Nonlinear Stochastic Partial Differential Equations Research Articles

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II

This paper is a continuation of our previous work (Part I, Stochastic Process. Appl. 93 (2001) 181–204), with the main purpose of establishing the uniqueness of the stochastic viscosity solution introduced there. We shall prove a comparison theorem between a stochastic viscosity solution and an ω-wise stochastic viscosity solution, which will lead to the uniqueness results. As the byproducts we extend the measurable section theorem of Dellacherie and Meyer (1978), and a fundamental lemma of Crandall et al. (1992)

On the behavior of solutions to certain parabolic SPDE's driven by wiener processes

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear stochastic partial differential equations driven by finite-dimensional Wiener processes. This class encompasses important equations that occur in the mathematical analysis of certain migration phenomena in population dynamics and population genetics. The solutions to such equations are generalized random fields whose long-time behavior we investigate in detail. In particular, we unveil the mechanism whereby these random fields approach the global attractor by proving that their asymptotic behavior is entirely controlled by that of their spatial average. We also show how to determine explicitly the corresponding Lyapunov exponents when the nonlinearities of the noise-term of the equations are subordinated to the nonlinearity of the drift-term in some sense. The ultimate picture that emerges from our analysis is one that displays a phenomenon of exchange of stability between the components of the global attractor. We provide a very simple interpretation of this phenomenon in the case of Fisher's equation of population genetics. Our method of investigation rests upon the use of martingale arguments, of a comparison principle and of some simple ergodic properties for certain Lebesgue- and Itô integrals.

Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations

We present here a general uniqueness result for stochastic viscosity solutions of fully nonlinear, second order possibly degenerate, stochastic partial differential equations.

Fully nonlinear stochastic pde with semilinear stochastic dependence

Abstract We study in this Note scalar fully nonlinear stochastic second order partial differential equations, possibly, degenerate with a semilinear stochastic dependence. We introduce a notion of stochastic viscosity solutions adapted to that class of equations. And we show general existence, uniqueness and regularity results.

Large Deviations for Stochastic Volterra Equations in the Plane

We prove a Large Deviation Principle for the family of solutions of Volterra equations in the plane obtained by perturbation of the driving white noise. One of the motivations for the study of such class of equations is provided by non-linear hyperbolic stochastic partial differential equations appearing in the construction of some path-valued processes on manifolds. The proof uses the method developped by Azencott for diffusion processes. The main ingredients are exponential inequalities for different classes of two-parameter stochastic integrals; these integrals are related to the representation of the stochastic term in the differential equation as a representable semimatringale.

On exponential stability criteria of stochastic partial differential equations

Some criteria for the mean square and almost sure exponential stability of nonlinear stochastic partial differential equations are shown in this paper. In particular, the main results obtained in Caraballo and Real (1994, Stochast. Anal. Appl. 12(5), 517–525) are improved, since the new coercivity condition introduced in this work permits the state independent term γ to be time dependent and nonnegative but of subexponential growth, while in Caraballo and Real (1994) this parameter is required to be constant and nonpositive. Several examples are studied to illustrate the theory.

Particle representations for a class of nonlinear SPDEs

An infinite system of stochastic differential equations for the locations and weights of a collection of particles is considered. The particles interact through their weighted empirical measure, V, and V is shown to be the unique solution of a nonlinear stochastic partial differential equation (SPDE). Conditions are given under which the weighted empirical measure has an L 2-density with respect to Lebesgue measure.

Dynamics of defect formation

A dynamic symmetry-breaking transition with noise and inertia is analyzed. Exact solution of the linearized equation that describes the critical region allows precise calculation (exponent and prefactor) of the number of defects produced as a function of the rate of increase of the critical parameter. The procedure is valid in both the overdamped and underdamped limits. In one space dimension, we perform quantitative comparison with numerical simulations of the nonlinear nonautonomous stochastic partial differential equation and report on signatures of underdamped dynamics.

Fully nonlinear stochastic partial differential equations: non-smooth equations and applications

In this Note, we extend the result described in a previous note to the case of nonsmooth Hamiltonians for fully nonlinear stochastic partial differential equations, and we present some applications of our theory to pathwise stochastic control and to the propagation of fronts in random environments.

Fully nonlinear stochastic partial differential equations

In this Note, we propose a new theory of “stochastic viscosity solutions” for fully nonlinear stochastic partial differential equations. This theory allows to handle a large class of equations which covers in particular various applications such as models of phase transitions and front propagation in random media and pathwise stochastic control. These applications will be detailed in a subsequent note.

On the Unique Solvability of Some Nonlinear Stochastic PDEs

The Cauchy problem for 1-dimensional nonlinear stochastic partial differential equations is studied. The uniqueness and existence of solutions in $cH^2_p(T)$-space are proved.

Pulsed quadrature-phase squeezing of solitary waves inχ(2)parametric waveguides

It is shown that coherent quantum simultons (simultaneous solitary waves at two different frequencies) can undergo quadrature-phase squeezing as they propagate through a dispersive chi((2)) waveguide. This requires a treatment of the coupled quantized fields including a quantized depleted pump field. A technique involving nonlinear stochastic parabolic partial differential equations using a nondiagonal coherent state representation in combination with an exact Wigner representation on a reduced phase space is outlined. We explicitly demonstrate that group-velocity matched chi((2)) waveguides which exhibit collinear propagation can produce quadrature-phase squeezed simultons. Quasi-phase-matched KTP waveguides, even with their large group-velocity mismatch between fundamental and second harmonic at 425 nm, can produce 3 dB squeezed bright pulses at 850 nm in the large phase-mismatch regime. This can be improved to more than 6 dB by using group-velocity matched waveguides.

