Pathwise uniqueness for stochastic reaction-diffusion equations in Banach spaces with an Hölder drift component (Q378034)

The authors consider the general class of reaction-diffusion equations in the Banach space \[ dx(t)= [Ax(t)+ F(X(t))+ B(X(t))]\,dt+ dW(t),\quad X(0)=x, \] where \(A\) is the Laplacian operator in the one-dimensional space domain \([0,1]\) with Dirichlet or Neumann boundary conditions, the Banach space \(E\) is the closure of \(D(A)\) in \(C([0, 1])\), \(x\in E\), \(F\) is a very general reaction-diffusion operator in \(E\), \(B: E\to E\) is a Hölder continuous and bounded operator, \(W(t)\) is a space-time Wiener process. The authors prove the pathwise uniqueness of the solution for the considered equation using the associated Kolmogorov equation and the perturbation technique. Unfortunately, the obtained results are not illustrated by an example.