Parabolic Stochastic Differential Equations Research Articles

Stability of Stochastic Partial Differential Equations

In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a Hilbert space. For the solution of the initial-boundary value problems (IBVPs), we obtain the stability estimates for stochastic parabolic equations with dependent coefficients in specific applications.

Well-Posedness for Nonlinear Parabolic Stochastic Differential Equations with Nonlinear Robin Conditions

In this paper, we present the existence and uniqueness of strong probabilistic solutions for nonlinear parabolic Stochastic Partial Differential Equations (SPDEs) with nonlinear Robin boundary conditions in a domain with holes. On the boundary of the holes, a nonlinear Robin condition is imposed, while a homogeneous Dirichlet condition is prescribed on the exterior boundary. The coefficient matrix is assumed to be symmetric, while the nonlinear random forces are assumed to satisfy some types of regularities. We use Galerkin’s approximation method, probabilistic compactness results and some results from stochastic calculus.

On temporal regularity of stochastic convolutions in $2$-smooth Banach spaces

Nous montrons que les trajectoires des solutions des equations aux derivees partielles stochastiques paraboliques ont la meme regularite en temps que le processus de Wiener (aussi loin que vont les connaissances actuelles en la matiere). La regularite temporelle est consideree dans l’espace de Besov–Orlicz $B^{1/2}_{\Phi _{2},\infty }(0,T;X)$ ou $\Phi _{2}(x)=\exp (x^{2})-1$ et $X$ est un espace de Banach $2$-lisse.

A Weak Galerkin Method with RT Elements for a Stochastic Parabolic Differential Equation

A Weak Galerkin Method with RT Elements for a Stochastic Parabolic Differential Equation

Regularity of the Mild Solution of a Parabolic Equation with Stochastic Measure

We study a stochastic parabolic differential equation generated by a general stochastic measure. The existence, uniqueness, and Holder regularity of the mild solution are established.

A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations

In this paper, we present a numerical scheme to solve the initial-boundary value problem for backward stochastic partial differential equations of parabolic type. Based on the Galerkin method, we approximate the original equation by a family of backward stochastic differential equations (BSDEs, for short), and then solve these BSDEs by the time discretization. Combining the truncation with respect to the spatial variable and the backward Euler method on time variable, we obtain the global $L^2$ error estimate.

Space–time fractional stochastic partial differential equations

We consider non-linear time-fractional stochastic heat type equation ∂tβut(x)=−ν(−Δ)α/2ut(x)+It1−β[σ(u)W⋅(t,x)] in (d+1) dimensions, where ν>0,β∈(0,1), α∈(0,2] and d<min{2,β−1}α, ∂tβ is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, It1−β is the fractional integral operator, W⋅(t,x) is space–time white noise, and σ:R→R is Lipschitz continuous. Time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove existence and uniqueness of mild solutions to this equation and establish conditions under which the solution is continuous. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in Foondun and Khoshnevisan (2009), Walsh (1986). In sharp contrast to the stochastic partial differential equations studied earlier in Foondun and Khoshnevisan (2009), Khoshnevisan (2014) and Walsh (1986), in some cases our results give existence of random field solutions in spatial dimensions d=1,2,3. Under faster than linear growth of σ, we show that time fractional stochastic partial differential equation has no finite energy solution. This extends the result of Foondun and Parshad (in press) in the case of parabolic stochastic partial differential equations. We also establish a connection of the time fractional stochastic partial differential equations to higher order parabolic stochastic differential equations.

Numerics of stochastic parabolic differential equations with stable finite difference schemes

In the present article, we focus on the numerical approximation of stochastic partial differential equations of Ito type with space-time white noise process, in particular, parabolic equations. For each case of additive and multiplicative noise, the numerical solution of stochastic diffusion equations is approximated using two stochastic finite difference schemes and the stability and consistency conditions of the considered methods are analyzed. Numerical results are given to demonstrate the computational efficiency of the stochastic methods.

