<mods:mods xmlns:mods="http://www.loc.gov/mods/v3" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-4.xsd"><mods:titleInfo><mods:title>“A Derivation and Analysis of the N-point Hierarchy For Conservation Laws And Multistable Dynamics with White Noise in the Allen-Cahn Equation</mods:title></mods:titleInfo><mods:name type="personal"><mods:namePart>Caginalp, Carey</mods:namePart><mods:role><mods:roleTerm type="text">creator</mods:roleTerm></mods:role></mods:name><mods:name type="personal"><mods:namePart>Menon, Govind</mods:namePart><mods:role><mods:roleTerm type="text">Advisor</mods:roleTerm></mods:role></mods:name><mods:name type="personal"><mods:namePart>Dafermos, Constantine</mods:namePart><mods:role><mods:roleTerm type="text">Reader</mods:roleTerm></mods:role></mods:name><mods:name type="personal"><mods:namePart>Kaspar, David</mods:namePart><mods:role><mods:roleTerm type="text">Reader</mods:roleTerm></mods:role></mods:name><mods:name type="corporate"><mods:namePart>Brown University. Department of Applied Mathematics</mods:namePart><mods:role><mods:roleTerm type="text">sponsor</mods:roleTerm></mods:role></mods:name><mods:originInfo><mods:copyrightDate>2017</mods:copyrightDate></mods:originInfo><mods:physicalDescription><mods:extent>xi, 106 p.</mods:extent><mods:digitalOrigin>born digital</mods:digitalOrigin></mods:physicalDescription><mods:note type="thesis">Thesis (Ph. D.)--Brown University, 2017</mods:note><mods:genre authority="aat">theses</mods:genre><mods:abstract>The first part of this thesis concerns conservation laws subject to random initial conditions. Using a&#13;
discrete set of values for the solution, one can derive a hierarchy of equations in terms of the states.&#13;
This hierarchy involves the n-point function, the probability that the solution takes on various&#13;
values at different positions, in terms the n+1-point function. A second approach involves obtaining&#13;
statistics on the n-point function with shocks. This approach yields a hierarchy of equations in&#13;
which the time derivative of the n-point function is expressed in terms of the spatial derivatives of&#13;
the n+1 and lower functions. The hierarchy of equations remains valid through collisions of shocks.&#13;
In the second part of this thesis, we study the influence of multistable dynamics under the influence&#13;
of white noise in a model of an ODE with a cubic nonlinearity. A prototypical bistable dynamic is&#13;
replaced by white noise to obtain a stochastic di¤erential equation. A key question is whether there&#13;
is a steady state, or invariant measure, and if so, what form it takes. We show that under the limit&#13;
of a vanishing amount of this white noise, there is indeed an invariant measure concentrated as a&#13;
Dirac type measure at the most stable equilibria if fluctations are uniform, or at multiple equilibria&#13;
if they are not.</mods:abstract><mods:subject authority="fast" authorityURI="http://id.worldcat.org/fast" valueURI="http://id.worldcat.org/fast/00893484"><mods:topic>Differential equations, Partial</mods:topic></mods:subject><mods:subject><mods:topic>Stochastics</mods:topic></mods:subject><mods:subject authority="fast" authorityURI="http://id.worldcat.org/fast" valueURI="http://id.worldcat.org/fast/01077737"><mods:topic>Probabilities</mods:topic></mods:subject><mods:language><mods:languageTerm authority="iso639-2b">English</mods:languageTerm></mods:language><mods:recordInfo><mods:recordContentSource authority="marcorg">RPB</mods:recordContentSource><mods:recordCreationDate encoding="iso8601">20170616</mods:recordCreationDate></mods:recordInfo><mods:identifier type="doi">10.7301/Z0N014ZJ</mods:identifier><mods:accessCondition type="rights statement" xlink:href="http://rightsstatements.org/vocab/InC/1.0/">In Copyright</mods:accessCondition><mods:accessCondition type="restriction on access">Collection is open for research.</mods:accessCondition><mods:typeOfResource authority="primo">dissertations</mods:typeOfResource></mods:mods>