Weak Solutions of Stochastic Differential Equations over the Field of p-Adic Numbers

Study of stochastic differential equations on the field of p-adic numbers was initiated by the second author and has been developed by the first author, who proved several results for the p-adic case, similar to the theory of ordinary stochastic integral with respect to Levy processes on the Euclidean spaces. In this article, we present an improved definition of a stochastic integral on the field and prove the joint (time and space) continuity of the local time for p-adic stable processes. Then we use the method of random time change to obtain sufficient conditions for the existence of a weak solution of a stochastic differential equation on the field, driven by the p-adic stable process, with a Borel measurable coefficient.Comment: To appear in Tohoku Math.

Spectral-element simulations of long-term fault slip: Effect of low-rigidity layers on earthquake-cycle dynamics

We develop a spectral element method for the simulation of long-term histories of spontaneous seismic and aseismic slip on faults subjected to tectonic loading. Our approach reproduces all stages of earthquake cycles: nucleation and propagation of earthquake rupture, postseismic slip and interseismic creep. We apply the developed methodology to study the effects of low-rigidity layers on the dynamics of the earthquake cycle in 2-D. We consider two cases: small (M ~ 1) earthquakes on a fault surrounded by a damaged fault zone and large (M ~ 7) earthquakes on a vertical strike-slip fault that cuts through shallow low-rigidity layers. Our results indicate how the source properties of repeating earthquakes are affected by the presence of a damaged fault zone with low rigidity. Compared to faults in homogeneous media, we find (1) reduction in the earthquake nucleation size, (2) amplification of slip rates during dynamic rupture propagation, (3) larger recurrence interval, and (4) smaller amount of aseismic slip. Based on linear stability analysis, we derive a theoretical estimate of the nucleation size as a function of the width and rigidity reduction of the fault zone layer, which is in good agreement with simulated nucleation sizes. We further examine the effects of vertically-stratified layers (e.g., sedimentary basins) on the nature of shallow coseismic slip deficit. Our results suggest that low-rigidity shallow layers alone do not lead to coseismic slip deficit. While the low-rigidity layers result in lower interseismic stress accumulation, they also cause dynamic amplification of slip rates, with the net effect on slip being nearly zero

Spectral-element modeling of spontaneous earthquake rupture on rate and state faults: Effect of velocity-strengthening friction at shallow depths

We develop a spectral-element methodology (SEM) for simulating dynamic rupture on rate and state faults and use it to study how the rupture is affected by a shallow fault region of steady-state velocity-strengthening friction. Our comparison of the developed SEM and a spectral boundary-integral method (BIM) for an anti-plane (two-dimensional) test problem shows that the two methods produce virtually identical solutions for the finest resolution we use and that the convergence with grid reduction of the developed SEM methodology is comparable to that of BIM. We also use the test problem to compare numerical resolution required for different state evolution laws and for linear slip-weakening friction. Using our three-dimensional implementation of the methodology, we find that a shallow velocity-strengthening fault region can significantly alter dynamic rupture and ground motion. The velocity-strengthening region suppresses supershear propagation at the free surface occurring in the absence of such region, which could explain the lack of universally observed supershear rupture near the free surface. In addition, the velocity-strengthening region promotes faster fall-off of slip velocity behind the rupture front and decreases final slip throughout the entire fault, causing a smaller average stress drop. The slip decrease is largest in the shallow parts of the fault, resulting in the depth profile of slip qualitatively consistent with observations of shallow co-seismic slip deficit. The shallow velocity-strengthening region also reduces the amplification of strong ground motion due to a low-velocity bulk structure

Discreteness-Induced Slow Relaxation in Reversible Catalytic Reaction Networks

Slowing down of the relaxation of the fluctuations around equilibrium is investigated both by stochastic simulations and by analysis of Master equation of reversible reaction networks consisting of resources and the corresponding products that work as catalysts. As the number of molecules $N$ is decreased, the relaxation time to equilibrium is prolonged due to the deficiency of catalysts, as demonstrated by the amplification compared to that by the continuum limit. This amplification ratio of the relaxation time is represented by a scaling function as $h = N \exp(-\beta V)$, and it becomes prominent as $N$ becomes less than a critical value $h \sim 1$, where $\beta$ is the inverse temperature and $V$ is the energy gap between a product and a resource

Synthetic Turing protocells: vesicle self-reproduction through symmetry-breaking instabilities

The reproduction of a living cell requires a repeatable set of chemical events to be properly coordinated. Such events define a replication cycle, coupling the growth and shape change of the cell membrane with internal metabolic reactions. Although the logic of such process is determined by potentially simple physico-chemical laws, the modeling of a full, self-maintained cell cycle is not trivial. Here we present a novel approach to the problem which makes use of so called symmetry breaking instabilities as the engine of cell growth and division. It is shown that the process occurs as a consequence of the breaking of spatial symmetry and provides a reliable mechanism of vesicle growth and reproduction. Our model opens the possibility of a synthetic protocell lacking information but displaying self-reproduction under a very simple set of chemical reactions