AN INTEGRATION BY PARTS FORMULA FOR STOCHASTIC HEAT EQUATIONS WITH FRACTIONAL NOISE

In this paper,we establish the integration by parts formula for the solution of fractional noise driven stochastic heat equations using the method of coupling.As an application,we also obtain the shift Harnack inequalities.

MARTINGALE REPRESENTATION AND LOGARITHMIC-SOBOLEV INEQUALITY FOR THE FRACTIONAL ORNSTEIN-UHLENBECK MEASURE

In this paper,we consider the measure determined by a fractional OrnsteinUhlenbeck process.For such a measure,we establish an explicit form of the martingale representation theorem and consequently obtain an explicit form of the Logarithmic-Sobolev inequality.To this end,we also present the integration by parts formula for such a measure,which is obtained via its pull back formula and the Bismut method.

An integration by parts formula on path space over manifolds carrying geometric flow

We establish an integration by parts formula on the path space with reference measure P, the law of the(reflecting) diffusion process on manifolds with possible boundary carrying geometric flow, which leads to the standard log-Sobolev inequality for the associated Dirichlet form. To this end, we first modify Hsu's multiplicative functionals to define the damp gradient operator, which links to quasi-invariant flows; and then establish the derivative formula for the associated inhomogeneous diffusion semigroup.

Shift Harnack inequality and integration by parts formula for semilinear stochastic partial differential equations

Shift Harnack inequality and integration by parts formula are established for semilinear stochastic partial differential equations and stochastic functional partial differential equations by modifying the coupling used by F. -Y. Wang [Ann. Probab., 2012, 42（3）： 994-1019]. Log-Harnack inequality is established for a class of stochastic evolution equations with non- Lipschitz coefficients which includes hyperdissipative Navier-Stokes/Burgers equations as examples. The integration by parts formula is extended to the path space of stochastic functional partial differential equations, then a Dirichlet form is defined and the log-Sobolev inequality is established.

Derivative Formula and Coupling Property for Linear SDEs Driven by Lévy Processes

In this paper we investigate an integration by parts formula for Lévy processes by using lower bound conditions of the corresponding Lévy measure. As applications, derivative formula and coupling property are derived for transition semigroups of linear SDEs driven by Lévy processes.

Second Order Nonlinear Evolution Inclusions Existence and Relaxation Results

This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x（t）. In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 （T, H）. Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 （T, H） to the solutions of the original convex problem （strong relaxation）. An example of a nonlinear hyperbolic optimal control problem is also discussed.

Some Liouville type results of Hessian equations

We obtain some Liouville type results of Hessian equations on Sn.