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Abstracts and Talk Materials
Random Dynamical Systems
October 22 - 26, 2012

Keywords of the presentation: Burgers equation, invariant distributions, variational principle, random minimizers, one force one solution principle, last passage percolation

The Burgers equation is a basic hydrodynamic model describing the evolution of the velocity field of sticky dust particles. When supplied with random forcing it turns into an infinite-dimensional random dynamical system that has been studied since late 1990's. The variational approach to Burgers equation allows to study the system by analyzing optimal paths in the random landscape generated by the random force potential. Therefore, this is essentially a random media problem. For a long time only compact cases of Burgers dynamics on the circle or a torus were understood well. In this talk I discuss the Burgers dynamics on the entire real line with no compactness or periodicity assumption. The main result is the description of the ergodic components and One Force One Solution principle on each component. Joint work with Eric Cator and Kostya Khanin.

Recent papers by Feng, Forde and Fouque (SIAM J Financial Math, 2010) and Feng, Fouque and Kumar (Ann. Appl. Prob, 2012) have obtained large deviation results for the small time asymptotic behavior for the log stock price in a fast mean-reverting stochastic volatility model. These results involve specific assumptions about the rate of growth of the speed parameter in the volatility process as the small time decreases to zero. It turns out that a simple time change argument converts the original problem into one concerning the large time asymptotic behavior of a related additive functional of the (rate 1) diffusion process. The scenarios described in the papers above correspond to large and moderate deviations of this additive functional away from its ergodic limit. We will discuss briefly some recent results for the case of superlarge deviations. This is joint work with Jerome Grand’Maison (University of Southern California).

On 2D stochastic dissipative Euler equation
October 22, 2012

Keywords of the presentation: Random attractors, Random dynamical systems

A 2D Euler equation containing a dissipation term and driven by an additive noise is studied. We prove the existence of a global flow and define a notion of weak attractor with appropriate topologies. If time permits, some results on stationary solutions will be given when the noise is of multiplicative type.

Keywords of the presentation: noisy observer, accuracy, stability, random attractor, stochastic Navier-Stokes equation

We consider a noisy observer for an unknown solution u(t) of a deterministic model. The observer is a stochastic model similar to the original one, and it arises as a limit of frequent observations in filtering. Here noisy observations of the low Fourier modes of u(t) together with knowledge of the deterministic model are used to track the unknown solution u(t).

In this talk as an example we discuss the simple 3DVAR-filter and the deterministic 2D-Navier Stokes equation. We establish stability and accuracy of the filter by studying this for the stochastic PDE describing the observer.

For stability one has to show that solutions of the observer starting from different initial conditions converge exponentially fast towards each other. This would also imply that the random attractor of the observer is at most a single random point.

Accuracy means that any solution of the observer converges exponentially fast to a small neighbourhood of the unknown solution u(t). In terms of random attractors one has to verify that the whole attractor is close to u(t).

Joint work with Andrew Stuart, Kody Law, and Konstantinos Zygalakis

Synchronization in stochastic pde systems
October 24, 2012

We first consider a system of semilinear parabolic stochastic partial differential equations with additive space-time noise on the union of thin bounded tubular domains with interaction via interface and give conditions which guarantee synchronized behaviour of solutions at the level of pullback attractors. Moreover, in the case of nondegenerate noise we obtain stronger synchronization phenomena in comparison with analogous results in the deterministic case. Then we deal with an abstract system of two coupled nonlinear stochastic (infinite dimensional) equations subjected to additive white noise type process. This kind of systems may describe various interaction phenomena in a continuum random medium. Under suitable conditions we prove the existence of an exponentially attracting random invariant manifold for the coupled system which means that we can observe (nonlinear) master-slave synchronization phenomena in the coupled system. Several examples from Mathematical Physics are discussed. Partially based on joint results with T. Caraballo, P. E. Kloeden, and B. Schmalfuss.

