By M. N. Feller
The Lévy Laplacian is an infinite-dimensional generalization of the well known classical Laplacian. the speculation has turn into well-developed in recent times and this publication is the 1st systematic therapy of the Lévy-Laplace operator. The booklet describes the infinite-dimensional analogues of finite-dimensional effects, and extra particularly these gains which seem basically within the generalized context. It develops a conception of operators generated through the Lévy Laplacian and the symmetrized Lévy Laplacian, in addition to a idea of linear and nonlinear equations concerning it. there are various difficulties resulting in equations with Lévy Laplacians and to Lévy-Laplace operators, for instance superconductivity conception, the idea of regulate platforms, the Gauss random box concept, and the Yang-Mills equation. The booklet is complemented by way of an exhaustive bibliography. the result's a piece that might be valued via these operating in practical research, partial differential equations and likelihood conception.
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Extra info for The Levy Laplacian
Example text
X 2H ) L V (x) = (2ϕ ( x 2 H )/ϕ( x 2 H ))V (x), V (x) = ϕ( x x then the function V (x) 2 H )S(x), where S(x) is an arbitrary harmonic function. Proof. , |V (x)| ln = 0. ϕ( x 2H ) |V (x)| = G(x), ϕ( x 2H ) x . 3 Self-adjointness of the non-symmetrized L´evy–Laplace operator 51 where G(x) is an arbitrary harmonic function. Hence, V (x) = ϕ( x 2 G(x) He = ϕ( x 2 H )S(x), where S(x) is an arbitrary harmonic function. We define an operator L in L2 (H, µ) with everywhere dense domain of definition D L , putting LU = D L = T.
1/2 (γm , ⊗y m ) H m , where the kernel γm ∈ ⊗H−m is such that γm (x) ∈ L2 (H, µ): (γm , y ⊗ . . ⊗ w)⊗H m is a symmetric m-linear continuous form (γm ∈ ⊗H−m , y, . . , w ∈ H+ ). )−1/2 (γm , ⊗x m )⊗H m for its measurable extension to H (although here x ∈ H, and not H+ ). The projection of such a form γm (x) onto m is a polynomial Pγm (x), which consists of γm (x) and the homogeneous forms of smaller degree ν (x, γm ), ν < m. In addition, (Pγm , Pγn )L2 (H,µ) = 0, Pγm ∈ for all m, for all Pγn ∈ n, n < m.
3. Functions of the form F(x) = f (U1 (x), . . , Um (x)), where f (u 1 , . . , u m ) is a twice continuously differentiable function of m variables over the set {U1 (x), . . , Um (x)} in Rm , and functions U j (x) are harmonic in the whole space H, j = 1, . . , m. Indeed, since L U j (x) = 0, j = 1, . . 4), m L F(x) = j=1 4. H = l 2, F(x) = ∞ ∂F ∂u j u j =U j (x) αk λ2m−2 xk2m k L U j (x) = 0. (2m + 2)(2m + 1) 2 m ≥ 1 is an integer, xk = (x, ek )l2 , ek = (0, . . , 1, 0, . ), K −1/2 ek = λk ek , k = 1, 2, .
