Homogeneous closed PDO are constructed which are analogous to the powers of (absolute value of ) ... more Homogeneous closed PDO are constructed which are analogous to the powers of (absolute value of ) infinite dimensional Laplacian and acting in Banach spaces of complexvalued functions defined on function spaces over a field of p-adic numbers. For elements of semigroups, for which these PDOs are generators, Feynman formulas and Feynman-Kac ones are obtained.
Latin American Journal of Probability and Mathematical Statistics, 2016
In this work, we establish pathwise functional Itô formulas for nonsmooth functionals of real-val... more In this work, we establish pathwise functional Itô formulas for nonsmooth functionals of real-valued continuous semimartingales. Under finite (p, q)variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition F t (X t) = F 0 (X 0
Properties of infinite dimensional pseudo-differential operators (PDO) are discussed; in particul... more Properties of infinite dimensional pseudo-differential operators (PDO) are discussed; in particular, the connection between two definitions of the PDO is considered: one given in terms of the Hamiltonian Feynman measure, and another introduced in this work in terms of the Kolmogorov integral.
We prove a version of the Stokes formula for differential forms on locally convex spaces announce... more We prove a version of the Stokes formula for differential forms on locally convex spaces announced in [10]. The main tool used for proving this formula is the surface layer theorem proved in the paper [6] by the author. Moreover, for differential forms of a Sobolev-type class relative to a differentiable measure [1], we compute the operator adjoint to the exterior differential in terms of standard operations of calculus of differential forms and the logarithmic derivative. Previously, this connection was established under essentially stronger assumptions on the space [8], the measure [7], or smoothness of differential forms [5]. See also [4].
We prove formulas that describe, under certain sufficiently general assumptions, the Fourier tran... more We prove formulas that describe, under certain sufficiently general assumptions, the Fourier transforms of distributions of processes and fields mentioned in the title. A similar method applied to complex Markov chains on a finite segment can be used to compute the Fourier transform of the complex Poisson measure from [1, Chap. IX] (where it is found by series expansion of the measure) given its finite-dimensional distributions. We assume that the value space of a random field is a Polish (i.e., complete, separable, and metrizable) locally convex space; however, the results seem to be new for real processes as well. An additional assumption on the processes is that their trajectories have no discontinuities of the second kind in the (weakened) topology of the value space; in particular, such are the Poisson jump processes. Earlier [2] such a formula was proved only for processes with strongly continuous trajectories. Also discussed in [2] are applications of a similar construction for continuous fields to the particular case of Wiener–Levy–Chentsov fields [3].
Transactions of the Moscow Mathematical Society, 2005
A family {M p } p∈R d of cylindrical measures is constructed on the space of functions (= traject... more A family {M p } p∈R d of cylindrical measures is constructed on the space of functions (= trajectories) f : [0, +∞) → R d such that for every t ≥ 0 the formula 2000 Mathematics Subject Classification. Primary 28B99; Secondary 46G10, 81Q05.
Homogeneous systems of matrix-valued transition measures similar to Markov systems and generating... more Homogeneous systems of matrix-valued transition measures similar to Markov systems and generating cylindrical countably additive measures on some function spaces are constructed and the Fourier transforms of the latter measures (= functional distributions) are evaluated; the case of complex values corresponds to Markov-Poisson chains introduced by V. P. Maslov in 1976. The state spaces of these systems (similar to the Markov ones) are measurable real vector spaces. These distributions (their Fourier transforms, respectively can be used to obtain Maslov-like formulas (Feynman-like formulas, respectively) to solve Schrödinger equations with effective potentials with values in complex matrix algebras. It is also shown that, if the values of the matrix potential commute, then in the same way, one can construct analogs of multitemporal Markov fields and evaluate the Fourier transforms of the corresponding functional distributions.
