Stochastic Functional (Partial) Differential Equations
In this chapter we investigate Harnack/shift Harnack inequalities and derivative formulas for stochastic functional differential equations. In this case, the strong or mild solution is no longer Markovian. These inequalities and formulas are therefore established for the semigroup associated with the functional (or segment) solutions. To this end, a time larger than the length of delay is necessary in order to construct a successful coupling by change of measure. Based on Bao et al. (Derivative formula and Harnack inequality for degenerate functional SDEs, to appear in Stochast. Dynam.; Bismut Formulae and Applications for Functional SPDEs, to appear in Bull. Math. Sci.), Shao et al. (Electron. J. Probab. 17:1–18, 2012), Wang and Yuan (Stoch. Process. Their Appl. 121:2692–2710, 2011), several specific models of elliptic SDDEs, semilinear SDPDEs, and degenerate SDDEs are considered.
KeywordsBrownian Motion Mild Solution Harnack Inequality Coupling Time Malliavin Calculus
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