Stochastic dynamics with singular lower order terms in finite and infinite dimensions

In this work, we aim to study several stochastic dynamics with singular coefficients. The results consist of three parts. In the first part, we study a class of second order parabolic equations $nabla (a(t,x) cdot nabla u(t,x))+b(t,x)cdot nabla u(t,x)+V(t,x) u(t,x)-partial_tu(t,x)=0 eqno (1)$ in the domain $[0,T]times mathbb{R}^{d}$, where $T<infty$. We assume that the matrix $a(t,x)=(a_{ij} t,x))$ is symmetric and uniformly elliptic, H"older continuous in $t$, $x$ and $frac{partial}{partial x_{i}}a_{ij}(t,x)$ are bounded and H"older continuous in $x$. The lower order terms are assumed to be in some proper time-dependent Kato classes. Then we prove that the parabolic equation (1) has a unique weak fundamental solution admitting two-sided Gaussian estimates. In the second part, we study the stochastic differential equation $ dX_t=dW_t+B(t, X_t)dt, X_s=x, eqno (2)$ where $W_{t}$ is a Brownian motion and $B(t,x)$ is a time-dependent singular drift. We assume $B(t,x)$ to be in the forward-Kato class $mathcal{F} mathcal{K}_{d-1}^{alpha}$ for some $alpha<frac{1}{2}$. The forward-Kato class $mathcal{F} mathcal{K}_{d-1}^{alpha}$ includes several important classes of functions. Then we prove that the stochastic differential equation (2) has a unique weak solution for every starting point $(s,x)$. In the last part, we consider an unbounded spin system on a simple and connected infinite graph $mathbb{G(V,E)}$ which is of bounded degree. We assume that the potential energy of each configuration $x in Omega:=mathbb{R}^{mathbb{V}}$ is given by the formal Hamiltonian $ H(x)=sum _{v in mathbb{V}}V_v(x_v)+frac{1}{2}sum_{substack{(v,v^prime) in mathbb{V} vsim v^prime}}W_vv^prime(x_v, x_v^prime). $ We show that under very mild conditions on the potentials we can still construct the corresponding Glauber dymamics.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Bielefeld

Year

2009

Author dc:creator

• Jin, Peng

Repository record source_url

https://pub.uni-bielefeld.de/record/2304858

OAI identifier oai:identifier

oai:pub.uni-bielefeld.de:2304858