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2014
Kouritzin, Michael, Ren, Y.X.
Let ℓ be Lebesgue measure and X=(Xt,t≥0;Pμ) be a supercritical, superstable process corresponding to the operator −(−Δ)α/2u+βu−ηu2 on Rd with constants β,η>0 and α∈(0,2]. Put View the MathML source, which for each smallθ is an a.s. convergent complexvalued martingale with limit View the MathML...

A law of the iterated logarithm for stochastic processes defined by differential equations with a small parameter
Download1994
Heunis, A.J., Kouritzin, Michael
Consider the following random ordinary differential equation: X˙ϵ(τ)=F(Xϵ(τ),τ/ϵ,ω)subject toXϵ(0)=x0, where {F(x,t,ω),t≥0} are stochastic processes indexed by x in Rd, and the dependence on x is sufficiently regular to ensure that the equation has a unique solution Xϵ(τ,ω) over the interval...

1997
Herein, an averaging theory for the solutions to Cauchy initial value problems of arbitrary order,εdependent parabolic partial differential equations is developed. Indeed, by directly developing bounds between the derivatives of the fundamental solution to such an equation and derivatives of the...

2002
In the context of simulating the transport of a chemical or bacterial contaminant through a moving sheet of water, we extend a wellestablished method of approximating reactiondiffusion equations with Markov chains by allowing convection, certain Poisson measure driving sources and a larger...

1997
Kouritzin, Michael, Dawson, Donald
A general Hilbertspacebased stochastic averaging theory is brought forth herein for arbitraryorder parabolic equations with (possibly long range dependent) random coefficients. We use regularity conditions onView the MathML sourcewhich are slightly stronger than those required to prove...

2004
Kouritzin, Michael, Long, H., Sun, W.
Herein, we consider direct Markov chain approximations to the Duncan–Mortensen–Zakai equations for nonlinear filtering problems on regular, bounded domains. For clarity of presentation, we restrict our attention to reflecting diffusion signals with symmetrizable generators. Our Markov chains are...

2008
In this paper, we give a direct derivation of the Duncan–Mortensen–Zakai filtering equation, without assuming right continuity of the signal, nor its filtration, and without the usual finite energy condition. As a consequence, the Fujisaki–Kallianpur–Kunita equation is also derived. Our results...

2013
Kouritzin, Michael, Wu, B., Newton, F.
Herein, we propose generating CAPTCHAs through random field simulation and give a novel, effective and efficient algorithm to do so. Indeed, we demonstrate that sufficient information about word tests for easy human recognition is contained in the site marginal probabilities and the...

2005
Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuousdiscrete filtering problems. Suppose that t→Xt is a Markov process and we wish to calculate the measurevalued process t→μt(⋅)≐P{Xt∈⋅σ{Ytk, tk≤t}}, where tk=kɛ and Ytk is a distorted,...

Rates of convergence in a central limit theorem for stochastic processes defined by differential equations with a small parameter
Download1992
Kouritzin, Michael, Heunis, A.J.
Let μ be a positive finite Borel measure on the real line R. For t ≥ 0 let et · E1 and E2 denote, respectively, the linear spans in L2(R, μ) of {eisx, s > t} and {eisx, s < 0}. Let θ: R → C such that ∥θ∥ = 1, denote by αt(θ, μ) the angle between θ · et · E1 and E2. The problems considered here...