Independent Diffusion Processes Research Articles

Conditioning two diffusion processes with respect to their first-encounter properties

We consider two independent identical diffusion processes that annihilate upon meeting in order to study their conditioning with respect to their first-encounter properties. For the case of finite horizon T < +∞, the maximum conditioning consists in imposing the probability P*(x, y, T) that the two particles are surviving at positions x and y at time T, as well as the probability γ*(z, t) of annihilation at position z at the intermediate times t ∈ [0, T]. The adaptation to various conditioning constraints that are less-detailed than these full distributions is analyzed via the optimization of the appropriate relative entropy with respect to the unconditioned processes. For the case of infinite horizon T = +∞, the maximum conditioning consists in imposing the first-encounter probability γ*(z, t) at position z at all finite times , whose normalization [1 − S*(∞)] determines the conditioned probability S*(∞) ∈ [0, 1] of forever-survival. This general framework is then applied to the explicit cases where the unconditioned processes are respectively two Brownian motions, two Ornstein–Uhlenbeck processes, or two tanh-drift processes, in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, the link with the stochastic control theory is described via the optimization of the dynamical large deviations at level 2.5 in the presence of the conditioning constraints that one wishes to impose.

Brownian snails with removal: epidemics in diffusing populations

Two stochastic models of susceptible/infected/removed (SIR) type are introduced for the spread of infection through a spatially-distributed population. Individuals are initially distributed at random in space, and they move continuously according to independent diffusion processes. The disease may pass from an infected individual to an uninfected individual when they are sufficiently close. Infected individuals are permanently removed at some given rate $\alpha$. Such processes are reminiscent of so-called frog models, but differ through the action of removal, as well as the fact that frogs jump whereas snails slither. Two models are studied here, termed the `delayed diffusion' and the `diffusion' models. In the first, individuals are stationary until they are infected, at which time they begin to move; in the second, all individuals start to move at the initial time $0$. Using a perturbative argument, conditions are established under which the disease infects a.s. only finitely many individuals. It is proved for the delayed diffusion model that there exists a critical value $\alpha_c\in(0,\infty)$ for the survival of the epidemic.

The coexistence of fast and slow diffusion processes in the life cycle of Aedes aegypti mosquitoes

A new model that describes the life cycle of mosquitoes of the species Aedes aegypti, main carriers of vector-borne diseases, is proposed. The novelty is to include in the model the coexistence of two independent diffusion processes, one fast which obeys the constitutive Fick’s law, the other slow which satisfies the Cattaneo evolution equation. The analysis of the corresponding ODE model shows the overall stability of the Mosquitoes-Free Equilibrium (MFE), together with the local stability of the other equilibrium point admitted by the system. Traveling wave type solutions have been investigated, providing an estimate of the minimal speed for which there are monotone waves that connect the homogeneous equilibria allowed by the system. A special section is dedicated to the analysis of the hyperbolic model obtained neglecting the fast diffusive contribution. This particular case is suitable to describe the biological process as it overcomes the paradox of infinite speed propagation, typical of parabolic systems. Several numerical simulations compare the existing models in the literature with those presented in this discussion, showing that although the generalized model is parabolic, the associated wave velocity admits upper bound represented by the speed of the waves linked to the classic parabolic model present in the published literature, so the presence of a slow flux together with a fast one slows down the speed with which a population spreads.

Ruin probability in the delayed renewal risk model perturbed by a diffusion process

In this paper, we consider the risk model perturbed by an independent diffusion process with a time delay in the arrival of the first claim. We derive the distribution of the counting process of the risk process, the integro-differential equations of the ruin probabilities and generalize its defective renewal equations. With claim amounts following exponential and mixed exponential distributions, explicit expressions and asymptotic properties of the ruin probabilities are derived. Numerical illustrations of the ruin probabilities are proposed when claim amounts are exponentially and mixed exponentially distributed. We extend the results to the case of the delayed renewal risk model with exchangeable risks which captures the possible dependence between inter-arrival times of the claims and derive the associated ruin probabilities.

