The Hölder and the Besov regularity of the density for the solution of a parabolic stochastic partial differential equation
Consider the stochastic partial differential equation \[ {\partial X\over\partial t} (t,x)= {\partial^2X\over\partial x^2} (t,x)+ \psi(X(t, x))+ \varphi(X(t, x))\dot W(t,x) \] on \([0,T]\times [0,1]\) with Neumann boundary conditions, where \(W\) is a space-time white noise. Assume that the coefficients together with all their derivatives are bounded and \(|\varphi(x)|\geq c>0\). Using the techniques of the Malliavin calculus, \textit{V. Bally} and \textit{E. Pardoux} proved [Potential Anal. 9, No. 1, 27-64 (1998; Zbl 0928.60040)] that the solution \(X(t,x)\) for any \((t,x)\in (0,T]\times [0,1]\) has an infinitely differentiable density \(p_{t,x}(y)\). The author proves that \(p_{t,x}(y)\) is \({1\over 2}\)-Hölder continuous in the variable \(t\) and \((1-\varepsilon)\)-Hölder continuous in the variable \(x\) for all \(\varepsilon\in (0,1)\). Moreover, as a function of \(x\) it belongs to the Besov space \(B_{1,\infty,\infty}\). The proof of these results is based on the integration by parts formula of Malliavin calculus and on some refined estimates for the Green kernel of the one-dimensional heat equation.
parabolic stochastic partial differential equations
scientific article; zbMATH DE number 1294461