We design a dynamic rate scheduling policy of Markov type via the solution (a social optimal Nash equilibrium point) to a utility-maximization problem over a randomly evolving capacity set for a class of generalized processor-sharing queues living in a random environment, whose job arrivals to each queue follow a doubly stochastic renewal process (DSRP). Both the random environment and the random arrival rate of each DSRP are driven by a finite state continuous time Markov chain (FS-CTMC). Whereas the scheduling policy optimizes in a greedy fashion with respect to each queue and environmental state and since the closed-form solution for the performance of such a queueing system under the policy is difficult to obtain, we establish a reflecting diffusion with regime-switching (RDRS) model for its measures of performance. Furthermore, we justify its asymptotic optimality through deriving the stochastic fluid and diffusion limits for the corresponding system under heavy traffic. In addition, we identify a cost function related to the utility function, which is minimized through minimizing the workload process in the diffusion limit. More importantly, our queueing model includes typical systems in the future wireless networks, such as, the J-user multi-input multi-output (MIMO) multiple access channel (MAC) and the broadcast channel (BC) under Markov fading with cooperation and admission control as special cases.

Dynamical systems defined on networks have applications in many fields, including computational and theoretical neuroscience. In particular, it is important to understand when networks exhibit synchronous or other types of coherent collective behaviors. Other questions include whether such coherent behavior is stable with respect to random perturbation, or what the detailed structure of this behavior is as it evolves. We will examine several models of networked dynamical systems and present a mixture of results that range from rigorous theorems for abstract models to quantitative comparisons of models and data.

Deterministic descriptions of dynamics of competing species with identical carrying capacities but distinct birth, death, and reproduction rates predict steady state coexistence with population ratios depending on initial conditions. Demographic fluctuations described by a Markovian birth-death model break this degeneracy. A novel large carrying capacity asymptotic theory confirmed by conventional analysis and simulations reveals a weak preference for longevity in the deterministic limit with finite-time extinction of one of the competitors on a time scale proportional to the total carrying capacity.

This is joint work with Yen Ting Lin and Hyejin Kim

Dynamical systems arising in biological, physical and chemical sciences are often subject to random influences, which are also known as “noise”. Stochastic differential equations are appropriate models for some of these systems. The noise in these stochastic differential equations may be Gaussian or non-Gaussian in nature. Non-Gaussianity of the noise manifests as nonlocality at a macroscopic level. In addition, randomness may have delicate, or even profound, impact on the overall evolution of dynamical systems. This poster will present two available theoretical and numerical techniques for analyzing stochastic dynamical systems, escape probability and mean exit time. The differences in dynamics under Gaussian and non-Gaussian noises are highlighted.

Joint work with Ting Gao, Xingye Kan, Xiaofan Li and Jinqiao Duan

Gaussian invariant measures in fluid dynamics
December 31, 1969

We consider invariant measures μ of Gaussian type for the equations of motions of fluids, both for the viscous case (Navier–Stokes equations) and for the ideal case (Euler equations). In the Navier–Stokes system (which is dissipative), a forcing term is required in order to get existence of invariant measures. Hence we deal with stochastic Navier–Stokes equations and with deterministic unforced Euler equations. Moreover, we consider shell models of turbulence, which are easier models from the mathematical point of view but shear many important fatures with the real fluid dynamical equations. We present results on existence and/or uniqueness of solutions having μ as invariant measure, and of uniqueness of this invariant measure. Finally, some open problems are presented.

Pathwise solutions to stochastic evolution equations
October 22, 2012

Keywords of the presentation: Stochastic PDEs, fractional Brownian motion, pathwise solutions, rough path theory

In this talk, we combine tools from classical fractional calculus and the Rough Path Theory to study the existence and uniqueness of mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in the interval $(1/3,1/2)$. The stochastic integral is given by a generalization of the well-known Young integral and can be defined independently of the initial condition. In order to formulate an operator equation solving the problem we need a second equation for the so called area in the space of tensors, and the key ingredient to get this is to construct a tensor depending on the noise path and also on the semigroup. We prove in a first step the existence of a unique Hölder continuous solution of the system of equations, consisting of the path and the area components, if the nonlinear term and the initial condition are sufficiently smooth. In a second part, we also prove similar results when considering more general initial states, by modifying accordingly the phase space.

The abstract theory is applicable to evolution equations driven by a fractional Brownian motion $B^H$ with Hurst parameter $H in (1/3,1/2]$. One important result is that the pathwise definition of the stochastic integral allows to prove that the solution process generates a random dynamical system.

This is a join work with Kening Lu (BYU, Provo) and Björn Schmalfuss (Universität Jena).

