TY  - THES
AB  - 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 $$nabla (a(t,x) cdot nabla u(t,x))+b(t,x)cdot nabla u(t,x)+V(t,x) u(t,x)-partial_{t}u(t,x)=0 eqno (1)$$ 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.
DA  - 2009
KW  - Stochastische Differentialgleichung
KW  - Lineare parabolische Differentialgleichung
KW  - Diffusionsprozess
KW  - Dirichlet-Raum
KW  - Second-order parabolic equations
KW  - Weak fundamental solutions
KW  - Stochastic differential equations
KW  - Diffusion processes
KW  - Glauber dynamics
LA  - eng
PY  - 2009
TI  - Stochastic dynamics with singular lower order terms in finite and infinite dimensions
UR  - https://nbn-resolving.org/urn:nbn:de:hbz:361-15941
Y2  - 2024-11-22T02:00:10
ER  -