| 研究実績の概要 |
The achievements are about the studies on anisotropic stochastic partial differential equations (SPDEs) and numerical schemes for stochastic differential equations (SDEs).
Under weak assumptions, especially locally monotonic one, the existence and uniqueness of probabilistically strong solutions to anisotropic SPDEs is obtained by the variational approach. The Galerkin approximation and compactness argument are developed. Such result is expressed by two special and important anisotropic SPDEs, one is the anisotropic stochastic reaction-diffusion and the other is anisotropic stochastic Navier-Stokes equation.
The convergence of numerical schemes for stochastic differential equations with singular drift and alpha-stable noise is studied. By choosing a suitable approximation method to simulate the segment process, the convergence rate of the Euler-Maruyama scheme associated with the weakly interacting system for path-distribution dependent SDEs is obtained. To overcome the difficulties from the singular drift and the multiplicative noise, we establish a deterministic inequality and refine the regularity of solutions to the associated Kolmogorov equation, and thus obtain strong convergence of Euler-Maruyama scheme associated stochastic systems considered.
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