Stochastic Diffusion Equations Research Articles

Existence and Uniqueness of Invariant Measures for Stochastic Evolution Equations with Weakly Dissipative Drifts

In this paper, a new decay estimate for a class of stochastic evolution equations with weakly dissipative drifts is established, which directly implies the uniqueness of invariant measures for the corresponding transition semigroups. Moreover, the existence of invariant measures and the convergence rate of corresponding transition semigroup to the invariant measure are also investigated. As applications, the main results are applied to singular stochastic $p$-Laplace equations and stochastic fast diffusion equations, which solves an open problem raised by Barbu and Da Prato in [Stoc. Proc. Appl. 120(2010), 1247-1266].

On the evolution of film roughness during magnetron sputtering deposition

The effect of long-range screening on the surface morphology of thin films grown with pulsed-dc (p-dc) magnetron sputtering is studied. The surface evolution is described by a stochastic diffusion equation that includes the nonlocal shadowing effects in three spatial dimensions. The diffusional relaxation and the angular distribution of the incident particle flux strongly influence the transition to the shadowing growth regime. In the magnetron sputtering deposition the shadowing effect is essential because of the configuration of the magnetron system (finite size of sputtered targets, rotating sample holder, etc.). A realistic angular distribution of depositing particles is constructed by taking into account the cylindrical magnetron geometry. Simulation results are compared with the experimental data of surface roughness evolution during 100 and 350 kHz p-dc deposition, respectively.

On the Approximate Solution of Non-Linear Stochastic Diffusion Equation Using Symbolic WHEP~!2009-08-04~!2009-11-28~!2010-08-23~!

On the Approximate Solution of Non-Linear Stochastic Diffusion Equation Using Symbolic WHEP~!2009-08-04~!2009-11-28~!2010-08-23~!

Note on coefficient matrices from stochastic Galerkin methods for random diffusion equations

In a recent work by Xiu and Shen [D. Xiu, J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, J. Comput. Phys. 228 (2009) 266–281], the Galerkin methods are used to solve stochastic diffusion equations in random media, where some properties for the coefficient matrix of the resulting system are provided. They also posed an open question on the properties of the coefficient matrix. In this work, we will provide some results related to the open question.

A variational approach to stochastic nonlinear parabolic problems

This work is concerned with an optimal control approach to stochastic nonlinear parabolic diffusion equations with monotonically increasing nonlinearity. This approach leads to sharper existence and uniqueness results under minimal growth conditions on nonlinear diffusion coefficients.

Laplace, Fourier, and stochastic diffusion

Stochastic diffusion equation, which attained prominence with Einstein's work on Brownian motion at the beginning of the twentieth century, was first formulated by Laplace a century earlier as part of his work on Central Limit Theorem. Between 1807 and 1811, Fourier's work on heat diffusion, and Laplace'swork on probability influenced and inspired each other. This brief period of interaction between these two illustrious figures must be considered remarkable for its profound impact on subsequent developments in mathematical physics, probability theory and pure analysis.

Invariant measures and the Kolmogorov equation for the stochastic fast diffusion equation

We prove the existence of an invariant measure μ for the transition semigroup P t associated with the fast diffusion porous media equation in a bounded domain O ⊂ R d , perturbed by a Gaussian noise. The Kolmogorov infinitesimal generator N of P t in L 2 ( H − 1 ( O ) , μ ) is characterized as the closure of a second-order elliptic operator in H − 1 ( O ) . Moreover, we construct the Sobolev space W 1 , 2 ( H − 1 ( O ) , μ ) and prove that D ( N ) ⊂ W 1 , 2 ( H − 1 ( O ) , μ ) .

Using Homotopy WHEP technique for solving a stochastic nonlinear diffusion equation

In this paper, the diffusion equation under square and cubic nonlinearities and stochastic nonhomogeneity is solved using the Homotopy WHEP technique. The homotopy perturbation method is introduced in the WHEP technique to deal with non-perturbative systems. The new technique is then used to solve the nonlinear diffusion equation by making comparisons with Homotopy perturbation method (HPM). The method of analysis is illustrated through case studies and figures.

A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations

The discretization of linear partial differential equations with random data by means of the stochastic Galerkin finite element method results in general in a large coupled linear system of equations. Using the stochastic diffusion equation as a model problem, we introduce and study a symmetric positive definite Kronecker product preconditioner for the Galerkin matrix. We compare the popular mean-based preconditioner with the proposed preconditioner which—in contrast to the mean-based construction—makes use of the entire information contained in the Galerkin matrix. We report on results of test problems, where the random diffusion coefficient is given in terms of a truncated Karhunen-Loeve expansion or is a lognormal random field.

Stochastic models of the grinding of disperse materials

A unified methodology for constructing stochastic models of the processes of the grinding of disperse materials is proposed. Using the fundamental relationships of the theory of stochastic Markovian processes, the integrodifferential and diffusion equations of the kinetics of grinding are rigorously derived. Particular solutions of these equations are obtained and their analysis is performed.

Stochastic Porous Media Equations and Self-Organized Criticality

The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized criticality behavior of stochastic nonlinear diffusion equations with critical states.

