Date of Completion

4-30-2013

Embargo Period

4-30-2013

Keywords

meyers inequality, stable-like processes, stability, pathwise uniqueness, SDEs with jumps

Major Advisor

Richard F. Bass

Associate Advisor

Maria Gordina

Associate Advisor

Iddo Ben-Ari

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

We consider a class of symmetric stable-like operators of order $\alpha\in (0,2)$. Let $$\sE(u,u)=\int_{\R^d}\int_{\R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+\al}} \, dy\, dx$$ be the Dirichlet form for a stable-like operator, let $$\Gamma u(x)=\Big(\int_{\R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+\al}} \, dy\Big)^{1/2},$$ let $\sL$ be the associated infinitesimal generator, and suppose $A(x,y)$ is jointly measurable, symmetric, bounded, and bounded below by a positive constant. We prove that if $u$ is the weak solution to $\sL u=h$, then $\Gamma u\in L^p$ for some $p>2$. As an application, we prove strong stability results for stable-like operators. If $A$ is perturbed slightly, we give explicit bounds on how much the semigroup and fundamental solution are perturbed.\\ For $\alpha\in (0,1)$, we consider the one-dimensional jump stochastic differential equation driven by one-sided stable processes of order $\alpha$: \[dX_t= \phi(X_{t-})\ dZ_t.\] We prove that pathwise uniqueness holds for this equation under the assumptions that $\phi$ is continuous, non-decreasing and positive on $\R$. A counter-example is given to show that the positivity of $\phi$ is crucial for pathwise uniqueness to hold. \\

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