The first passage time (FPT) problem for Brownian motion has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the boundary itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of new integrals via its differentiation or integration. In this thesis, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of new Volterra type equations and prove uniqueness for a subclass. The uniqueness result is then employed to demonstrate how certain functional transforms of the boundary affect the density function. Furthermore, we generalize a class of Fredholm integral equations and show its fundamental connection to the new class of Volterra equations. The Fredholm equations are then shown to provide a unified approach for computing the FPT distribution for linear, square root and quadratic boundaries. In addition, through the Fredholm equations, we analyze a polynomial expansion of the FPT density and employ a regularization method to solve for the coefficients. Moreover, the Volterra and Fredholm equations help us to examine a modification of the classical FPT under which we randomize, independently, the starting point of the Brownian motion. This randomized problem seeks the distribution of the starting point and takes the boundary and the (unconditional) FPT distribution as inputs. We show the existence and uniqueness of this random variable and solve the problem analytically for the linear boundary. The randomization technique is then drawn on to provide a structural framework for modeling mortality. We motivate the model and its natural inducement of 'risk-neutral' measures to price mortality linked financial products. Finally, we address the inverse FPT problem and show that in the case of the scale family of distributions, it is reducible to nding a single, base boundary. This result was applied to the exponential and uniform distributions to obtain analytical approximations of their corresponding base boundaries and, through the scaling property, for a general boundary.

The first passage time (FPT) problem for Brownian motion has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the boundary itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of new integrals via its differentiation or integration. In this thesis, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of new Volterra type equations and prove uniqueness for a subclass. The uniqueness result is then employed to demonstrate how certain functional transforms of the boundary affect the density function. Furthermore, we generalize a class of Fredholm integral equations and show its fundamental connection to the new class of Volterra equations. The Fredholm equations are then shown to provide a unified approach for computing the FPT distribution for linear, square root and quadratic boundaries. In addition, through the Fredholm equations, we analyze a polynomial expansion of the FPT density and employ a regularization method to solve for the coefficients. Moreover, the Volterra and Fredholm equations help us to examine a modification of the classical FPT under which we randomize, independently, the starting point of the Brownian motion. This randomized problem seeks the distribution of the starting point and takes the boundary and the (unconditional) FPT distribution as inputs. We show the existence and uniqueness of this random variable and solve the problem analytically for the linear boundary. The randomization technique is then drawn on to provide a structural framework for modeling mortality. We motivate the model and its natural inducement of 'risk-neutral' measures to price mortality linked financial products. Finally, we address the inverse FPT problem and show that in the case of the scale family of distributions, it is reducible to nding a single, base boundary. This result was applied to the exponential and uniform distributions to obtain analytical approximations of their corresponding base boundaries and, through the scaling property, for a general boundary.

The heterogeneous condensation of a vapour onto a substrate is a key step in a wide range of chemical and physical process that occur in both nature and technology. For example, dust and pollutant aerosol particles, ranging in size from several microns down to just a few nanometers, serve as cloud condensation nuclei in the atmosphere, and nanoscale structured surfaces provide templates for the controlled nucleation and growth of variety of complex materials. While much is known about the general features of heterogeneous nucleation onto macroscopic surfaces, much less is understood about both the dynamics and thermodynamics of nucleation involving nanoscale heterogeneities. The goal of this thesis is to understand the general features of condensation of vapours onto different types of nanoscale heterogeneity that range in degree of solubility from being insoluble, to partially miscible through to completely miscible. The heterogeneous condensation of the Lennard-Jones vapour onto an insoluble nanoscale seed particle is studied using a combination of molecular dynamics simulations and thermodynamic theory. The nucleation rate and free energy barrier are calculated from molecular dynamics using the mean first passage time method. These results show that the presence of a weakly interacting seed has no effect on the formation of small cluster embryos but accelerates the rate by lowering the free energy barrier of the larger clusters. A simple phenomenological model of film formation on a small seed is developed by extending the capillarity based liquid drop model. It captures the general features of heterogeneous nucleation, but a comparison with the simulation results show that the model significantly overestimates the height of the nucleation barrier while providing good estimates of the critical film size. A non-volatile liquid drop model that accounts for solution non-ideality is developed to describe the thermodynamics of partially miscible and fully miscible droplets in a solvent vapour. The model shows ideal solution drops dissolve always spontaneously, but partially miscible drops exhibit a free energy surface with two minima, associated with a partially dissolved drop and a fully dissolved drop, separated by a free energy barrier. The solubility transition between the two drops is shown to follow a hysteresis loop as a function of system volume similar to that observed in deliquescence. A simple lattice gas model describing the absorption of mono-layers of vapour onto the particle is also developed. Finally, molecular dynamics simulation of miscible and partially miscible binary Lennard-Jones mixtures are also used to study this system. For all cases studied, condensation onto the drop occurs spontaneously. Sub-monolayers of the solvent phase form when the system volume is large. At smaller system volumes, complete film formation is observed and the dynamics of film growth are dominated by cluster-cluster coalescence. Some degree of mixing into the core of the particle is observed for the miscible mixtures for all volumes. However, mixing of the solvent into the particle core only occurs below an onset volume for the partially miscible case, suggesting the presence of a solubility transition similar to the one described by the thermodynamic model.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics.

