The first exit time of a biharmonic pseudo process and its application to the boundary value problems

半线弯曲梁由结构端固定，其表示为四阶偏微分方程θ_tu(t,x)=-θ^4_xu(t,x) t>,x>0θ^j_xu(t,0)=0 θ^k_xu(t,0) t>=0 t>0，其中（j,k）取自x=0处结构端导出的组合之一：(j,k)=(0,1)；固定端，(j,k)=(0,2)；固定端，(j,k)=(1,3)；侧端，(j,k)=(2,3)；对于每一个（j,k），我们构造了满足随机边界条件的双调和伪过程{X^<(jk)>(t)}，它对应于上式。然后，在分析了相应的{X^<(jk)>(t)}在边界处的行为后，我们研究了弯曲梁的精确运动。

A half line bending beam is fixed by a structural end, what is represented by the 4-th order partial differential equationθ_tu(t,x)=-θ^4_xu(t,x) t>0,x>0θ^j_xu(t,0)=0 θ^k_xu(t,0) t>=0 t>0,where (j,k) is taken from one of such combinations derived from structural ends at x=0 :(j,k)=(0,1) ; The fixed end, (j,k)=(0,2) ; The pinned end,(j,k)=(1,3) ; The side end, (j,k)=(2,3) ; The free endFor each (j,k), we constructed the diharmonic pseudo processes {X^<(jk)>(t)} which satisfies a stochastic boundary condition and corresponds to the above equation. Then, we investigated a precise motion of the bending beam, after analyzing behaviors of the corresponding {X^<(jk)>(t)} at the boundary.