{"id":{"repo_id":"bielefeld","oai_identifier":"oai:pub.uni-bielefeld.de:2304858"},"canonical_url":"https://search.dev.ndltd.org/etd/bielefeld/oai:pub.uni-bielefeld.de:2304858","repository":{"repo_id":"bielefeld","name":"Universität Bielefeld","base_url":"https://pub.uni-bielefeld.de/oai"},"display":{"title":"Stochastic dynamics with singular lower order terms in finite and infinite dimensions","abstract":"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.","abstract_html":"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 $<span class=\"etd-inline-math\">nabla (a(t,x) cdot nabla u(t,x))+b(t,x)cdot nabla u(t,x)+V(t,x) u(t,x)-partial<sub>t</sub>u(t,x)=0 eqno (1)</span>$ in the domain $[0,T]times mathbb{R}^{d}$, where $T&lt;infty$. We assume that the matrix $a(t,x)=(a_{ij} t,x))$ is symmetric and uniformly elliptic, H&quot;older continuous in $t$, $x$ and $frac{partial}{partial x_{i}}a_{ij}(t,x)$ are bounded and H&quot;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 $<span class=\"etd-inline-math\"> dX<sub>t</sub>=dW<sub>t</sub>+B(t, X<sub>t</sub>)dt, X<sub>s</sub>=x, eqno (2)</span>$ 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 $alpha&lt;frac{1}{2}$. 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 $<span class=\"etd-inline-math\"> H(x)=sum <sub>v in mathbb{V}</sub>V<sub>v</sub>(x<sub>v</sub>)+frac{1}{2}sum<sub>substack{(v,v<sup>prime</sup>) in mathbb{V} vsim v<sup>prime</sup>}</sub>W<sub>vv<sup>prime</sup></sub>(x<sub>v</sub>, x<sub>v<sup>prime</sup></sub>). </span>$ We show that under very mild conditions on the potentials we can still construct the corresponding Glauber dymamics.","abstract_has_math":true,"creators":["Jin, Peng"],"institution":"Universität Bielefeld","degree_name":null,"degree_level":"thesis.doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-11-12","date_published":"2009-11-12","updated_at":"2026-07-27T18:49:52Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://pub.uni-bielefeld.de/record/2304858","outbound_label":"Repository record","outbound_source":"source_url"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Jin, Peng"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["Universitätsbibliothek Bielefeld"]},{"key":"dc:type","label":"Dc Type","values":["doctoralThesis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["thesis.doctoral"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Universität Bielefeld"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Stochastic dynamics with singular lower order terms in finite and infinite dimensions"]}]}],"canonical_facts":{"dc:creator":["Jin, Peng"],"dc:description.abstract":["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."],"dc:format.medium":["application/pdf"],"dc:publisher":["Universitätsbibliothek Bielefeld"],"dc:title":["Stochastic dynamics with singular lower order terms in finite and infinite dimensions"],"dc:type":["doctoralThesis"],"thesis:degree_level":["thesis.doctoral"],"thesis:institution_name":["Universität Bielefeld"]},"updated_at":"2026-07-27T18:49:52Z"}