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The relatively new subject of stochastic differential
equations has
increasing importance in both theory and
applications. The subject
draws upon two main sources, probability/stochastic
processes and
differential equations/dynamical systems. There
exists a significant
``culture gap" between the corresponding research
communities. The
objective of the dissertation project is to present a
concise yet
mostly self-contained theory of stochastic
differential equations
from the differential equations/dynamical systems
point of view,
primarily incorporating semigroup theory and
functional analysis
techniques to study the solutions. Prerequisites from
probability/stochastic processes are developed as needed.
For continuous-time stochastic
processes whose random variables are (Lebesgue)
absolutely
continuous, the Fokker-Planck equation is employed to
study the
evolution of the densities, with applications to
predator-prey
models with noisy coefficients.
equations has
increasing importance in both theory and
applications. The subject
draws upon two main sources, probability/stochastic
processes and
differential equations/dynamical systems. There
exists a significant
``culture gap" between the corresponding research
communities. The
objective of the dissertation project is to present a
concise yet
mostly self-contained theory of stochastic
differential equations
from the differential equations/dynamical systems
point of view,
primarily incorporating semigroup theory and
functional analysis
techniques to study the solutions. Prerequisites from
probability/stochastic processes are developed as needed.
For continuous-time stochastic
processes whose random variables are (Lebesgue)
absolutely
continuous, the Fokker-Planck equation is employed to
study the
evolution of the densities, with applications to
predator-prey
models with noisy coefficients.