Lyapunov functionals vs Lyapunov functions for various types of stability of hybrid stochastic differential equation

In this paper, the effects of using Lyapunov like-functionals vs using Lyapunov like-functions coupled with partial differential inequalities on various types of stability criteria of hybrid stochastic parabolic differential equation of Itô type are investigated. A concluding remark will highlight the advantages of using the two approaches and the major differences between the utilization of two approaches for this specific type of differential equations with diffusion term.

On the Cauchy-Dirichlet Problem in the Half Space for Parabolic SPDEs in Weighted Hoelder Spaces

We study the Cauchy-Dirichlet problem for a second order linear parabolic stochastic differential equation (SPDE) with constant coefficients in a half-space. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Holder classes with weights.

On Cauchy—Dirichlet Problem for Parabolic Quasilinear SPDEs

We study the Cauchy–Dirichlet problem for a second-order quasilinear parabolic stochastic differential equation (SPDE) in a domain with a zero order noise term driven by a cylindrical Brownian motion. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Holder classes with weights.

On Cauchy–Dirichlet problem in half-space for parabolic SPDEs in weighted Hölder spaces

We study the Cauchy–Dirichlet problem for a second-order linear parabolic stochastic differential equation in the half-space with a zero-order noise term driven by a cylindrical Brownian motion. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes with weights.

On the existence of stochastic optimal control of distributed state system

An optimal control problem governed by a semilinear parabolic stochastic differential equation with nonlinear diffusion coefficient is studied and the existence of a quasi-optimal (nonrelaxed) control is proved without assuming convexity of the coefficients. The control state space is not compact and of infinite dimension, the stochastic equation is driven by a cylindrical Brownian motion.

On the Cauchy Problem for Stochastic Stokes Equations

We extend Krylov's Lp-solvability theory of the second order quasi-linear parabolic stochastic differential equations to the stochastic Stokes equation. Some additional integrability and regularity properties are also presented.

On the Cauchy problem for parabolic SPDEs in Hölder classes

We study Cauchy’s problem for certain second-order linear parabolic stochastic differential equation (SPDE)driven by a cylindrical Brownian motion.Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes.

A parabolic stochastic differential equation with fractional Brownian motion input

An existence and uniqueness theorem is proved for a quasilinear stochastic evolution equation with an additive noise in the form of a stochastic integral with respect to a Hilbert space-valued fractional Borwnian motion. Ideas of the finite-dimensional approximation by the Galerkin method are used.

Control problem for diffusion-type random fields

In this chapter we derive the conditions which guarantee the existence of optimal or e-optimal controls for stochastic systems described by stochastic parabolic differential equation. For random processes similar problems were investigated in [26]. Control problem for some types of processes and fields was discussed also in [18]. Our references for this chapter are [13, 15, 46].

Weak infinitesimal generator for a stochastic partial differential equation with time delay

In this paper, we consider the Markov solution process for a stochastic parabolic differential equation with time delay. Under the Lipschitz condition and boundedness on the drift and diffusion coefficient, properties of the weak infinitesimal generator of the associated Markov operators are established. Actions of the weak infinitesimal generator on the space of quasi-tame functions are also investigated.

Stochastic reaction-diffusion equations with continuous reaction

We study stochastic parabolic differential equations with an additive (i.e. state independent) Gaussian noise which is white in time and coloured in space. For continuous reaction functions we show...

Influence of natural phase fluctuations on active mode locking

An analysis is made of the influence of the spontaneous radiation noise on the operation of a mode-locked laser with an inhomogeneously broadened line. Mode locking is achieved by modulation of the losses. The system of equations for the perturbations of the phases of individual modes is replaced by a parabolic stochastic differential equation in which the mode number is considered to be an independent continuous variable. The boundary conditions and requirements for a Langevin source in the parabolic equation are formulated. It is shown that the phase of the nth mode drifts from that of the mth mode in accordance with the random walk law. A limit is derived for the total number of modes which can be locked when the fluctuation strength (the spectral width of a free-running mode) and the modulation depth are fixed.