Quantitative Propagation of Chaos in a Bimolecular Chemical Reaction-Diffusion Model

We study a stochastic system of $N$ interacting particles which models bimolecular chemical reaction-diffusion. In this model, each particle $i$ carries two attributes: the spatial location $X_t^i\in \mathbb{T}^d$, and the type $\Xi_t^i\in \{1,\ldots,n\}$. While $X_t^i$ is a standard (independent) diffusion process, the evolution of the type $\Xi_t^i$ is described by pairwise interactions between different particles under a series of chemical reactions described by a chemical reaction network. We prove that, as $N \to \infty$, the stochastic system has a mean field limit which is described by a nonlocal reaction-diffusion partial differential equation. In particular, we obtain a quantitative propagation of chaos result for the interacting particle system. Our proof is based on the relative entropy method used recently by Jabin and Wang [Invent. Math., 214 (2018), pp. 523--591]. The key ingredient of the relative entropy method is a large deviation estimate for a special partition function, which was proved previously by combinatorial estimates. We give a simple probabilistic proof based on a novel martingale argument.

Finite element analysis of the water diffusion behaviour in pigmented epoxy coatings under alternating hydrostatic pressure

The water diffusion behaviour in pigmented epoxy coatings under alternating hydrostatic pressure (AHP) has been calculated by the finite element method. The water diffusion in the coating under AHP follows the non-Fickian diffusion, which is caused by cracks and superposed by the occurrences of many independent Fickian diffusion processes. The cracks largely contribute to the water absorption of the coating. If the occurrence time of the cracks is measured by the experiment, the lifetime of the coating under AHP can be quickly determined.

On Intersections of Independent Nondegenerate Diffusion Processes

Let X(1)={X(1)(s),s∈ℝ+}andX(2)={X(2)(t),t∈ℝ+} be two independent nondegenerate diffusion processes with values in ℝd. The existence and fractal dimension of intersections of the sample paths of X(1) and X(2) are studied. More generally, let E1,E2⊆(0,∞)andF⊂ℝd be Borel sets. A necessary condition and a sufficient condition for P{X(1)(E1)∩X(2)(E2)∩F≠∅}>0 are proved in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 × E2 × F in the metric space (ℝ+×ℝ+×ℝd,pˆ),wherepˆ is an unsymmetric metric defined in ℝ+×ℝ+×ℝd. Under reasonable conditions, results resembling those of Browian motion are obtained.

Preliminary test and estimation in some multifactor diffusion processes

We consider the general inference problem of the drift parameters matrices of m independent multivariate diffusion processes. In particular, we develop preliminary test and Stein-type estimation theory for m-parameters matrices which are suspected to satisfy some constraints. By using asymptotic distributional quadratic risk criterion, we analyze the relative performance of the pretest and shrinkage estimators. Further, we generalize some inference problems recently studied for a single multifactor Vasicek model. Finally, we carry out some simulation studies whose results corroborate our theoretical findings for observation periods of small and moderate lengths of time.

Shrinkage strategies in some multiple multi-factor dynamical systems

In this paper, we are interested in estimation problem for the drift parameters matrices of m independent multivariate diffusion processes. More specifically, we consider the case where the m -parameters matrices are supposed to satisfy some uncertain constraints. Given such an uncertainty, we develop shrinkage estimators which improve over the performance of the maximum likelihood estimator (MLE). Under an asymptotic distributional quadratic risk criterion, we study the relative dominance of the established estimators. Further, we carry out simulation studies for observation periods of small and moderate lengths of time that corroborate the theoretical finding for which shrinkage estimators outperform over the MLE. The proposed method is useful in model assessment and variable selection.