Efficient stochastic Galerkin methods for random diffusion equations

We discuss in this paper efficient solvers for stochastic diffusion equations in random media. We employ generalized polynomial chaos (gPC) expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by Galerkin projection. Although the resulting system of diffusion equations are coupled, we show that one can construct fast numerical methods to solve them in a decoupled fashion. The methods are based on separation of the diagonal terms and off-diagonal terms in the matrix of the Galerkin system. We examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters. Numerical examples are provided, for both steady and unsteady random diffusions, to support the analysis.

A perturbation theory for large deviation functionals in fluctuating hydrodynamics

We study a large deviation functional of density fluctuation by analyzing stochastic nonlinear diffusion equations driven by the difference between the densities fixed at the boundaries. By using a fundamental equality that yields the fluctuation theorem, we first relate the large deviation functional with a minimization problem. We then develop a perturbation method for solving the problem. In particular, by performing an expansion with respect to the average current, we derive the lowest order expression for the deviation from the local equilibrium part. This expression implies that the deviation is written as the spacetime integration of the excess entropy production rate during the most probable process of generating the fluctuation that corresponds to the argument of the large deviation functional.

Balance of datum land prices among cities based on the city gravitation model and stochastic diffusion equation

A balance of urban datum land prices is achieved to harmonize regional land prices and make the prices truly reflect different economic development levels and land prices among cities. The current piecewise linear interpolation balance method widely used has two drawbacks that always lead to an unsatisfactory balance among some cities. When the excess of land price in the central city to the surrounding zone reaches a certain degree, land price in the circumjacent city is not only consistent with the local land grade and land use level, but also influenced by the diffusion of land price in the central city. Thus, a new balanced scheme of datum land prices based on the city gravitation model and stochastic diffusion equation is brought forward. Finally, the new method is examined in the practice of datum land price balance in Hubei Province, China.

Stability of diffusion stochastic functional differential equations with Markov parameters

The paper justifies the second Lyapunov method for diffusion stochastic functional differential equations with Markov parameters, which generalize stochastic diffusion equations without aftereffect. Analogs of Lyapunov stability theorems, which generalize the results for systems with finite aftereffect, are proved.

Convergence in Nonlinear Filtering for Stochastic Delay Systems

We study an approximation scheme for a nonlinear filtering problem when the state process X is the solution of a stochastic delay diffusion equation and the observation process is a noisy function of $X(s)$ for $s\in [t-\tau,t]$, where $\tau$ is a constant. The approximating state is the piecewise linear Euler–Maruyama scheme, and the observation process is a noisy function of the approximating state. The rate of convergence of this scheme is computed.

GARCH modelling of covariance in dynamical estimation of inverse solutions

The problem of estimating unobserved states of spatially extended dynamical systems poses an inverse problem, which can be solved approximately by a recently developed variant of Kalman filtering; in order to provide the model of the dynamics with more flexibility with respect to space and time, we suggest to combine the concept of GARCH modelling of covariance, well known in econometrics, with Kalman filtering. We formulate this algorithm for spatiotemporal systems governed by stochastic diffusion equations and demonstrate its feasibility by presenting a numerical simulation designed to imitate the situation of the generation of electroencephalographic recordings by the human cortex.

Irreversible Liouville equation and nonperturbative approach in quantum computing

Having analyzed the role of time-ordering boundary conditions in deriving irreversible kinetic equation a general irreversible Liouville equation is proposed. By this equation a variety of irreversible kinetic equations, such as a stochastic diffusion equation, a irreversible project Liouville equation can be constructed. We also develop a nonperturbative method to solve the eigenvalue problem for this equation. Application in controlling decoherence for quantum computing is also studied.

Stochastic income and wealth

As is well known, the deterministic version of the maximum principle allows us to state rigorously the relationship between deterministic income (essentially the Hamiltonian) and deterministic wealth (essentially the state evaluation function). As a background motivating point of departure, consider the simple story of an infinitely long-lived individual whose sole wealth consists of a bank deposit paying a constant rate of interest. For this simple parable, whether we define and measure income in the spirit of Fisher as being the return on wealth, or in the spirit of Lindahl as being consumption plus net value of investment, or in the spirit of Hicks as being the largest permanently maintainable level of consumption, we get the same answer to the question ‘‘what is income?’’. The theory of the deterministic maximum principle shows is that this fundamental identity of the three seemingly different definitions of income is much broader and goes much deeper than the simple bank-account parable. We now seek to extend the investigation to cover uncertainty by dealing with the case where net capital accumulation is described by a stochastic diffusion equation in place of a deterministic differential equation. Stochastic diffusion equations introduce mathematical subtleties and complexities, which we will only deal with casually in this treatment. The introduction of uncertainty ratchets up the required level of analysis another several notches on the scale of mathematical complexity. To treat the subject of this paper fully rigorously and also at the level of generality of a multi-dimensional economic growth problem could easily turn this paper into a book and would be out of all proportion to its www.elsevier.com/locate/econbase Japan and the World Economy 16 (2004) 277–301

On conformal field theory and stochastic Loewner evolution

We describe Stochastic Loewner Evolution on arbitrary Riemann surfaces with boundary using Conformal Field Theory methods. We propose in particular a CFT construction for a probability measure on (clouded) paths, and check it against known restriction properties. The probability measure can be thought of as a section of the determinant bundle over moduli spaces of Riemann surfaces. Loewner evolutions have a natural description in terms of random walk in the moduli space, and the stochastic diffusion equation translates to the Virasoro action of a certain weight-two operator on a uniformised version of the determinant bundle.