In this work, we establish pathwise functional It\^o formulas for non-smooth functionals of real-... more In this work, we establish pathwise functional It\^o formulas for non-smooth functionals of real-valued continuous semimartingales. Under finite $(p,q)$-variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition $$F_t(X_t) = F_0(X_0)+ \int_0^t\nabla^hF_s(X_s)ds + \int_0^t\nabla^wF_s(X_s)dX(s) - \frac{1}{2}\int_{-\infty}^{+\infty}\int_0^t(\nabla^w_xF_s)(^{x}X_s)d_{(s,x)}\ell^x(s)$$ Here, $X_t= \{X(s); 0\le s\le t\}$ is the continuous semimartingale path up to time $t\in [0,T]$, $\nabla^h$ is the horizontal derivative, $(\nabla^w_x F_s)(^{x}X_s)$ is a weak derivative of $F$ with respect to the terminal value $x$ of the modified path $^{x}X_s$ and $\nabla^w F_s(X_s) = (\nabla^w_x F_s)(^{x} X_s)|_{x=X(s)}$. The double integral is interpreted as a space-time 2D-Young integral with differential $d_{(s,x)}\ell^x(s)$, where $\ell$ is the local-time of $X$. Under less restrictive joint variation assumptions on $...
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2007
Cauchy problems for Schrödinger equations with matrix-valued potentials are explicitly solved und... more Cauchy problems for Schrödinger equations with matrix-valued potentials are explicitly solved under following assumptions:. — equations are written in momentum form;. — the potentials are Fourier transformed matrix-valued measures with, in general, noncommuting values;. — initial Cauchy data are good enough. The solutions at time t are presented in form of integrals over some spaces of piecewise continuous mappings of the segment [0, t] to a finite-dimensional momentum space, and measures of the integration are countably additive but matrix-valued (resulting in matrices of ordinary Lebesgue integrals). Known results gave solutions either when the measure had commuting (complex) values, or when the integration over infinite dimensional spaces was quite symbolic, or when such integrating was of chronological type and hence more complicated. The method used below is based on technique of matrix-valued transition amplitudes.
As the main step, the method used by V. P. Maslov for representing a solution of the initial-valu... more 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.
A class of the Schroedinger type equations, with respect to functions defined on the product of [... more A class of the Schroedinger type equations, with respect to functions defined on the product of [0∞) (``time'') and of the p-adic field (``space''), is considered. In the equation a Vladimirov operator substitutes the standard Lalacian. The main aim of the paper is to get representations of solutions of Cauchy problems for such equations by integrals over Feynman pseudomeasures on paths in the p-adic field (which is spacial domains of the solutions). The significant role is played by some p-adic analogs of Poisson-Maslov measures which are constructed in the paper and are used to get the representations.
Proceedings of the Steklov Institute of Mathematics, 2009
ABSTRACT We obtain Feynman formulas in the momentum space and Feynman-Kac formulas in the momentu... more ABSTRACT We obtain Feynman formulas in the momentum space and Feynman-Kac formulas in the momentum and phase spaces for a p-adic analog of the heat equation in which the role of the Laplace operator is played by the Vladimirov operator. We also present the Feynman and Feynman-Kac formulas in the configuration space that have been proved in our previous papers under additional constraints. In all these formulas, integration is performed with respect to countably additive measures. The technique developed in the paper is fundamentally different from that used by the authors when studying path integrals in configuration spaces. In particular, the paper extensively uses the infinite-dimensional Fourier transform.
ABSTRACT Feynman-Kac formulas for heat conduction equations with Vladimirov’s operator (acting he... more ABSTRACT Feynman-Kac formulas for heat conduction equations with Vladimirov’s operator (acting here as a Laplace operator) are proved; it is assumed that unknown functions are determined on the product of the real axis and a space over the field of p-adic numbers and take real or complex values. Similar formulas may be obtained for Schrödinger type equations. Such equations may be useful in construction both mathematical models of processes whose scales are characterized by the Planck length and time and mathematical models describing phenomenology in chemistry, mechanics of continua, and also in psychology.
