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Semilinear Stochastic Evolution Equations with Lévy Noise

Salavati, Erfan | 2015

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 47170 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Zohuri Zangeneh, Bijan
  7. Abstract:
  8. Semilinear stochastic evolution equations with multiplicative Lévy noise and monotone nonlinear drift are considered. A novel method of proof for establishing existence and uniqueness of the mild solution is proposed. We also prove the continuous dependence of the mild solution with respect to initial conditions and also on coefficients. As corollaries of the continuity result, we derive sufficient conditions for asymptotic stability of the solutions, we show that Yosida approximations converge to the solution and we prove that solutions have Markov property. Examples on stochastic partial differential equations and stochastic delay differential equations are provided to demonstrate the theory developed. The main tool in our study is an inequality which gives a pathwise bound for the norm of stochastic convolution integrals. We also study the pth moments of solutions of stochastic evolution euations with Lévy noise and monotone drift and we prove the existence of the mild solution in Lp and provide a sufficient condition for exponential asymptotic stability of the solutions in Lp. The main tool in the study of pth moments, is a new inequality for the pth power of the norm of a stochastic convolution integral in a Hilbert space, which we state and prove. This inequality is stronger than analogues inequalities in the literature in the sense that it is pathwise and not in expectation
  9. Keywords:
  10. Levy Process ; Stochastic Evolution Equation ; Semigroups Theory ; Stochastic Partial Differential Equation ; Monotone Operators ; Stochastic Convolution Integrals ; Stochastic Maximal Inequalities

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