Short-time Asymptotic Behavior Research Articles

Short-time asymptotic behavior of the Brox diffusion

Short-time asymptotic behavior of the Brox diffusion

ON THE TRANSITION DENSITY FUNCTION OF THE DIFFUSION PROCESS GENERATED BY THE GRUSHIN OPERATOR

The short-time asymptotic behavior of the transition density function of the diffusion process generated by the general Grushin operator will be investigated, by using its explicit expression in terms of expectation. Further the dependence on of the asymptotics will be seen.

An inverse problem in time-fractional diffusion equations with nonlinear boundary condition

This paper deals with the 1D time-fractional diffusion equations with nonlinear boundary condition. We first give an integral equation of the solution via the theta-function and eigenfunction expansion and establish the short time asymptotic behavior of the solution. We then verify the uniqueness of the inverse problem in determining the fractional order by the one endpoint observation whether the nonlinear boundary condition is known or not. Furthermore, in the case when the nonlinear boundary condition is known, we can establish the Lipschitz stability of the fractional order with respect to the measured data at the one endpoint by using the Lipschitz continuity of the solution with respect to the fractional order.

Concentration for Coulomb gases on compact manifolds

We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of Chafa\"i, Hardy and Ma\"ida on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green function. This last ingredient is not available on a manifold. We solve this problem by using the heat kernel and its short-time asymptotic behavior.

Very singular solution and short time asymptotic behaviors of nonnegative singular solutions for heat equation with nonlinear convection

In this paper we study the following Cauchy problem:ut=uxx+(un)x,(x,t)∈R×(0,∞),u(x,0)=0,x≠0, where parameter n≥0. Its nonnegative solution is called singular solution when u(x,t) satisfies the equation in the sense of distribution, initial conditions in the classical sense and also u(x,t) exhibits a singularity at the origin (0,0). As we know, the singular solution is called source-type solution if the initial is Mδ(x), where δ(x) is Dirac measure and constant M>0. The solution is called very singular solution if it possesses more singularity than that of source-type solution at the origin. Here we focus on what happens in the interactive effect between the diffusion and convection in a whole physical process. We find critical values n2<n1<n0 such that there exists unique source-type solution in the exponent range of 0<n<n0, while there exists no nonnegative singular solution if n≥n0. Only in the case of n2<n<n1 there exists a very singular solution, but in the case of n≥n1 or n≤n2 there is no solution that exhibits more singular than source-type solution at the origin. Furthermore we describe the short time asymptotic behavior of the singular solutions when such Cauchy problem is solvability for source-type solution or very singular solution.

Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension

In this paper we consider the following equation $$ u_t=(u^m)_{xx}+(u^n)_x, \ \ (x, t)\in \mathbb{R}\times(0, \infty) $$with a Dirac measure as initial data, i.e., $u(x,0)=\delta(x)$. The solution of the Cauchy problem is well-known assource-type solution. In the recent work [11] the authorstudied the existence and uniqueness of such kind of singularsolutions and proved that there exists a number $n_0=m+2$ suchthat there is a unique source-type solution to the equation when$0 \leq n < n_0$. Here our attention is focused on the nonexistenceand asymptotic behavior near the origin for a short time. We provethat $n_0$ is also a critical number such that there exits nosource-type solution when $n \geq n_0$ and describe the short timeasymptotic behavior of the source-type solution to the equationwhen $0 \leq n < n_0$. Our result shows that in the case ofexistence and for a short time, the source-type solution of suchequation behaves like the fundamental solution of the standardporous medium equation when $0 \leq n < m+1$, the uniqueself-similar source-type solution exists when $n = m+1$, and thesolution does like the nonnegative fundamental entropy solution inthe conservation law when $m+1 < n < n_0$, while in the case ofnonexistence the singularity gradually disappears when $n \geq n_0$that the mass cannot concentrate for a short time and no such a singular solutions exists. The results of previouswork [11] and this paper give a perfect answer to suchtopical researches.