Homogeneous closed PDO are constructed which are analogous to the powers of (absolute value of ) ... more Homogeneous closed PDO are constructed which are analogous to the powers of (absolute value of ) infinite dimensional Laplacian and acting in Banach spaces of complexvalued functions defined on function spaces over a field of p-adic numbers. For elements of semigroups, for which these PDOs are generators, Feynman formulas and Feynman-Kac ones are obtained.
Latin American Journal of Probability and Mathematical Statistics, 2016
In this work, we establish pathwise functional Itô formulas for nonsmooth functionals of real-val... more In this work, we establish pathwise functional Itô formulas for nonsmooth functionals of real-valued continuous semimartingales. Under finite (p, q)variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition F t (X t) = F 0 (X 0
Properties of infinite dimensional pseudo-differential operators (PDO) are discussed; in particul... more Properties of infinite dimensional pseudo-differential operators (PDO) are discussed; in particular, the connection between two definitions of the PDO is considered: one given in terms of the Hamiltonian Feynman measure, and another introduced in this work in terms of the Kolmogorov integral.
We prove a version of the Stokes formula for differential forms on locally convex spaces announce... more We prove a version of the Stokes formula for differential forms on locally convex spaces announced in [10]. The main tool used for proving this formula is the surface layer theorem proved in the paper [6] by the author. Moreover, for differential forms of a Sobolev-type class relative to a differentiable measure [1], we compute the operator adjoint to the exterior differential in terms of standard operations of calculus of differential forms and the logarithmic derivative. Previously, this connection was established under essentially stronger assumptions on the space [8], the measure [7], or smoothness of differential forms [5]. See also [4].
We prove formulas that describe, under certain sufficiently general assumptions, the Fourier tran... more We prove formulas that describe, under certain sufficiently general assumptions, the Fourier transforms of distributions of processes and fields mentioned in the title. A similar method applied to complex Markov chains on a finite segment can be used to compute the Fourier transform of the complex Poisson measure from [1, Chap. IX] (where it is found by series expansion of the measure) given its finite-dimensional distributions. We assume that the value space of a random field is a Polish (i.e., complete, separable, and metrizable) locally convex space; however, the results seem to be new for real processes as well. An additional assumption on the processes is that their trajectories have no discontinuities of the second kind in the (weakened) topology of the value space; in particular, such are the Poisson jump processes. Earlier [2] such a formula was proved only for processes with strongly continuous trajectories. Also discussed in [2] are applications of a similar construction for continuous fields to the particular case of Wiener–Levy–Chentsov fields [3].
Transactions of the Moscow Mathematical Society, 2005
A family {M p } p∈R d of cylindrical measures is constructed on the space of functions (= traject... more A family {M p } p∈R d of cylindrical measures is constructed on the space of functions (= trajectories) f : [0, +∞) → R d such that for every t ≥ 0 the formula 2000 Mathematics Subject Classification. Primary 28B99; Secondary 46G10, 81Q05.
Homogeneous systems of matrix-valued transition measures similar to Markov systems and generating... more Homogeneous systems of matrix-valued transition measures similar to Markov systems and generating cylindrical countably additive measures on some function spaces are constructed and the Fourier transforms of the latter measures (= functional distributions) are evaluated; the case of complex values corresponds to Markov-Poisson chains introduced by V. P. Maslov in 1976. The state spaces of these systems (similar to the Markov ones) are measurable real vector spaces. These distributions (their Fourier transforms, respectively can be used to obtain Maslov-like formulas (Feynman-like formulas, respectively) to solve Schrödinger equations with effective potentials with values in complex matrix algebras. It is also shown that, if the values of the matrix potential commute, then in the same way, one can construct analogs of multitemporal Markov fields and evaluate the Fourier transforms of the corresponding functional distributions.
