a fundamental solution even if the coefficients of the lower order terms are unbounded (and satisfy growth estimates of the type \(|b_i(t,x)|\leq K_b(1+|x|^\beta)\), \(|c(t,x)|\leq K_c(1+|x|^\beta)\), \(x\in \mathbb{R}^d\), for a suitable \(\beta< 1\)). In Appendix A, corresponding uniqueness results are developed.NEWLINENEWLINENEWLINEIn Chapter 2, sample path properties of random fields, in particular Gaussian random fields, are studied and sufficient conditions on the covariance \(\Gamma\) of the Gaussian random field \(\Phi\) are found so that the operator \(\partial_t- L-\varepsilon^{-1} \Phi\nabla\) may satisfy the hypotheses of the first chapter. Finally, in Chapter 3, theorems on tightness of laws of the solutions \(\{u^\varepsilon, \varepsilon> 0\}\) to (1) on the space \(C([0, T]; H^{-r}(\mathbb{R}^d))\) are established for \(r> 0\) sufficiently large, \(H^{-r}\) beig a Sobolev space of negative order. As a consequence, it follows that the laws of \(u^\varepsilon\) converge weakly to the unique solution of a stochastic partial differential equation NEWLINE\[NEWLINEdu= \Biggl({\nu\over 2} \sum^d_{i,j= 1} \partial^2_{x_i x_j}+{1\over a} \sum^d_{i,j=1} \Gamma_{ij}(0)- {1\over a} \sum^d_{i,j=1} \partial_{x_i} \Gamma_{ij}(0) \partial_{x_j}\Biggr) u dt+ K^{1/2}_u dWNEWLINE\]NEWLINE in \(L^2(\mathbb{R}^d)\), where \(W\) is a cylindrical Wiener process and for \(f\in L^2(\mathbb{R}^d)\) the operator \(K_f: H^r\to H^{-r}\) is defined by NEWLINE\[NEWLINE(\xi, K_f\zeta)= {2\over a} \sum^d_{i,j=1} \int_{\mathbb{R}^d\times \mathbb{R}^d} f(x) f(y) \partial_{x_i} \partial_{y_j} [\xi(x) \zeta(y) \Gamma_{ij}(x- y)] dx dy.NEWLINE\]