THE LÉVY LAPLACIAN

By CHARLES SMITH

CONTENTS INTRODUCTION THE L´EVY LAPLACIAN DE?NITION OF THE IN?NITE-DIMENSIONAL LAPLACIAN EXAMPLES OF LAPLACIANS FOR FUNCTIONS ON IN?NITE- DIMENSIONAL SPACES GAUSSIAN MEASURES L´EVY–LAPLACE OPERATORS IN?NITE-DIMENSIONAL ORTHOGONAL POLYNOMIALS THE SECOND-ORDER DIFFERENTIAL OPERATORS GENERATED BY THE L´EVY LAPLACIAN DIFFERENTIAL OPERATORS OF ARBITRARY ORDER GENERATED BY THE L´EVY LAPLACIAN SYMMETRIC L´EVY–LAPLACE OPERATOR THE SYMMETRIZED L´EVY LAPLACIAN ON FUNCTIONS FROM THE DOMAIN OF DE?NITION OF THE L´EVY–LAPLACE OPERATOR THE L´EVY LAPLACIAN ON FUNCTIONS FROM THE DOMAIN OF DE?NITION OF THE SYMMETRIZED L´EVY–LAPLACE OPERATOR SELF-ADJOINTNESS OF THE NON-SYMMETRIZED L´EVY–LAPLACE OPERATOR HARMONIC FUNCTIONS OF IN?NITELY MANY VARIABLES ARBITRARY SECOND-ORDER DERIVATIVES ORTHOGONAL AND STOCHASTICALLY INDEPENDENT SECOND-ORDER DERIVATIVES TRANSLATIONALLY NON-POSITIVE CASE CONTENTS LINEAR ELLIPTIC AND PARABOLIC EQUATIONS WITH L´EVY LAPLACIANS THE DIRICHLET PROBLEM FOR THE L´EVY–LAPLACE AND L´EVY–POISSON EQUATIONS THE DIRICHLET PROBLEM FOR THE L´EVY–SCHR¨ODINGER STATIONARY EQUATION THE RIQUIER PROBLEM FOR THE EQUATION WITH ITERATED L´EVY LAPLACIANS THE CAUCHY PROBLEM FOR THE HEAT EQUATION QUASILINEAR AND NONLINEAR ELLIPTIC EQUATIONS WITH L´EVY LAPLACIANS THE DIRICHLET PROBLEM FOR THE EQUATION LU(X) = F (U(X)) THE DIRICHLET PROBLEM FOR THE EQUATION F (U(X), LU(X)) = F(X) THE RIQUIER PROBLEM FOR THE EQUATION LU(X) = F (U(X)) THE RIQUIER PROBLEM FOR THE EQUATION F (U(X), LU(X)) = LU(X) THE RIQUIER PROBLEM FOR THE EQUATION F (U(X), LU(X), LU(X)) = NONLINEAR PARABOLIC EQUATIONS WITH L´EVY LAPLACIANS THE CAUCHY PROBLEM FOR THE EQUATIONS ?U(T, X)/?T = F (LU(T, X)) AND ?U(T, X)/?T = F (T, LU(T, X)) THE CAUCHY PROBLEM FOR THE EQUATION ?U(T, X)/?T = F (U(T, X), LU(T, X)) THE CAUCHY PROBLEM FOR THE EQUATION ?(T, ?U(T, X)/?T) = F (F(X), LU(T, X)) THE CAUCHY PROBLEM FOR THE EQUATION F (U(T, X), ?U(T, X)/?T, LU(T, X)) =