Research On The Existence For Solutions Of Stochastic Parabolic Anderson Models

Stochastic partial differential equation is an important tool to analysis and simulate reality system. In recent years, it is widely used in hydrodynamics, financial mathematics, chemistry, biology, control problem and many other fields, and obtained the very good development. Parabolic Anderson model is a classic example. The model was first characterized by Nobel physics prize winner P. W. Anderson on the electrons transport in some random lattices. Physical background of this equation is profound, and often appear in the field of kinetics and population dynamics.In this dissertation, we mainly discuss the existence of solutions of stochas-tic parabolic Anderson model with non-Gaussian potential under different in-tegral sense, and give the conditions of the existence of mild solution and weak solution respectively. In addition, we obtain the Cole-Hopf solution of KPZ e-quation with Gaussian noise independent of time by Cole-Hopf transformation and Feynman-Kac formula of parabolic Anderson model. This dissertation is organized as follows:Chapter1:We summarize the physical background and application of the parabolic Anderson model and KPZ equation. On the other hand, we describe the research status, progress and problems of two models briefly.Chapter2:We summarize the main theory of stochastic differential equation involved in this dissertation. Chapter3:We study the existence of solutions to the following type of parabolic Anderson model in the sense of Ito where u0(x) is a bounded measurable function, V(x) is potential, x∈Rd, and W(t) is1-dimensional Brownian motion. This model can be seen as a special parabolic Anderson model with random potential V(x)W(t)For parabolic Anderson model, a related problem is Brownian motion in random potentials, which is used to describe the trajectory of a Brownian particle that is trapped by the obstacles randomly distributed in the space. Let nonnegative K(x) be a properly chosen shape function and u(x) a random measure on R.d. We define the random function which heuristically represents the total trapping energy at x∈Rd generated by the obstacles.In our model (0.0.7), we investigate Brownian motion in random potential, and the system is affected by W(t). That is to say, some obstacles are a family of particles distributed in the space as the centered Gaussian field w(dx), where w(dx) is a Gaussian random measure with Ew[|w(dx)|2]=dx. An additional particle executes random movement in the space Rd whose trajectory Bs(s>0) is a d-dimensional Brownian motion. W(t) denotes the external random perturbation. Throughout, w(dx), Bs and W(s) are independent each other and the notations Ew, Eo and EW are used for the expectations with respect to the Gaussian field w(dx), Brownian motion Bs and W(s), respectively.In this setting, V(x) in (0.0.8) represents the total trapping energy x∈Rd generated by the Gaussian obstacles. The random integral represents the total action from both the Gaussian obstacles and external force W(s) and over the Brownian particle up to the time t.Firstly, we get the the definition of the Gaussian integral based on the infinitely divisible measure, and list the properties of the Gaussian integral.Lemma1A Borel-measurable function f(x) is integrable on Rd with respect to w(dx) if and only ifLemma2Under the assumption (0.0.9)Let K(x)≥0be a function on Rd. We consider the random integral as the (random) function in x. By the shifting invariance of Lebesgue measure, the above integral is well defined for every x∈Rd under the assumptionNext, we consider the continuity of the Gaussian potential V(x).Take M∈R,[-M,M]d is a compact domain in Rd, we consider a metric space ([-M, M]d,ρ) with the metric p(x, y)=C|x-y|θ, where0<θ≤1, and|·|is the Euclidean norm.Lemma3Assume that there is an α>0and C independent of x∈Rd such that Then{V(x);x∈Rd} has a continuous modification.In the following, we still use the notation V(x) for the continuous modi-fication of the Gaussian integral given in (0.0.10) and still writeBy the shifting invariance of the Gaussian field w(dx), we have that for any z∈Rd,If u(t, x) has a measurable version, the mild solution of the equation (0.0.7) is defined as the solution of the following integral equation if exists where pt(x,y) is the heat kernel of the Laplace operator1/2Δ on Rd: and the stochastic integral on the right hand side is taken in the sense of Ito.If the integrals∫t0V2(Bs)ds and∫t0V(Bs)dW(s) are well defined and the random field exists, we show that the u(t, x) in (0.0.11) is a mild solution to the equation (0.0.7) through Ito formula conditionally and Markov property.The following is the main results of this chapter.Theorem4Suppose that K(x) satisfies Then for t＜1/∫RdK2(x)dx,we have Consequently, the random field u(t, x) in (0.0.11) is a mild solution to equation (0.0.7).In addition, We study the existence of solutions to the KPZ equation under Gaussian noise independent of time, and we translate KPZ equation into parabolic Anderson model using Cole-Hopf transformation firstly, then obtain the solution of the KPZ model through the study of solution to parabolic Anderson model.Chapter4:we consider to solve the equation (0.0.7) under different definition of stochastic integral. We will use suitable approximation technique to show that the Feynman-Kac representation is a weak solution to model (0.0.7) in the sense that for any C∞and compactly supported function φ on Rd, where the last term is a Stratonovich stochastic integral.The following is the main result of this chapter.Theorem5Suppose that K(x) satisfies0<∫Rd K2(x)dx<+∞and lemma3. Then for the random field u(t, x) in (0.0.12) is a weak solution to equation(0.0.7).