On the expected penalty function for a risk model perturbed by diffusion

In this paper, a continuous time risk model perturbed by an independent diffusion (Wiener) process is considered. The claim number process is assumed to be a generalised Erlang(2) process. Both the expected penalty functions with zero initial surplus and the Laplace transforms of the expected penalty functions are obtained from an integro-differential equations system. The analytic expressions for expected penalty functions are derived when the claim amounts are exponentially distributed. Moreover, the closed form expressions for the ruin probabilities are proposed. Finally, numerical results are also provided to illustrate the applicability of the main result.

Interacting Particle Systems and Yaglom Limit Approximation of Diffusions with Unbounded Drift

We study the existence and the exponential ergodicity of a general interacting particle system, whose components are driven by independent diffusion processes with values in an open subset of $\mathbb{R}^d$, $d\geq1$. The interaction occurs when a particle hits the boundary: it jumps to a position chosen with respect to a probability measure depending on the position of the whole system. Then we study the behavior of such a system when the number of particles goes to infinity. This leads us to an approximation method for the Yaglom limit of multi-dimensional diffusion processes with unbounded drift defined on an unbounded open set. While most of known results on such limits are obtained by spectral theory arguments and are concerned with existence and uniqueness problems, our approximation method allows us to get numerical values of quasi-stationary distributions, which find applications to many disciplines. We end the paper with numerical illustrations of our approximation method for stochastic processes related to biological population models.

Calcium Independent Substrate and Product Diffusion Process of Secretary Phospholipase A2 from Taiwan Cobra

Understanding how membrane lipids can be delivered to the active site distant from the interfacial binding surface of the enzyme and how the products got released from the active site are important to depict the interfacial enzyme reaction mechanism. Based on the crystal structure of the trimeric complex structure of the cobra phospholipase A2 (PLA2) from Naja atra with the enyzymatic substrate of diacylheptanoyl phosphatidylcholine (PC) and products of lyso PC and fatty acids (minic by SDS), we suggest that promiscuous bindings of phospholipids to the interfacial enzyme may boost the lipid desorption process via a cooperative hydrophobic interaction among the hydrocarbon chains of phospholipids and that with the interfacial surface of the enzyme. We also show that phospholipids in membranes surface with high curvature can promote the diffusion of the lipid into the substrate binding hydrophobic channel of cobra PLA2 in a calcium independent manner. The PC substrate binding site within the channel without calcium is distinct from that in the presence of calcium as one compares its binding position with that of transition binding intermediates. Interestingly, calcium appears to destabilize the binding of both substrate and product binding at the hydrophobic channel even though it is required for the enzymatic catalysis. Our results suggest that the calcium independent lipid diffusion process play an important role in the interfacial binding activation of secretary PLA2 and shed new light for the future depiction of the energy landscape.

On a multi-channel change-point problem

A continuous change-point problem is studied in which N independent diffusion processes X j are observed. Each process X j is associated with a “channel”, each has an unknown piecewise constant drift and the unit diffusion coefficient. All the channels are connected only by a common change-point of drift. As the result, a change-point problem is defined in which the unknown and unidentifiable drift forms a 2N-dimensional nuisance parameter. The asymptotics of the minimax rate in estimating the change-point is studied as N → ∞. This rate is compared with the case of the known drift. This problem is a special case of an open change-point detection problem in the high-dimensional diffusion with nonparametric drift.

Diffusion mechanism of sodium ions in compacted montmorillonite under different NaCl concentration

Compacted bentonite, the major mineral being montmorillonite, is a candidate buffer material for geological disposal of high-level radioactive waste. The diffusion behavior of radionuclides in the compacted montmorillonite is an important issue to be clarified for the safety assessment of geological disposal. In this study, one-dimensional, non-steady diffusion experiments using 22Na at different diffusion temperatures and the measurement of the basal spacings by XRD were conducted for Na-montmorillonite saturated with NaCl solutions of different concentrations. Some basal spacings of the montmorillonite were found to decrease from 1.88 to 1.56 nm as the NaCl concentration increased from 0 to 0.1 M. The apparent self-diffusion coefficients at 298 K obtained in this study slightly increased as the NaCl concentration increased from 0 to 0.5 M, while the activation energies, calculated from the temperature dependences of the diffusion coefficients, were found to be 14, 22, and 17 kJ mol −1 at NaCl concentrations of 0, 0.1, and 0.5 M, respectively. These NaCl concentration dependences of the diffusion coefficients and their activation energies can be explained by assuming three independent diffusion processes (diffusion in pore water, on external surface of montmorillonite, and in interlayer of montmorillonite).

