344 research outputs found
Central Limit Results for Jump-Diffusions with Mean Field Interaction and a Common Factor
A system of weakly interacting particles whose dynamics is given in terms
of jump-diffusions with a common factor is considered. The common factor is
described through another jump-diffusion and the coefficients of the evolution
equation for each particle depend, in addition to its own state value, on the
empirical measure of the states of the particles and the common factor. A
Central Limit Theorem, as , is established. The limit law is
described in terms of a certain Gaussian mixture. An application to models in
Mathematical Finance of self-excited correlated defaults is described
Infinite Dimensional Forward-Backward Stochastic Differential Equations and the KPZ Equation
Kardar-Parisi-Zhang (KPZ) equation is a quasilinear stochastic partial
differential equation(SPDE) driven by a space-time white noise. In recent years
there have been several works directed towards giving a rigorous meaning to a
solution of this equation. Bertini, Cancrini and Giacomin have proposed a
notion of a solution through a limiting procedure and a certain renormalization
of the nonlinearity. In this work we study connections between the KPZ equation
and certain infinite dimensional forward-backward stochastic differential
equations. Forward-backward equations with a finite dimensional noise have been
studied extensively, mainly motivated by problems in mathematical finance.
Equations considered here differ from the classical works in that, in addition
to having an infinite dimensional driving noise, the associated SPDE involves a
non-Lipschitz (namely a quadratic) function of the gradient. Existence and
uniqueness of solutions of such infinite dimensional forward-backward equations
is established and the terminal values of the solutions are then used to give a
new probabilistic representation for the solution of the KPZ equation
A Numerical Scheme for Invariant Distributions of Constrained Diffusions
Reflected diffusions in polyhedral domains are commonly used as approximate
models for stochastic processing networks in heavy traffic. Stationary
distributions of such models give useful information on the steady state
performance of the corresponding stochastic networks and thus it is important
to develop reliable and efficient algorithms for numerical computation of such
distributions. In this work we propose and analyze a Monte-Carlo scheme based
on an Euler type discretization of the reflected stochastic differential
equation using a single sequence of time discretization steps which decrease to
zero as time approaches infinity. Appropriately weighted empirical measures
constructed from the simulated discretized reflected diffusion are proposed as
approximations for the invariant probability measure of the true diffusion
model. Almost sure consistency results are established that in particular show
that weighted averages of polynomially growing continuous functionals evaluated
on the discretized simulated system converge a.s. to the corresponding
integrals with respect to the invariant measure. Proofs rely on constructing
suitable Lyapunov functions for tightness and uniform integrability and
characterizing almost sure limit points through an extension of Echeverria's
criteria for reflected diffusions. Regularity properties of the underlying
Skorohod problems play a key role in the proofs. Rates of convergence for
suitable families of test functions are also obtained. A key advantage of
Monte-Carlo methods is the ease of implementation, particularly for high
dimensional problems. A numerical example of a eight dimensional Skorohod
problem is presented to illustrate the applicability of the approach
Inviscid Large deviation principle and the 2D Navier Stokes equations with a free boundary condition
Using a weak convergence approach, we prove a LPD for the solution of 2D
stochastic Navier Stokes equations when the viscosity converges to 0 and the
noise intensity is multiplied by the square root of the viscosity. Unlike
previous results on LDP for hydrodynamical models, the weak convergence is
proven by tightness properties of the distribution of the solution in
appropriate functional spaces
Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise
The Freidlin-Wentzell large deviation principle is established for the
distributions of stochastic evolution equations with general monotone drift and
small multiplicative noise. As examples, the main results are applied to derive
the large deviation principle for different types of SPDE such as stochastic
reaction-diffusion equations, stochastic porous media equations and fast
diffusion equations, and the stochastic p-Laplace equation in Hilbert space.
The weak convergence approach is employed in the proof to establish the Laplace
principle, which is equivalent to the large deviation principle in our
framework.Comment: 31 pages, published in Appl. Math. Opti
Ultrasonic Bioreactor as a Platform for Studying Cellular Response
The need for tissue-engineered constructs as replacement tissue continues to grow as the average age of the world’s population increases. However, additional research is required before the efficient production of laboratory-created tissue can be realized. The multitude of parameters that affect cell growth and proliferation is particularly daunting considering that optimized conditions are likely to change as a function of growth. Thus, a generalized research platform is needed in order for quantitative studies to be conducted. In this article, an ultrasonic bioreactor is described for use in studying the response of cells to ultrasonic stimulation. The work is focused on chondrocytes with a long-term view of generating tissue-engineered articular cartilage. Aspects of ultrasound (US) that would negatively affect cells, including temperature and cavitation, are shown to be insignificant for the US protocols used and which cover a wide range of frequencies and pressure amplitudes. The bioreactor is shown to have a positive influence on several factors, including cell proliferation, viability, and gene expression of select chondrocytic markers. Most importantly, we show that a total of 138 unique proteins are differentially expressed on exposure to ultrasonic stimulation, using mass-spectroscopy coupled proteomic analyses. We anticipate that this work will serve as the basis for additional research which will elucidate many of the mechanisms associated with cell response to ultrasonic stimulation
Ultrasonic Bioreactor as a Platform for Studying Cellular Response
The need for tissue-engineered constructs as replacement tissue continues to grow as the average age of the world’s population increases. However, additional research is required before the efficient production of laboratory-created tissue can be realized. The multitude of parameters that affect cell growth and proliferation is particularly daunting considering that optimized conditions are likely to change as a function of growth. Thus, a generalized research platform is needed in order for quantitative studies to be conducted. In this article, an ultrasonic bioreactor is described for use in studying the response of cells to ultrasonic stimulation. The work is focused on chondrocytes with a long-term view of generating tissue-engineered articular cartilage. Aspects of ultrasound (US) that would negatively affect cells, including temperature and cavitation, are shown to be insignificant for the US protocols used and which cover a wide range of frequencies and pressure amplitudes. The bioreactor is shown to have a positive influence on several factors, including cell proliferation, viability, and gene expression of select chondrocytic markers. Most importantly, we show that a total of 138 unique proteins are differentially expressed on exposure to ultrasonic stimulation, using mass-spectroscopy coupled proteomic analyses. We anticipate that this work will serve as the basis for additional research which will elucidate many of the mechanisms associated with cell response to ultrasonic stimulation
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