In this work, we establish pathwise functional It\^o formulas for non-smooth functionals of real-... more In this work, we establish pathwise functional It\^o formulas for non-smooth functionals of real-valued continuous semimartingales. Under finite $(p,q)$-variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition $$F_t(X_t) = F_0(X_0)+ \int_0^t\nabla^hF_s(X_s)ds + \int_0^t\nabla^wF_s(X_s)dX(s) - \frac{1}{2}\int_{-\infty}^{+\infty}\int_0^t(\nabla^w_xF_s)(^{x}X_s)d_{(s,x)}\ell^x(s)$$ Here, $X_t= \{X(s); 0\le s\le t\}$ is the continuous semimartingale path up to time $t\in [0,T]$, $\nabla^h$ is the horizontal derivative, $(\nabla^w_x F_s)(^{x}X_s)$ is a weak derivative of $F$ with respect to the terminal value $x$ of the modified path $^{x}X_s$ and $\nabla^w F_s(X_s) = (\nabla^w_x F_s)(^{x} X_s)|_{x=X(s)}$. The double integral is interpreted as a space-time 2D-Young integral with differential $d_{(s,x)}\ell^x(s)$, where $\ell$ is the local-time of $X$. Under less restrictive joint variation assumptions on $...
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2007
Cauchy problems for Schrödinger equations with matrix-valued potentials are explicitly solved und... more Cauchy problems for Schrödinger equations with matrix-valued potentials are explicitly solved under following assumptions:. — equations are written in momentum form;. — the potentials are Fourier transformed matrix-valued measures with, in general, noncommuting values;. — initial Cauchy data are good enough. The solutions at time t are presented in form of integrals over some spaces of piecewise continuous mappings of the segment [0, t] to a finite-dimensional momentum space, and measures of the integration are countably additive but matrix-valued (resulting in matrices of ordinary Lebesgue integrals). Known results gave solutions either when the measure had commuting (complex) values, or when the integration over infinite dimensional spaces was quite symbolic, or when such integrating was of chronological type and hence more complicated. The method used below is based on technique of matrix-valued transition amplitudes.
As the main step, the method used by V. P. Maslov for representing a solution of the initial-valu... more 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.
A class of the Schroedinger type equations, with respect to functions defined on the product of [... more A class of the Schroedinger type equations, with respect to functions defined on the product of [0∞) (``time'') and of the p-adic field (``space''), is considered. In the equation a Vladimirov operator substitutes the standard Lalacian. The main aim of the paper is to get representations of solutions of Cauchy problems for such equations by integrals over Feynman pseudomeasures on paths in the p-adic field (which is spacial domains of the solutions). The significant role is played by some p-adic analogs of Poisson-Maslov measures which are constructed in the paper and are used to get the representations.
Proceedings of the Steklov Institute of Mathematics, 2009
ABSTRACT We obtain Feynman formulas in the momentum space and Feynman-Kac formulas in the momentu... more ABSTRACT We obtain Feynman formulas in the momentum space and Feynman-Kac formulas in the momentum and phase spaces for a p-adic analog of the heat equation in which the role of the Laplace operator is played by the Vladimirov operator. We also present the Feynman and Feynman-Kac formulas in the configuration space that have been proved in our previous papers under additional constraints. In all these formulas, integration is performed with respect to countably additive measures. The technique developed in the paper is fundamentally different from that used by the authors when studying path integrals in configuration spaces. In particular, the paper extensively uses the infinite-dimensional Fourier transform.
ABSTRACT Feynman-Kac formulas for heat conduction equations with Vladimirov’s operator (acting he... more ABSTRACT Feynman-Kac formulas for heat conduction equations with Vladimirov’s operator (acting here as a Laplace operator) are proved; it is assumed that unknown functions are determined on the product of the real axis and a space over the field of p-adic numbers and take real or complex values. Similar formulas may be obtained for Schrödinger type equations. Such equations may be useful in construction both mathematical models of processes whose scales are characterized by the Planck length and time and mathematical models describing phenomenology in chemistry, mechanics of continua, and also in psychology.
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Papers by N. Shamarov