Parameter estimation
December 31, 1969

Suppose we are attempting to describe a system with a model that depends on one or several parameters. Such parameters typically represent physical quantities we would like to measure. It is therefore desirable to develop ‘inverse’ methods to accurately calibrate these parameters by observing incomplete or noisy data. On the other hand if the parameters are already known but we lack confidence in the underlying model such ‘inverse’ methods represent a means of testing and validating a model. Not surprisingly ‘parameter estimation’ problems arise frequently in applications and represent an important direction for the analysis of both deterministic and random dynamical systems.

We describe some recent work in this direction for the stochastic Navier-Stokes equations and other nonlinear SPDEs. We consider the problem of determining the viscosity nu and use the Girsanov theorem to derive a class of maximum likelihood type estimators based on an observation of the first N Fourier modes of a single sample path observed on a finite time window. We rigorously establish the consistency and asymptotic normality of these estimators via a splitting method. The analysis treats strong, pathwise solutions for both the periodic and bounded domain cases in the presence of an additive white (in time) noise.

A small particle with diameter on the order of nano-meters to micrometers undergoes random motion when in a fluid. The motion is described by a stochastic Newton equation, using stochastic differential equations (SDE), relating acceleration with the forces acting on the particle. Experimentally, it is difficult to measure the instantaneous velocity of the particle. Therefore, valid approximations to the SDE are useful for applications. One such approximation is the small mass, also called the Smoluchowski-Kramers, approximation. This work describes a new way to identify and prove this limit for systems of arbitrary dimension and gives applications of this theorem to systems of experimental interest.

Keywords of the presentation: Chaotical Behavior, positive entropy, compact random sets, random dynamical systems, stochastic partial differential

In this talk, we present an answer to the long standing problem on the implication of positive entropy of a random dynamical system. We study C^0 infinite dimensional random dynamical systems in a Polish space, do not assume any hyperbolicity, and prove that chaos and weak horseshoe exist inside the random invariant set when its entropy is positive. This result is new even for finite dimensional random dynamical systems and infinite dimensional deterministic dynamical systems generated by either parabolic PDEs or hyperbolic PDEs. We mention that in general one does not expect to have a horseshoe without assuming hyperbolicity. For example, consider the product system of a circle diffeomorphism with an irrational rotation number and a system with positive entropy. This product system has positive entropy and a weak horseshoe, but has no horseshoe.

This is a joint work with Prof. Kening Lu

Invariant densities for dynamical systems with random switching
December 31, 1969

Consider a finite set of smooth vector fields on a smooth manifold. Given an initial point on the manifold, along with an initial vector field, one can define a stochastic process that follows the trajectory induced by the point and the vector field for an exponentially distributed time, then switches to a new vector field, again following the corresponding trajectory for an exponentially distributed time. The two-component process that captures the position on the manifold, as well as the driving vector field, is Markov. Under Hörmander-type hypoellipticity conditions on the vector fields, Yuri Bakhtin and I showed the following: Provided that the Markov semigroup has an invariant measure, it is unique and absolutely continuous with respect to Lebesgue measure.

We construct Lévy processes with discontinuous jump characteristics in form of weak solutions of appropriate stochastic differential equations, or related martingale problems with non-local operators. For this purpose we prove a general existence theorem for martingale problems in which a sequence of operators generating Feller processes approximates an operator with a range containing discontinuous functions. The approach crucially depends on uniform estimates for the probability densities of the approximating processes derived from properties of the associated symbols. The theorem is applicable to stable like processes with discontinuous stability index. This talk is based on work with N. Willrich (WIAS Berlin).

Lyapunov exponents play an important role in the study of the behavior of dynamical systems. They measure the average rate of separation of orbits starting from nearby initial points. They are used to describe the local stability of orbits and chaotic behavior of systems. Multiplicative Ergodic Theorem provides the theoretical foundation of Lyapunov exponents, which gives the fundamental information of Lyapunov Exponents and their associates invariant subspaces.

In this talk, I will report the work on Multiplicative Ergodic Theorem (with Kening Lu), which is applicable to infinite dimensional random dynamical systems in a separable Banach space. The system could be generated by, for example, random partial differential equations.

On reduction to random invariant manifolds
December 31, 1969

We introduce an explicit formula for the first term of the asymptotic expansion of the local random invariant manifold (LRIM) about the origin for a broad class of nonlinear stochastic evolution equations with linear multiplicative white noise. This provides an explicit reduction procedure to reduce a given SPDE to the corresponding LRIM, and it can be treated as a first step towards the study of the stability of the trivial stationary solution and the bifurcation from it by reducing the system to its corresponding LRIM.

