We establish stochastic functional integral representations for solutions of Oberbeck-Boussinesq equations in the form of McKean-Vlasov-type mean field equations, which can be used to design numerical schemes for calculating solutions and for implementing Monte-Carlo simulations of Oberbeck-Boussinesq flows. Our approach is based on the duality of conditional laws for a class of diffusion processes associated with solenoidal vector fields, which allows us to obtain a novel integral representation theorem for solutions of some linear parabolic equations in terms of the Green function and the pinned measure of the associated diffusion. We demonstrate via numerical experiments the efficiency of the numerical schemes, which are capable of revealing numerically the details of Oberbeck-Boussinesq flows within their thin boundary layer, including B{\'e}nard's convection feature.

翻译：我们建立了奥伯贝克-布辛涅斯克方程解在McKean-Vlasov型平均场方程形式下的随机泛函积分表示，可用于设计求解数值方案并实现奥伯贝克-布辛涅斯克流动的蒙特卡罗模拟。该方法基于与无散向量场相关的一类扩散过程的条件对偶性，从而利用格林函数和相关扩散过程的钉扎测度，获得一类线性抛物型方程解的新型积分表示定理。通过数值实验验证了数值方案的效率，该方案能够数值揭示奥伯贝克-布辛涅斯克流动在薄边界层内的细节，包括贝纳德对流特征。