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Mathematical Analysis of Large-Scale Biological Neural Networks with Delay

Mehri, Sima | 2019

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 52199 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Zohuri-Zangeneh, Bijan
  7. Abstract:
  8. It is well-known that the components of solution to a system of N interacting stochastic differential equations with an averaged sum of interaction terms and with independent identically distributed (chaotic) initial values , as $N$ tends to infinity , converge to the solutions of Vlasov-McKean equations , in which the averaged sum is replaced by the expectation . Since the solutions to the corresponding Vlasov-McKean equations are independent , this phenomenon is called propagation of chaos . This thesis is about well-posedness of path-dependent stochastic differential equations , propagation of chaos for spatially structured neural network with delay and existence and uniqueness of weak solutions to Vlasov-McKean equations . In Chapter 2, existence and uniqueness of strong solution to path-dependent stochastic differential equations driven by martingale noise under local monotonicity and coercivity assumptions with controls with respect to supremum norm are obtained . Because the noise coefficient is not separately coercive and local monotone , using ordinary Gronwall lemma together with Burkholder-Davis-Gundy theorem is impossible . As a solution to this issue , a stochastic Gronwall lemma for c\`adl\`ag martingales is proved . This result is obtained in joint work with Michael Scheutzow . In Chapter 3 , we consider spatially structured neural networks driven by martingale noise with monotone coefficients , fully path dependent delay and with a disorder parameter . Well-posedness of the network equations is implied by the first result . Well-posedness for the associated Vlasov-McKean equation and a corresponding propagation of chaos result in the infinite population limit are proven . Our existence result for the Vlasov-McKean equation is based on the Euler approximation , that is applied to this type of equation for the first time . This result is obtained in joint work with Michael Scheutzow , Wilhelm Stannat , and Bijan Zohuri-Zangeneh . In Chapter 4 , we present a Lyapunov type approach to the problem of existence and uniqueness of general law-dependent stochastic differential equations . In the existing literature , most results concerning existence and uniqueness are obtained under regularity assumptions of the coefficients with respect to the Wasserstein distance . Some existence and uniqueness results for irregular coefficients have been obtained by considering the total variation distance . Here we extend this approach to the control of the solution in some weighted total variation distance , that allows us now to derive a rather general weak uniqueness result , merely assuming measurability and certain integrability on the drift coefficient and some non-degeneracy on the dispersion coefficient . We also present an abstract weak existence result for the solution of law-dependent stochastic differential equations with merely measurable coefficients , based on an approximation with law-dependent stochastic differential equations with regular coefficients under Lyapunov type assumptions . This result is obtained in joint work with Wilhelm Stannat
  9. Keywords:
  10. Chaos Propagation ; Neural Networks ; Path-Dependent Stochastic Differential Equations ; Law-Dependent Stochastic Differential Equations ; Vlosov-McKean Equation

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