In this talk, we will introduce support properties of solutions to nonlinear stochastic reaction-diffusion equations driven by random noise ˙W : ∂tu = aijuxixj + biuxi + cu + ξσ(u) ˙W , (ω, t, x) ∈ Ω × R+ × Rd; u(0, ·) = u0, where aij , bi, c and ξ are bounded and random coefficients. The noise ˙W is spacetime white noise or spatially homogeneous colored noise satisfying reinforced Dalang’s condition. We present examples of conditions on σ(u) that guarantee the compact support property of the solution. In addition, we suggest potential generalization of these conditions. This is joint work with Kunwoo Kim and Jaeyun Yi.

This presentation focuses on unbiased simulation methods for quantities associated with sample paths from stochastic differential equations. Unbiased simulation methods can be found by changing probability measures with appropriately selecting the Radon-Nikodym derivative processes. I propose an unbiased Monte-Carlo simulation method that can be used even when the Girsanov kernel for the change of probability measures is not bounded. Then, I illustrate its practical application through an example involving unbiased Monte Carlo simulation for pricing the continuously averaging arithmetic Asian options under the Black-Scholes model.

Weak discuss existence of singular solutions for Stokes and the Navier-Stokes equations in the half-space. We construct their solutions whose normal derivatives are unbounded for the Stokes and Navier-Stokes equations near boundary away from support of singular data. 

Atoms and molecules aim to minimize surface energy, while crystals exhibit directional preferences. Starting from the classical isoperimetric problem, we investigate the evolution of volume-preserving crystalline mean curvature flow. Defining a notion of viscosity solutions, we demonstrate the preservation of geometric properties associated with the Wulff shape. We establish global-in-time existence and regularity for a class of initial data. Furthermore, we discuss recent findings on the long-time behavior of the flow towards the critical point of the anisotropic perimeter functional in a planar setting.