Keywords of the presentation: Stochastic wave equation, hyperbolic equations, second-moment stability

While stochastic parabolicity condition is well-known, there is no immediate analogue for stochastic equations that are second-order in time. The objective of this paper is to present a general approach to the study of well-posedness and stability in Sobolev spaces on $bRd$ of the initial value problem for second-order in time stochastic evolution equations with constant coefficients and multiplicative time-only Gaussian white noise, and to illustrate the results on some particular examples. In general, there is no single condition for well-posedness in terms of the coefficients of the equation, but analysis of certain classes of equations leads to analogues of the stochastic parabolicity condition.

Waves in lattices with imperfections
October 22, 2012

We study wave motion in a lattice differential equation, specifically, in a spatially discrete Allen-Cahn equation with a bistable nonlinearity. The system may, but need not, be near the continuum limit. Superimposed on the regular (periodic) lattice is a spatial variation which varies in an arbitrary (generally non-periodic) fashion; it can either be a variation in the coupling (diffusion) constants between consectutive lattice points, or in the nonlinearities at each point. Existence of generalized traveling waves is shown, as well as phenomena of pinning versus transmission of waves.

This is joint work with Shiu-Nee Chow, Kening Lu, and Wenxian Shen.

Symmetric fractional Brownian motion
December 31, 1969

We consider a random symmetric matrix whose entries are independent fractional Brownian motions with the same Hurst paramter H>1/2. First we show that at all positive times the eigenvalues do not collide almost surely. Secondy, we derive a stochastic differential equation for the stochastic process of the ordered eigenvalues using the Skorohod integral. In this equation the drift term is similar to the Dyson Brownian motion, but the stochastic integral term is more involved. We show that this integral term is H-self-similar and it has the same 1/H-variation as the fractional Brownian motion with Hurst parameter H.

Keywords of the presentation: stochastic differential equations, Levy processes, heavy tails, first exit time, multiplicative noise

We discuss the behaviour of dynamical systems perturbed by small Levy noise with heavy tails. In particular we determine the asymptotics of first exit times for systems with a) additive noise with power tails, b) additive noise with sub- and super-exponential tails; c) multiplicative noises of Ito, Stratonovich and Marcus type. As examples we consider $alpha$-stable Levy ratchet and a motion of a charged Levy particle in an external magnetic field.

Invariance of the Gibbs measure for the periodic quartic gKdV
December 31, 1969

We prove invariance of the Gibbs measure for the (gauge transformed) periodic quartic gKdV. The Gibbs measure is supported on H^s for s less than 1/2, and the quartic gKdV is analytically ill-posed in this range. In order to consider the flow in the support of the Gibbs measure, we combine a probabilistic argument with the second iteration and construct local-in-time solutions to the (gauge transformed) quartic gKdV almost surely in the support of the Gibbs measure. Then, we use Bourgain's idea to extend these local solutions to global solutions, and prove the invariance of the Gibbs measure under the flow. Finally, Inverting the gauge, we construct almost sure global solutions to the (ungauged) quartic gKdV below H^{1/2}

Keywords of the presentation: total variation flow, stochastic singular diffusions, parabolic 1-Laplace equation, stochastic variational inequalities, stochastic partial differential equations

This is joint work with Viorel Barbu (Romanian Academy, Iasi).

We extend the approach of variational inequalities (VI) to partial differential equations (PDE) with singular coefficients, to the stochastic case. As a model case we concentrate on the parabolic 1-Laplace equation (a PDE with highly singular diffusivity) on a bounded convex domain in N-dimensional Euclidean space, perturbed by linear multiplicative noise, where the latter is given by a function valued (infinite dimensional) Wiener process. We prove existence and uniqueness of solutions for the corresponding stochastic variational inequality (SVI) in all space dimensions N and for any square-integrable initial condition, thus obtaining a stochastic version of the (minimal) total variation flow. One main tool to achieve this, is to transform the SVI and its approximating stochastic PDE into a deterministic VI, PDE respectively, with random coefficients, thus gaining sharper spatial regularity results for the solutions. We also prove finite time extinction of solutions with positive probability in up to N = 3 space dimensions.

