M. N. Feller

We develop a method to construct a solution to a boundary problem and an initial-boundary value problem in a fundamental domain of a Hilbert space for a class of nonlinear parabolic equations not containing explicitly the unknown function, $$\frac{\partial U(t,x)}{\partial t}=f(t,\Delta_LU(t,x)),$$ where $\Delta _L$ is the infinite dimensional Lévy Laplacian.

We present solutions to some boundary value and initial-boundary value problems for the "wave" equation with the infinite dimensional L\'evy Laplacian $\Delta _L$ $$\frac{\partial^2 U(t,x)}{\partial t^2}=\Delta_LU(t,x)$$ in the Shilov class of functions.

We construct solutions to initial, boundary and initial-boundary value problems for quasilinear parabolic equations with an infinite dimensional Lévy Laplacian $\Delta _L$, $$\frac{\partial U(t,x)}{\partial t}=\Delta_LU(t,x)+f_0(U(t,x)),$$ in fundamental domains of a Hilbert space. The solution is defined in the functional class where a solution of the corresponding problem for the heat equation $\frac {\partial U(t,x)}{\partial t}=\Delta_LU(t,x)$ exists.

We suggest a method to solve boundary and initial-boundary value problems for a class of nonlinear parabolic equations with the infinite dimensional L'evy Laplacian $\Delta _L$ $$f\Bigl(U(t,x),\frac{\partial U(t,x)}{\partial t},\Delta_LU(t,x)\Bigl)=0$$ in fundamental domains of a Hilbert space.

A Dirichlet form associated with the infinite dimensional symmetrized Levy-Laplace operator is constructed. It is shown that there exists a natural connection between this form and a Markov process. This correspondence is similar to that studied in a previous paper by the same authors for the non-symmetric Levy Laplacian.