Mild solutions of SPDE's driven by Poisson noise in infinite dimensions and their dependence on initial conditions

Knoche C (2005)
Bielefeld (Germany): Bielefeld University.

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Bielefeld Dissertation | English
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Röckner, Michael
Alternative Title
Milde Lösungen von SPDEs mit Poisson-Rauschen in unendlich dimensionalen Hilberträumen und ihre Abhängigkeit vom Anfangswert
Abstract
We investigate stochastic partial differential equations in an infinite dimensional Hilbert space H of the following form: dX(t)=[AX(t)+F(X(t))]dt+B(X(t),y)q(dt,dy), X(0)=g, where q(dt,dy) is a compensated Poisson random measure associated to a stationary Poisson point process on a [sigma]-finite measure space (U,C,m), A is the generator of a strongly continuous semigroup, F is a measurable mapping from H to H, B is a measurable mapping from the product space HxU to H and g is an H-valued random variable. We are interested in the existence and uniqueness of a so called mild solution. But, apart from the existence and uniqueness of mild solutions our main interest is directed towards their regularity w.r.t. the initial datum g. Under Lipschitz assumptions on the drift and the diffusion we show that for every initial condition g, which is a p-integrable random variable, there exists a unique mild solution in the space of predictable p-integrable processes, where p>=2. In the case that p equals two this is done by Banach's fixed point theorem and the help of the isometric property of the stochastic integral. The case p>2 is crucial for the analysis of the dependence on the initial condition. For the proof of the existence of a unique mild solution in the space of predictable p-integrable processes, where p>2, one needs an appropriate estimate for the p'th moment of the stochastic integral. In this paper we prove a generalized Burkholder-Davis-Gundy-inequality, for estimating the p'th moment of the stochastic integral. This inequality enables us to prove the existence of a unique mild solution in the space of predictable p-integrable processes, where p>2. Under the main assumption that the coefficients F and B are Gâteaux differentiable we prove the Gâteaux differentiability of the mild solution as a mapping from the space of square integrable random variables to the space of predictable square integrable processes. Already the Gâteaux differentiability enables us to give, under the additional assumptions that F is dissipative, the intensity measure m of q is finite and B( ,y) is constant, an [omega]-wise estimate for the Gâteaux derivative of the mild solution and gradient estimates for the resolvent corresponding to the mild solution. Under stronger assumptions than needed for the statement about the Gâteaux differentiability, we prove our main results: the first and second order Fréchet differentiability of the mild solution. Under the main assumption that the coefficients F and B are Fréchet differentiable, twice Fréchet differentiable, respectively, we prove that the mild solution as a mapping from the space of q-integrable random variables to the space of predictable p-integrable processes is Fréchet-differentiable if q>p>=2, twice Fréchet-differentiable if q>4p>=8, respectively.
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Knoche C. Mild solutions of SPDE's driven by Poisson noise in infinite dimensions and their dependence on initial conditions. Bielefeld (Germany): Bielefeld University; 2005.
Knoche, C. (2005). Mild solutions of SPDE's driven by Poisson noise in infinite dimensions and their dependence on initial conditions. Bielefeld (Germany): Bielefeld University.
Knoche, C. (2005). Mild solutions of SPDE's driven by Poisson noise in infinite dimensions and their dependence on initial conditions. Bielefeld (Germany): Bielefeld University.
Knoche, C., 2005. Mild solutions of SPDE's driven by Poisson noise in infinite dimensions and their dependence on initial conditions, Bielefeld (Germany): Bielefeld University.
C. Knoche, Mild solutions of SPDE's driven by Poisson noise in infinite dimensions and their dependence on initial conditions, Bielefeld (Germany): Bielefeld University, 2005.
Knoche, C.: Mild solutions of SPDE's driven by Poisson noise in infinite dimensions and their dependence on initial conditions. Bielefeld University, Bielefeld (Germany) (2005).
Knoche, Claudia. Mild solutions of SPDE's driven by Poisson noise in infinite dimensions and their dependence on initial conditions. Bielefeld (Germany): Bielefeld University, 2005.
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