A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := {x ∈ [0, 1]N: ∑i≥1xi = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := { x ∈ [0, 1] N : ∑ i≥1 x i = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

Limit theorems for number of diffusion processes, which did not absorb by boundaries

We have random number of independent diffusion processes with absorption on boundaries in some region at initial time t = 0. The initial numbers and positions of processes in region is defined by the Poisson random measure. It is required to estimate the number of the unabsorbed processes for the fixed time τ > 0. The Poisson random measure depends on τ and τ → ∞.

Dual Diffusion Model for Single-Cell Recording Data From the Superior Colliculus in a Brightness-Discrimination Task

Monkeys made saccades to one of two peripheral targets based on the brightness of a central stimulus. Task difficulty was manipulated by varying the ratio of stimulus black-and-white pixels. Correct response probability for two monkeys varied directly with difficulty. Deep layer SC neurons exhibited robust presaccadic activity the magnitude of which was unaffected by task difficulty when the stimulus specified a saccade toward a target within the neuron's response field. Activity after stimuli specifying saccades to targets outside the response field was affected by task difficulty, increasing as the task became more difficult. A quantitative model derived from studies of human decision-making was fit to the behavioral data. The model assumes that information from the stimulus drives two independent diffusion processes. Simulated paths from the model were compared with neuron activity, assuming that firing rate is linearly related to position in the accumulation process. The firing rate data show delayed availability of discriminative information for fast, intermediate, and slow decisions when activity is aligned on the stimulus and very small differences in discriminative information when aligned on the saccade. The model produces exactly these patterns of results. The accumulation process is highly variable, allowing the process both to make errors, as is the case for the behavioral performance, and also to account for the firing rate results. Thus the dual diffusion model provides a quantitative account for both the behavior in a simple decision-making task as well as the patterns of activity in competing populations of neurons.

Prescriptions for optimizing intensities of short-lived RIBs at ISOL based research facilities

In order to generate useful intensities of short-lived radioactive species at Isotope Separator On-Line (ISOL) based research facilities, species of interest must be diffused from the interior of the production target-material, effusively transported to an ion source, ionized, extracted, mass-analyzed, and accelerated to research energies in times commensurate with their lifetimes. The intensities at such facilities are principally limited by decay losses associated with times required for diffusion-release from target materials and transport from the target to the ion source and by the maximum permissible primary beam power depositional density that can be tolerated without deleteriously affecting target integrity or ion source efficiency. Consequently, it is imperative to minimize delay times associated with the independent diffusion and effusive-flow processes by choosing small dimensioned (short-diffusion length) highly refractory target materials and formatting them in highly permeable mechanically and thermally robust structures that can withstand high-power beam irradiation for extended periods of time. This article provides basic information on methods, procedures and principles for selecting target materials; for designing fast diffusion-release targets; for designing fast effusive-flow vapor transport systems; and introduces methods for controlling target temperatures as required for optimizing intensities of short-lived radioactive ion beams at ISOL based research facilities.

The Gerber–Shiu function in a Sparre Andersen risk process perturbed by diffusion

We consider a Sparre Andersen risk process that is perturbed by an independent diffusion process, in which claim inter-arrival times have a generalized Erlang(n) distribution (i.e. as the sum of n independent exponentials, with possibly different means). This leads to a generalization of the defective renewal equations for the expected discounted penalty function at the time of ruin given by Tsai and Willmot [10,11] and Gerber and Shiu [21,22]. The limiting behavior of the expected discounted penalty function is studied, when the dispersion coefficient goes to zero. Finally, explicit results are given for the case where n=2.