HIGH‐ORDER SHORT‐TIME EXPANSIONS FOR ATM OPTION PRICES OF EXPONENTIAL LÉVY MODELS

AbstractThe short‐time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In this work, a novel second‐order approximation for at‐the‐money (ATM) option prices is derived for a large class of exponential Lévy models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second‐order term, in time‐t, is of the form , with d2 only depending on Y, the degree of jump activity, on σ, the volatility of the continuous component, and on an additional parameter controlling the intensity of the “small” jumps (regardless of their signs). This extends the well‐known result that the leading first‐order term is . In contrast, under a pure‐jump model, the dependence on Y and on the separate intensities of negative and positive small jumps are already reflected in the leading term, which is of the form . The second‐order term is shown to be of the form and, therefore, its order of decay turns out to be independent of Y. The asymptotic behavior of the corresponding Black–Scholes implied volatilities is also addressed. Our method of proof is based on an integral representation of the option price involving the tail probability of the log‐return process under the share measure and a suitable change of probability measure under which the pure‐jump component of the log‐return process becomes a Y‐stable process. Our approach is sufficiently general to cover a wide class of Lévy processes, which satisfy the latter property and whose Lévy density can be closely approximated by a stable density near the origin. Our numerical results show that the first‐order term typically exhibits rather poor performance and that the second‐order term can significantly improve the approximation's accuracy, particularly in the absence of a Brownian component.

Spectral asymptotics for Laplacians on self-similar sets

Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace–Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function.

Short-time behavior of mixed diffusion-barrier controlled adsorption

This paper focuses on the short-time adsorption kinetics of nonionic surfactants onto water/air surfaces, analyzed in the context of the mixed diffusion-barrier controlled adsorption modeling framework. Specifically, we reconcile the apparent contradiction between theoretical prediction and experimental observations on the adsorption kinetics mechanism at short times: while the mixed diffusion-barrier controlled model predicts a barrier-controlled adsorption, as well as the impossibility of a diffusion-controlled adsorption at asymptotic short times, the short-time experimental dynamic surface tension (DST) behavior of many nonionic surfactants has been interpreted to result from diffusion-controlled adsorption at asymptotic short times. This is because the short-time experimental DST of these surfactants displays a t variation, which is considered as a fingerprint for the existence of diffusion-controlled adsorption, based on the short-time asymptotic behavior of the diffusion-controlled adsorption model. As a result of this interpretation, the fundamental physical nature of the energy barrier has been proposed to be associated with high surfactant surface concentrations. In this paper, we derive a new nonasymptotic short-time formalism of the mixed diffusion-barrier controlled model to describe surfactant adsorption onto a spherical pendant-bubble surface, including determining the ranges of time and surfactant surface concentration values where the short-time formalism is applicable. Based on this formalism, we find that one can expect to observe an apparent t variation of the DST at short times even for the mixed diffusion-barrier controlled adsorption model. We analyze the consequence of this finding by re-evaluating the existing notions of the energy barrier. We conclude that the energy barrier is associated with the adsorption of a single surfactant molecule onto a clean surface.

On Short Time Asymptotic Behavior of Some Symmetric Diffusions on General State Spaces

For conservative symmetric diffusions on a general state space (X,m), the short time asymptotic behavior of tlog∫ X 1 A ⋅T t 1 B dm is investigated, where T t is the associated semigroup and A and B are measurable subsets of X. It is proved that the superior limit is dominated by the inferior limit up to some absolute constant. When Γ2 of the associated Dirichlet form is lower bounded, it is shown that the limit exists for any A and B, and is described by the intrinsic metric between them. Applications to infinite-dimensional spaces and fractals are given.

Short-Time Asymptotics of the Heat Kernel on a Concave Bboundary

A probabilistic method is used to study short-time asymptotic behavior of heat kernel in the exterior of an insulated smooth convex body. The expansion of the heat kernel $p(t,a,b)$ when both a and b are on the boundary is obtained by reducing the problem to the computation of a Wiener functional on a Brownian bridge. The leading terms of $\log p(t,a,b)$ are proved to be \[ - \frac{{\rho ^2 }} {{2t}} - \frac{{\mu _1 \rho ^{{1 / 3}} }} {{t^{{1 / 3}} }}\int_0^\rho {N(s)^{{2 / 3}} ds - \left( {\frac{d} {2} + \frac{1} {6}} \right)\log t + C_0 + o(1)} \] where $\rho $ is the distance between a and b, $C_0 $ is the normal curvature of the geodesic joining a and b, and $C_0 $ is an explicitly identified constant.

Sorption in inhomogeneous slabs

We find the exact, short time asymptotic behavior of the diffusion controlled sorption of a Fickian penetrant into an inhomogeneous slab. We also investigate the long time uptake of such a penetrant into an inhomogeneous slab when the inhomogeneity can either be treated as a small perturbation or when it is bounded from above by means of a variational principle. We discuss and compare the kind of information provided by permeation and sorption studies for a simple type of inhomogeneity.