An averaging principle for stochastic systems on foliated manifolds
December 31, 1969

Consider a SDE on a foliated manifold whose trajectories lay on compact leaves. We investigate the effective behaviour of a small transversal perturbation of order $epsilon$. An average principle is shown to hold such that the vertical (transversal to the leaves) component converges to the solution of a deterministic ODE, according to the average of the perturbation vector field with respect to invariant measures on the leaves, as $epsilon$ goes to zero. An estimate of the rate of convergence is given. This result is in the direction of a generalization of X.-M. Li' s article on an average principle for completely integrable stochastic Hamiltonian system (Nonlinearity, 2008).

We consider stochastic differential equations on a Euclidean space driven by a Kunita-type semimartingale field satisfying a one-sided local Lipschitz condition. We address questions of local and global existence and uniqueness of solutions as well as existence of a local or global semiflow. Further, we will provide sufficient conditions for strong $p$-completeness, i.e. almost sure non-explosion for subsets of dimension $p$ under the local solution semiflow.

Traveling waves in diffusive random media
October 22, 2012

Keywords of the presentation: traveling wave solutions, random media, bistable or monostable nonlinearity

The current talk is concerned with traveling wave solutions of reaction diffusion equations in random media. I will first introduce the concept of random traveling wave solutions and present a general theorem about the existence of such traveling wave solutions. I will then discuss the existence, uniqueness, and stability of random traveling wave solutions of reaction diffusion equations with bistable or monostable nonlinearity in temporally random media. I will end the talk with some remarks and open problems.

Stability of travelling waves in stochastic Nagumo equations
October 23, 2012

Keywords of the presentation: stochastic stability of travelling waves, functional inequalities

Stability of travelling waves for the Nagumo equation on the whole line is proven using a new approach via functional inequalities and an implicitely defined phase adaption. The approach can be carried over to obtain pathwise stability of travelling wave solutions in the case of the stochastic Nagumo equation as well. The noise term considered is of multiplicative type with variance proportional to the distance of the solution to the orbit of the travelling wave solutions.

The motivation for our study is to understand the stability properties of the action potential travelling along the nerve axon under the influence of thermal noise, that can be modelled in a mathematical idealized way with the help of stochastic FitzHugh Nagumo systems. In our talk we will demonstrate how the stochastic stability for the action potential leads to a simple computational approach for estimate the probability of propagation failure in nerve axons.

Lennard-Jones particles tend to aggregate in tetrahedra that can then form larger clusters with icosahedral symmetry. For clusters with certain number of particles (e.g. 38), the icosahedral structure is not the one with lowest energy, which is an FCC-based structure with octahedral symmetry. We may ask: How does the system reorganize itself after self-assembly to reach this ground state of its energy and how long does this process take? The dynamics of this reorganization can be modeled by Markov chain, i.e. a random walk on a network whose nodes are the local energy minima of the cluster and whose edges are the minimum energy paths connecting these minima with weights that depend on the energy barrier to be crossed to hop from one local minima to one of its neighbor on the network. It is natural to think about using large deviation theory (LDT) to understand the pathways of reorganization on such a network. Here, however, we show that the predictions of LDT are only valid at very low temperature when the time-scale of reorganization is extremely large. At more moderate temperatures, the system remains highly metastable (in the sense that there exists low lying eigenvalues in the spectrum of the chain) but reorganization occurs by a pathway that is different than that predicted by LDT and can be quantified precisely using different tools that will be presented in the talk. These tools are applicable to other Markov chains displaying metastability over two or more states and show that LDT results must be applied with care when the dimensionality of the system is large and entropic effects are important.

Bifurcation of non-autonomous stochastic equations
October 23, 2012

Keywords of the presentation: Random periodic solution, random almost periodic solution, random automorphic solution, random attractor, bifurcation, non-autonomous stochastic equation.

This talk is concerned with bifurcation of random dynamical systems generated by non-autonomous stochastic equations. We first introduce definitions of pathwise random almost periodic and almost automorphic solutions for stochastic equations, which are corresponding counterparts of non-autonomous deterministic systems. We then discuss pitchfork bifurcation of random periodic (almost periodic, almost automorphic) solutions of equations with multiplicative noise. We also demonstrate that additive white noise could destroy bifurcation of non-autonomous deterministic equations. Finally, we discuss bifurcation of random periodic solutions of a class of stochastic parabolic equations on bounded domains.

