Browsing by Author "Halme, Eetu"

Solving partial differential equations using methods of probability theory has a long history. In this thesis we show that the solutions of the conductivity equation in Lipschitz domain D with Neumann boundary conditions and uniformly elliptic, measurable conductivity parameter $\kappa$, can be represented using a Feynman-Kac formula for reflecting diffusion processes X on the domain D. We begin with history and connection to statistical experiments in Chapter 1. Chapter 2 starts by introducing Banach and Hilbert spaces with spectral theory of bounded operators, together with Hölder and Sobolev spaces. Sobolev spaces provide the right properties for the boundary data and solutions. In Chapter 3, we introduce the basics of stochastic processes, martingale and continuous semimartingales. We also need the theory of Markov processes, which Hunt and Feller processes are based on. Hunt processes are because of their correspondence with Dirichlet forms to define the reflecting diffusion process X. We also introduce the concept of local time of a process. In Chapter 4 we introduce Dirichlet forms, their correspondence with self-adjoint operators and Revuz measure. In Chapter 5, we introduce the conductivity equation and the Dirichlet-to-Neumann map $\Lambda_\kappa$. The goal of the Calderón's problem is to reconstruct the conductivity parameter $\kappa$ from the map $\Lambda_\kappa$, which is a difficult, non-linear and ill-posed inverse problem. The Chapter 6 constitutes the main body of the thesis, and here we prove the Feynman-Kac formula for solutions of the conductivity equation. We use the correspondence between the Dirichlet forms and selfadjoint operators, to define a semigroup (T_t) of solutions to a abstract Cauchy equation and from the semigroup (T_t) which we can associate by the Dunford-Pettis theorem a transition density function p for the reflecting diffusion process X. We show that using De-Giorgi-Nash-Moser estimates that p is Hölder continuous and defined everywhere. We also prove p converges exponentially to stationary distribution. We generalise the concept of boundary local times using the Revuz measure, and prove the occupation formula. These results together with the Skorohod decomposition for Lipschitz conductivities is used in the four part proof of Feynman-Kac formula. In Chapter 7, we introduce the boundary trace process, which is a pure jump process corresponding to the hitting times on the boundary of the reflecting diffusion process. We state that the trace process is the infinitesimal generator of the Dirichlet-to-Neumann map -\Lambda_\kappa and thus provides a probabilistic interpetation for Calderón's problem. We end with discussion on applications of the theory and potential directions for new research. The main references of the thesis are the articles of Piiroinen and Simon ''From Feynman–Kac Formulae to Numerical Stochastic Homogenization in Electrical Impedance Tomography'' and ''Probabilistic interpretation of the Calderón problem''.