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 and quasi-Feynman formulas for evolution equations

New methods for obtaining representations of solutions of the Cauchy problem for linear evolution equations, i.e., equations of the form u _t '(t, x) = Lu(t, x), where the operator L is linear and depends only on the spatial variable x and does not depend on time t, are proposed. A solution of the Cauchy problem, that is, the exponential of the operator tL, is found on the basis of constructions proposed by the author combined with Chernoff’s theorem on strongly continuous operator semigroups.     

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

Doklady Mathematics -     

Representations for the solutions and coefficients of evolution equations

We give new representations for the solutions and coefficients of evolution equations in the linear case. The obtained formulas contain some functional arbitrariness that can be used in identification problems. We also give classes of hyperbolic equations that admit the generalized functionally invariant solutions.     

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.     

Representations of solutions of nonlocal boundary-value problems for parabolic equations

We study two-point boundary-value problems for parabolic equations whose solutions are representable in terms of Green's functions of the Cauchy problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 947–951, July, 1994.     

Asymptotic formulas for nonoscillatory solutions of conditionally oscillatory half-linear equations

We establish asymptotic formulas for nonoscillatory solutions of a special conditionally oscillatory half-linear second order differential equation, which is seen as a perturbation of a general nonoscillatory half-linear differential equation

Representations for the solutions and coefficients of second-order differential equations

New representations are given for the solutions and coefficients of second-order evolution differential equations in the linear and nonlinear cases. The formulas for linear equations have wide arbitrariness which can be used in identification problems. We study the questions of running-wave type for nonlinear one-dimensional equations.     

Representations of solutions to linear integral equations with difference kernels

For linear integral equations with difference kernels, we give a formula for the inverse in terms of the solutions to two specific problems. If the equation is self-adjoint, these solutions are simply related.     

Filtration formulas for solutions of nonlinear equations with random right-hand sides

Explicit filtration formulas are obtained for the solutions of nonlinear differential equations with random right-hand sides. In the case of a Gaussian random process, these formulas are simplified.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 586–596, May, 1994.     

Product formulas for solutions of initial value partial differential equations,I

Families of operators which approximate semi-groups or evolution systems generated by partial differential operators are constructed. Product formulas are used to recover these semi-groups or evolution systems through product integrals. Conditions on generators are provided under which its semi-group or evolution system can be approximated in this way by families of specific types of operators.