Small perturbations in a hyperbolic stochastic partial differential equation

We study the existence and properties of the density for the law of the solution to a nonlinear hyperbolic stochastic partial differential equation, driven by a two-parameter white noise. We also analyze the asymptotic behavior of the density for the law of the solution to the equation obtained by perturbing the noise. Under unrestricted Hörmander's-type conditions on the coefficients, we establish Varadhan's estimates.

Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality

A stochastic nonlinear partial differential equation is built for two different models exhibiting self-organized criticality, the Bak, Tang, and Wiesenfeld (BTW) sandpile model and the Zhang's model. The dynamic renormalization group (DRG) enables to compute the critical exponents. However, the nontrivial stable fixed point of the DRG transformation is unreachable for the original parameters of the models. We introduce an alternative regularization of the step function involved in the threshold condition, which breaks the symmetry of the BTW model. Although the symmetry properties of the two models are different, it is shown that they both belong to the same universality class. In this case the DRG procedure leads to a symmetric behavior for both models, restoring the broken symmetry, and makes accessible the nontrivial fixed point. This technique could also be applied to other problems with threshold dynamics.

Fully Nonlinear Stochastic Partial Differential Equations

The authors study a class of fully nonlinear stochastic partial differential equations by the reduction to a family of deterministic fully nonlinear equations using the stochastic characteristic method.

Remarks on Nonlinear Stochastic Partial Differential Equations: An Application of the Splitting-Up Method

The objective of this article is to apply the splitting-up method to the existence theorem for nonlinear stochastic partial differential equations. With use of this method an approximating sequence is constructed. By the compactness argument a convergent subsequence can be extracted, and this fact provides the solution.

An approximation theorem of wong-zakai type for nonlinear stochastic partial differential equations

We present an extension of the Wong-Zakai approximation theorem for nonlinear 984 given by the Wiener process and a martingale. By approximating these disturbances we obtain in the limit equation the Ito correction term for the infinite dimensional case. Such form of the correction term connected with the Wiener process was proved in the author's papers [21] and [22], where the approximation theorem for semilinear stochastic evolution equations in Hilbert spaces was studied. Our model here is similar as the one considered by Pardoux [17]

Nonlinear stochastic integral equations in the plane

Let W = { W ζ , ζ ϵ T} be the two-parameter Wiener process on T = [0, 1] 2. Consider the nonlinear stochastic partial differential equation: ∂ 2X s, t ∂s ∂t = a 3(X, s, t) ∂W 2 s, t ∂s ∂t +a 4(X, s, t)+a 1(s, t) ∂X s, t ∂s +a 2(s, t) ∂X s, t ∂t . We give a rigorous meaning to the notion of solution for this equation, by rewriting it as an integral equation which involves a stochastic integral term and mixed-line integrals with respect to the representable semimartingale X. Under some assumptions on the coefficients a 1, we prove the existence and uniqueness of solution for this stochastic integral equation.

Dynamics of a phi4 kink in the presence of strong potential fluctuations, dissipation, and boundaries.

We have carried out a number of simulations to study the dynamical behavior of kinks in the ${\mathrm{\ensuremath{\varphi}}}^{4}$ model in the presence of strong fluctuations of its double-well potential. Our work widens the computational and analytical knowledge of this system in four directions. First, we describe in detail a numerical procedure that can be easily generalized to other stochastic, soliton-bearing equations. We demonstrate that it exhibits consistency features never found in previous research on nonlinear stochastic partial differential equations. Second, we fix the range of validity of theoretical approaches based on secular perturbative expansions. We show how this range depends on a combination of noise strength and duration. Third, we numerically study the model beyond the applicability of analytical methods. We compute the main characteristics of kink dynamics in this regime and discuss their stability under this random perturbation. Finally, we introduce dissipation and boundaries in the dynamically disordered model. We establish that the essential consequence of friction action is to soften the noise effects, while boundaries give rise to a critical velocity below which kinks cannot enter the noisy zone.

Stochastic overland flows

A theoretical solution framework to the nonlinear stochastic partial differential equations (SPDE) of the kinematic wave and diffusion wave models of overland flows under stochastic inflows/outflows, stochastic surface roughness field and stochastic state of flows was obtained. This development was realized by means of an eigenfunction representation of the time-space overland flow depths, and by transforming the problem into the phase space. By using Van Kampen's lemma and the cumulant expansion theory of Kubo-Van Kampen-Fox, the deterministic partial differential equation (PDE) for the evolutionary probability density function (pdf) of overland flow depths was finally obtained. Once this deterministic PDE is solved for the time-varying pdf of overland flow depths, then the time-space varying pdf of overland flow depths can be obtained by a transformation given in the text. In this solution framework it is possible to incorporate the stochastic dynamic behavior of the parameters and of the forcing functions of the overland flow process. For example, not only the individual rainfall duration and fluctuating rain intensity characteristics but also the sequential behavior of rainfall patterns is incorporated into the evolutionary probability density function of overland flow depths.