In this thesis we develop and use a continuum random walk framework to solve problems that are usually studied using a discrete random walk on a discrete lattice. Problems studied include; the time it takes for a random walker to be absorbed at a trap on a fractal lattice, the calculation of the spectral dimension for several different classes of networks, the calculation of the density of states for a multi-layered Bethe lattice and the relationship between diffusion exponents and a resistivity exponent that occur in relevant power laws. The majority of the results are obtained by deriving an expression for a Laplace transformed Green’s function or first passage time, and then using Tauberian theorems to find the relevant asymptotic behaviour. The continuum framework is established by studying the diffusion equation on a 1-d bar with non-homogeneous boundary conditions. The result is extended to model diffusion on networks through linear algebra. We derive the transformation linking the Green’s functions and first passage time results in the continuum and discrete settings. The continuum method is used in conjunction with renormalization techniques to calculate the time taken for a random walker to be absorbed at a trap on a fractal lattice and also to find the spectral dimension of new classes of networks. Although these networks can be embedded in the d- dimensional Euclidean plane, they do not have a spectral dimension equal to twice the ratio of the fractal dimension and the random walk dimension when the random walk on the network is transient. The networks therefore violate the Alexander-Orbach law. The fractal Einstein relationship (a relationship relating a diffusion exponent to a resistivity exponent) also does not hold on these networks. Through a suitable scaling argument, we derive a generalised fractal Einstein relationship which holds for our lattices and explains anomalous results concerning transport on diffusion limited aggregates and Eden trees.

Organisms have movements that are usually modeled by particles’ random walks. Under some mathematical technical assumptions the movements are described by diffusion equations. However, empirical data often show that the movements are not simple random walks. Instead, they are correlated random walks and are described by telegraph equations. This thesis considers telegraph equations with and without bias corresponding to correlated random walks with and without bias. Analytical solutions to the equations with absorbing boundary conditions and their mean passage times are obtained. Numerical simulations of the corresponding correlated random walks are also performed. The simulation results show that the solutions are approximated very well by the corresponding correlated random walks and the mean first passage times are highly consistent with those from simulations on the corresponding random walks. This suggests that telegraph equations can be a good model for organisms with the movement pattern of correlated random walks. Furthermore, utilizing the consistency of mean first passage times, we can estimate the parameters of telegraph equations through the mean first passage time, which can be estimated through experimental observation. This provides biologists an easy way to obtain parameter values. Finally, this thesis analyzes the velocity distribution and correlations of movement steps of amoebas, leaving fitting the movement data to telegraph equations as future work.

We consider the motion of a Brownian particle trapped in an arbitrary bounded two or three-dimensional domain, whose boundary is reflecting except for a small absorbing window through which the particle can escape. We use the method of matched asymptotic expansions to calculate the mean first passage time, defined as the time taken for the Brownian particle to escape from the domain through the absorbing window. This is known as the narrow escape problem. Since the mean escape time diverges as the window shrinks, the calculation is a singular perturbation problem. We extend our results to include N absorbing windows of varying length in two dimensions and varying radius in three dimensions. We present findings in two dimensions for the unit disk, unit square and ellipse and in three dimensions for the unit sphere. The narrow escape problem has various applications in many fields including finance, biology, and statistical mechanics. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis solves estimation and control problems in computational neuroscience, mathematically dealing with the first-passage times of diffusion stochastic processes. We first derive estimation algorithms for model parameters from first-passage time observations, and then we derive algorithms for the control of first-passage times. Finally, we solve an optimal design problem which combines elements of the first two: we ask how to elicit first-passage times such as to facilitate model estimation based on said first-passage observations. The main mathematical tools used are the Fokker-Planck partial differential equation for evolution of probability densities, the Hamilton-Jacobi-Bellman equation of optimal control and the adjoint optimization principle from optimal control theory. The focus is on developing computational schemes for the solution of the problems. The schemes are implemented and are tested for a wide range of parameters.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21543号 / 理博第4450号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 福島 竜輝, 教授 熊谷 隆, 教授 牧野 和久 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM