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The paper is concerned with a class of stochastic reaction-diffusion equations arising from a spatial population growth model in random environments. Under some sufficient conditions, Theorem 3.1 shows that the equation has a unique positive global solution in space $H^1(D)$. Then it is proven in Theorem 4.1 that the solution, as the population size, is ultimately bounded in the mean $L^2-$norm as the time tends to infinity. An almost-sure upper bound is also obtained for the long run time-average of the exponential rate of the population growth in $L^2-$norm together with the $L^p-$moment of the population size with $p \geq 2.$ It is also shown in Theorem 4.3 that there is a unique invariant measure that leads to a stationary population distribution. For illustration, an example is given.

This paper is concerned with the problem of explosive solutions for a class of stochastic differential equations. Our main results are presented as two theorems. Theorem 1 is concerned with the existence of explosive solutions with positive probability under certain sufficient conditions. With some additional mild conditions, it is shown in Theorem 2 that the explosion will occur almost surely. The methods of auxiliary functions and cycles are used in the proofs. Several remarks about their applications are given.

This article is concerned with explosive solutions of the initial-boundary problem for a class of nonlinear stochastic wave equations in a domain 𝒟 ⊂ ℝ d . Under appropriate conditions on the initial data, the nonlinear term and the noise intensity, it is proved in Theorem 3.4 that there cannot exist a global solution and the local solution will blow up at a finite time in the mean L p − norm for p ≥ 1. An example is given to show the application of this theorem.

This work is concerned with a class of semilinear stochastic functional parabolic differential equations of retarded type. We first establish conditions to ensure the existence of a unique non-negative solution of the stochastic delay partial differential equation under investigation. Subsequently, the problem of explosive or blow-up solutions in mean L p -norm sense, p ≥ 1 , in a finite time to the stochastic system is considered. Several examples are given to illustrate the theory established in the work.

A semidiscrete Milstein scheme for stochastic partial differential equations of Zakai type on a bounded domain of ℝd is derived. It is shown that the order of convergence of this scheme is 1 for convergence in mean square sense. For almost sure convergence, the order of convergence is proved to be 1 - e for any e > 0. © 2010 Taylor & Francis.

This article is concerned with the Kolmogorov equation associated to a stochastic partial differential equation with an additive noise depending on a small parameter e > 0. As e vanishes, the parabolic equation degenerates into a first-order evolution equation. In a Gauss–Sobolev space setting, we prove that, as e ↓ 0, the solution of the Cauchy problem for the Kolmogorov equation converges in L 2(μ, H) to that of the reduced evolution equation of first-order, where μ is a reference Gaussian measure on the Hilbert space H.

The paper is concerned with the problem of explosive solutions to a class of nonlinear parabolic It^ o equations. Under some sufficient conditions on the initial state and the nonlinear coefficients, it will be shown that the solutions will explode innite time almost surely. Theorem 3.1 is concerned with the existence of a unique local strong solution. The main result is presented in Theorem 3.2. With the aid of Lemma 3.3 and Lemma 3.4, this theorem is proved by adopting the method of auxiliary functionals combined with the method of cycles. An example is given to illustrate the application of the main theorem.

Large-time asymptotic properties of solutions to a semilinear stochastic beam equation with damping in a bounded domain is considered. First an energy inequality and the exponential bound for a linear stochastic beam equation is established. Under appropriate conditions, the existence and uniqueness theorem for the nonlinear stochastic beam equation is proved. Next the main results on the boundedness of global solutions and the exponential stability of an equilibrium solution, in the mean-square sense, are given. Two examples are presented to illustrate some applications of the theorems.

Preliminaries Introduction Some Examples Brownian Motions and Martingales Stochastic Integrals Stochastic Differential Equations of Ito Type Levy Processes and Stochastic Integrals Stochastic Differential Equations of Levy Type Comments Scalar Equations of First Order Introduction Generalized Ito's Formula Linear Stochastic Equations Quasilinear Equations General Remarks Stochastic Parabolic Equations Introduction Preliminaries Solution of Stochastic Heat Equation Linear Equations with Additive Noise Some Regularity Properties Stochastic Reaction-Diffusion Equations Parabolic Equations with Gradient-Dependent Noise Nonlinear Parabolic Equations with Levy-Type Noise Stochastic Parabolic Equations in the Whole Space Introduction Preliminaries Linear and Semilinear Equations Feynman-Kac Formula Positivity of Solutions Correlation Functions of Solutions Stochastic Hyperbolic Equations Introduction Preliminaries Wave Equation with Additive Noise Semilinear Wave Equations Wave Equations in an Unbounded Domain Randomly Perturbed Hyperbolic Systems Stochastic Evolution Equations in Hilbert Spaces Introduction Hilbert Space-Valued Martingales Stochastic Integrals in Hilbert Spaces Ito's Formula Stochastic Evolution Equations Mild Solutions Strong Solutions Stochastic Evolution Equations of the Second Order Asymptotic Behavior of Solutions Introduction Ito's Formula and Lyapunov Functionals Boundedness of Solutions Stability of Null Solution Invariant Measures Small Random Perturbation Problems Large Deviations Problems Further Applications Introduction Stochastic Burgers and Related Equations Random Schroedinger Equation Nonlinear Stochastic Beam Equations Stochastic Stability of Cahn-Hilliard Equation Invariant Measures for Stochastic Navier-Stokes Equations Spatial Population Growth Model in Random Environment HJMM Equation in Finance Diffusion Equations in Infinite Dimensions Introduction Diffusion Processes and Kolmogorov Equations Gauss-Sobolev Spaces Ornstein-Uhlenbeck Semigroup Parabolic Equations and Related Elliptic Problems Characteristic Functionals and Hopf Equations Bibliography Index

The paper is concerned with strong solutions of bilinear stochastic wave equations in ℛ d , of which the coefficients contain semimartingale white noises with spatial parameters. For the Cauchy problems, the existence and spatial regularity of solutions in Sobolev spaces are proved under appropriate conditions. The dependence of solution regularity on the smoothness of the random coefficients is ascertained. The proofs are based on stochastic energy inequalities, the semigroup method and certain submartingale inequalities. Regularity results are also obtained for the special case of Wiener semimartingales.

We consider an estimation problem for time dependent spatial signal observed in a presence of additive cylindrical Gaussian white noise of a small intensity e . Under known a priori smoothness of the signal estimators with asymptotically the best in the mimimax sense risk convergence rate in e to zero are proposed. Moreover, on-line estimators for the signal and its derivatives in t are also created.

The paper is concerned with solutions of a multi–dimensional wave equation perturbed by a spatially dependent white noise. A spherical mean representation of the solutions is given. Based on such a representation, the regularity properties of the classical and generalized solutions are studied.The dependence of the solution regularity on the smoothness of the noise and the spatial dimension is determined.

For linear partial differential equations, some inverse source problems are treated statistically based on nonparametric estimation ideas. By observing the solution in a small Gaussian white noise, the kernel type of estimators is used to estimate the unknown source function and its partial derivatives.. It is proved that such estimators are consistent as the noise intensity tends to zero. Depending on the principal part of the differential operator, the optimal asymptotic rate of convergence is ascertained within a wide class of risk functions in a minimax sense.

For the observation model “signal + white Gaussian noise”, an on-line tracking algorithm for signal and its derivatives is proposed. The tracking algorithm applies to a class of signals with derivative up to the k th order. The asyptotic optimality in the minimax sense, with respect to small intensity of noise, is established.

General theorems concerning the existence and uniqueness of invariant measures are proved for a certain class of regular diffusion processes in Separable Banach spaces under some weak compactness and other conditions. Then, based on these theorems, some verifiable sufficient conditions are obtained to ensure the existence and uniqueness of an invariant distribution for the strong solution to some nonlinear evolution equations in a Hilbert space. The results are applied to certain monotone parabolic Ito equations as well as to the 2-D Navier-Stokes equations under random perturbations

We consider the Kolmogorov equation in infinite dimensions associated with a nonlinear stochastic evolution equation in some Hilbert space. By introducing a proper class of Gauss-Sobolev spaces, a L2 -theory of the Kolmogorov equation is developed. Under some suitable conditions, the existence and uniqueness of solutions of the Cauchy problem and the related elliptic problem are proved in such a Gauss-Sobolev space setting

We prove the existence and regularity of solutions to stochastic partial differential equations of parabolic Ito type in Holder spaces under the usual sublinear growth and local Lipschitz conditions. Some examples are given to which our main theorems apply.

It is proved that, under the usual measurability and the square-integrability conditions, the stochastic integral of an operator-valued integrand with respect to a continuous local martingale in Hilbert spaces can be defined for almost every sample path via a sequence of simple progressively measurable approximations. The result generalizes McKeanls construction of the It6 integral with respect to a one-dimensional Brownian motion

The paper is concerned with the problem of explosive solutions for a class of nonlinear stochastic wave equations in a domain $\mathcal{D}\subset\mathbb{R}^d$ for $d\leq3$. Under appropriate conditions on the initial data, the nonlinear term and the noise intensity is proved in Theorem 3.1 that the $L^2$-norm of the solution will blow up at a finite time in the mean-square sense. An example is given to show an application of the theorem., Published in at http://dx.doi.org/10.1214/09-AAP602 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Asymptotic Analysis Involving Stochastic Differential Equations.- Some Recent Results on Averaging Principle.- Cramer's Theorem for Nonnegative Multivariate Point Processes with Independent Increments.- On Bounded Solutions of the Balanced Generalized Pantograph Equation.- Numerical Methods for Non-Zero-Sum Stochastic Differential Games: Convergence of the Markov Chain Approximation Method.- Nonparametric Estimation.- On the Estimation of an Analytic Spectral Density Outside of the Observation Band.- On Oracle Inequalities Related to High Dimensional Linear Models.- Hypothesis Testing under Composite Functions Alternative.- Stochastic Partial Differential Equations.- On Parabolic Pdes and Spdes in Sobolev Spaces W P 2 without and with Weights.- Stochastic Parabolic Equations of Full Second Order.

The paper is concerned with a large deviation type of estimates for some diffusion processes in Hilbert spaces, including the Brownian motion, the ltd integral and the solution of a stochastic evolution equation. Exponential upper bounds in exit probability are obtained and proved for such processes to leave a ball of radius r before a finite time t. An application to parabolic Ito equations is given as an example

The paper is concerned with a class of parabolic equations with a gradient-dependent nonlinear term in a Gauss-Sobolev space setting. Un- der a local Lipschitz continuity condition, it is shown in Theorem 4.2 that there exists a unique strong solution of such a semilinear parabolic equa- tion for which a certain energy inequality holds. The theorem is applied to show the existence of the strong solutions to the Kolmogorov equation and the Hamilton-Jacobi-Bellman equation arising from the control problem for stochastic partial difierential equations.

Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are established. Under appropriate conditions, the existence theorem for a unique global solution is given. Next the questions of bounded solutions and the exponential stability of an equilibrium solution, in mean-square and the almost sure sense, are studied. Then, under some sufficient conditions, the existence of a unique invariant measure is proved. Two examples are presented to illustrate some applications of the theorems., Comment: Published at http://dx.doi.org/10.1214/105051606000000141 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This paper is concerned with a class of nonlinear stochastic wave equations in $\mathbb{R}^d$ with $d \leq 3$, for which the nonlinear terms are polynomial of degree $m$. As an example of the nonexistence of a global solution in general, it is shown that there exists an explosive solution of some cubically nonlinear wave equation with a noise term. Then the existence and uniqueness theorems for local and global solutions in Sobolev space $H_1$ are proven with the degree of polynomial $m \leq 3$ for $d = 3$, and $m \geq 2$ for $d = 1$ or 2.

In Equation (10) of Ref. [5], the mixed terms of the stochastic increments are missing, as has been pointed out in Refs [3,4]. Equation (10) should read This recursion can be efficiently simulated ...

The paper is concerned with some approximate solutions for random partial differential equations of parabolic type via discretization in time or by an eigenfunction expansion. In contrast with L p -convergence, it is shown that, under suitable conditions, the approximate solutions converge almost surely to the strong solution of a parabolic Ito equation.

This paper proves that several initial-boundary value problems for a wide class of nonlinear reaction-diffusion equations have solutions ci(x, t), 1 ⩽ i ⩽ N (with ci(x, t) representing the concentration of the ith species at position x in a set Ω at time t ⩾ 0), which exist for all t ⩾ 0 and are unique, smooth, nonnegative, and strictly positive for t > 0. The Volterra-Lotka predator-prey model with diffusion (to which the results above are proved to apply) is then studied in more detail. It is proved that any bounded solution of this model loses its spatial dependence and behaves like a periodic function of time alone as t → ∞. It is proved that if the spatial dimension is one or if the diffusion coefficients of the two species are equal, then all solutions are bounded.

Some stochastic partial differential equations arising from a turbulent transport model are studied using Hida's theory of Brownian functionals. For the spatially homogeneous case, the solutions are constructed as a regular or generalized Brownian functional, depending on a small parameter. The regularity property of such solutions is also determined. However, for the spatially nonhomogeneous equations, only generalized solutions in a series form involving iterated singular Wiener integrals are found.