Feynman formulas for nonlinear evolution equations

Transformations of measures, generalized measures, and functions generated by evolution differential equations on a Hilbert space E are studied. In particular, by using Feynman formulas, a procedure for averaging nonlinear random flows is described and an analogue of the law of large number for such flows is established (see [1, 2]).     

Feynman-Kac and Feynman formulas for infinite-dimensional equations with Vladimirov operator

Doklady Mathematics -     

Representations of solutions of Lindblad equations by randomized Feynman formulas

Representations of solutions of Lindblad equations by randomized Feynman integrals over trajectories are obtained by averaging similar representations for solutions of stochastic Schrödinger equations (Schrödinger–Belavkin equations). An approach based on the application of Chernoff’s theorem is applied. First, (randomized) Feynman formulas approximating Feynman path integrals are obtained; these formulas contain integrals over finite Cartesian powers of the space of values of the functions over which the Feynman integrals are taken.     

Superposition formulas for integrable vector evolution equations

We use the Bäcklund transformation to construct superposition formulas for solutions of some integrable vector equations.     

Change of variable formulas for Feynman pseudomeasures

We derive formulas describing the transformations of Feynman pseudomeasures generated by nonlinear permutations of the phase space. In particular, we obtain analogues of the Ramer formula for the Gauss measures and of the change of variable formula proved by Elworthy and Truman. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 355–367, June, 1999.     

Maslov-Poisson measure and Feynman formulas for the solution of the Dirac equation

As the main step, the method used by V. P. Maslov for representing a solution of the initial-value problem for the classical Schrödinger equation and admitting an application to the Dirac equation includes the construction of a cylindrical countably additive measure (that is an analog of the Poisson distribution) on a certain space of functions (= trajectories in the impulse space) whose Fourier transform coincides with the factor in the formula for representation of the solution of the Schrödinger equation by the integral in the so-called cylindrical Feynman (pseudo) measure (in the trajectory space of the configurational space for the classical system). On the other hand, in the Maslov formula for the solution of the Schrödinger equation, the exponential factor is (with accuracy up to a shift) the Fourier transform of the Feynman pseudomeasure. In the case of the Dirac equation, historically, for the first time, there arose formulas for the impulse representation that use countably additive functional distributions of the Poisson-Maslov measure type but with noncommuting (matrix) values. The paper finds generalized measures whose Fourier transforms coincide with an analog of the exponential factor under the integral sign in the Maslov-type formula for the Dirac equation and integrals with respect to which yield solutions of the Cauchy problem for this equation in the configurational space.     

Representation of solutions to evolution equations with Vladimirov operator in terms of Feynman path integrals

Doklady Mathematics -     

Feynman formulas and functional integrals for diffusion with drift in a domain on a manifold

We obtain representations for the solution of the Cauchy-Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as limits of integrals over the Cartesian powers of the domain; the integrands are elementary functions depending on the geometric characteristics of the manifold, the coefficients of the equation, and the initial data. It is natural to call such representations Feynman formulas. Besides, we obtain representations for the solution of the Cauchy-Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as functional integrals with respect to Weizsäcker-Smolyanov surface measures and the restriction of the Wiener measure to the set of trajectories in the domain; such a restriction of the measure corresponds to Brownian motion in a domain with absorbing boundary. In the proof, we use Chernoff’s theorem and asymptotic estimates obtained in the papers of Smolyanov, Weizsäcker, and their coauthors.     

Subsolutions for abstract evolution equations

We study different notions of subsolutions for an abstract evolution equation du/dt+Auf where A is an m-accretive nonlinear operation in an ordered Banach space X with order-preserving resolvents. A first notion is related to the operator d/dt+A in the ordered Banach space L ¹(0, T; X); a second one uses the evolution equation du/dt+A uf where A :x{y;zy for some zAx}; other notions are also considered.