Fractal properties of stochastic heat equations
December 31, 1969

By applying methods for studying Gaussian random fields, we investigate various analytic and fractal properties of the solutions of the stochastic heat equations driven by space-time white noise or fractional colored noise. These include fractal dimensions, exact modulus of continuity, hitting probabilities and existence of intersections. The proofs of these results are based on the properties of strong local nondeterminism. (Based on the on-going joint works with R. Dalang, C. Mueller and C. Tudor.)

Under the local Lipschitz conditions in $C$ space, the well-posedness for the mild solutions of a class of infinite dimensional stochastic differential equations with delays (IDSDEs) is developed. Firstly, we establish a local existence-uniqueness theorem for the IDSDEs without the linear growth condition, which shows that the mild solution of the considered equation must either explode at the end of the maximum existing interval or exist globally. Then, global existence theorems for IDSDEs are obtained by improving the classical Ito's formula and applying the Razumikhin technique, which extend the classical Winter theorem on global existence for ODEs and show that the results of Xu et al. (JDE,245(2008),1681-1703) under the conditions in $L^2$ still hold in $C$. Finally, the continuous dependence on initial data for SFDEHSs is studied. These results are new even for finite dimensional stochastic differential equations. Joint work with Xiaohu Wang, Zhiguo Yang, Lingying Teng, Bing Li and Shujun Long

Keywords of the presentation: mechanical particle systems, non-equilibrium steady states, sub-exponential mixing

Rigorous derivations of macroscopic heat conduction laws from the microscopic dynamics of mechanical particle systems coupled to heat reservoirs require good mixing properties of the stationary distributions. For many such systems in nonequilibrium, i.e., with two or more unequal heat reservoirs, the proof of the mere existence of stationary distributions is nontrivial due to the non-compactness of the phase space. It is relatively easy to envision scenarios under which a particles slow down (freezing) or speed up (heating), which may push initial distributions towards zero or infinite energy levels and ultimately violate convergence. We consider a class of mechanical particle systems interacting with thermostats. Particles move freely between collisions with disk-shaped thermostats arranged periodically on the torus. Upon collision, an energy exchange occurs, in which a particle exchanges its tangential component of the velocity for a randomly drawn one from the Gaussian distribution with the variance proportional to the temperature of the thermostat; the normal component of the velocity changes sign. We show that a stationary distribution exists, is unique, and is absolutely continuous with respect to the Lebesgue measure. In addition, all initial distributions converge to the stationary distribution and a large subclass of initial distributions does so at sub-exponential rates. The sub-exponential rates of convergence are primarily due to the influence of slow particles on the system.

We discuss the existence, uniqueness, asymptotic behavior and other qualitative properties of traveling wave solutions of Allen-Cahn equation with fractional Laplacians where the double well potential has unequal depths. The existence is proved by using a continuity argument, where a key ingredient is the estimate of the speed of the traveling wave in terms of the potential in order to get the uniform estimates of the solutions. The upper bound of the speed is given explicitly in terms of potential. The method can be applied to other elliptic operators such as the standard Laplacian and the convolution-form operators, etc.. Some open problems are posted at the end of the talk.

This is a joint work with Professor Changfeng Gui.

Random Periodic solutions of SDEs and SPDEs
October 25, 2012

I will talk about the random periodic solutions of random dynamical systems generated by stochastic differential equations and stochastic partial differential equations. I will start with definition and motivations of studying a random periodic solution, and discuss its connection with periodic measure. I will demonstrate that to find a random periodic solution for a hyperbolic system is equivalent to solve a solution of coupled infinite horizon stochastic integral equations. This works for non-dissipative stochastic systems. We then discuss mathematical tools, mainly Wiener-Sobolev compact embedding, to solve such coupled infinite horizon integral equations. This talk is based on a number of works, joint mainly with Z. Zheng and C. Feng respectively.

The well-posedness of stochastic hyperbolic equations with unbounded damping operators is established. The Skorohod integral with respect to Gaussian noise is used as the driving random source, which covers time-only, space-only or space-time noises. A procedure is described for defining a generalized solution using the Cameron-Martin version of the Wiener chaos expansion, which is also an effective algorithm in numerical computations.

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