Approximation and blow-up problems in stochastic differential equations

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Publication Type dissertation
School or College College of Science
Department Mathematics
Author Bezdek, Pavel
Title Approximation and blow-up problems in stochastic differential equations
Date 2016
Description In the first part of this work, we will show weak convergence of probability measures. The measure corresponding to the solution of the following one-dimensional nonlinear stochastic heat equation $\frac{\partial}{\partial t} u_{t}(x) = \frac{\kappa}{2} \frac{\partial^2}{\partial x^2} u_{ t}(x) + \sigma(u_{t}(x))\eta_\alpha$ with colored noise $\eta_\alpha$ will converge to the measure corresponding to the solution of the same equation but with white noise $\eta$, as $\alpha \uparrow 1$. Function $\sigma$ is taken to be Lipschitz and the Gaussian noise $\eta_\alpha$ is assumed to be colored in space and its covariance is given by $\operatorname{E} \left[ \eta_\alpha(t,x) \eta_\alpha(s,y) \right] = \delta(t-s) f_\alpha(x-y)$ where $f_\alpha$ is the Riesz kernel $f_\alpha(x) \propto 1/\left|x\right|^\alpha$. We will work with the classical notion of weak convergence of measures, that is convergence of probability measures on a space of continuous functions with compact domain and sup-norm topology. We will also state a result about continuity of measures in $\alpha$, for $\alpha \in (0,1)$. In the second part of this work, we will show existence and blow-up of the solution to $\frac{\partial}{\partial t} u_{t}(x) = \mathcal{L} u_{ t}(x) + \sigma(u_{t}(x))\eta$ on a circle with white noise $\eta$. The operator $\mathcal{L}$ is taken to be the generator of a L\'evy process and $\sigma$ is a nonlinear function of form $\sigma(x) \propto \left|x\right|^\gamma$, for $\gamma>1$. We will develop precise condition for existence or blow-up of the solution in terms of $\gamma$ and the L\'evy process corresponding to $\mathcal{L}$.
Type Text
Publisher University of Utah
Subject approximation; blow-up; Levy generator; stochastic heat equation
Dissertation Name Doctor of Philosophy
Language eng
Rights Management ©Pavel Bezdek
Format Medium application/pdf
Format Extent 683,112 bytes
Identifier etd3/id/4182
ARK ark:/87278/s6x385ts
Setname ir_etd
ID 197728
Reference URL https://collections.lib.utah.edu/ark:/87278/s6x385ts
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