EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE ULTRAPARABOLIC HAMILTON-JACOBI EQUATION

Michael D. Marcozzi

doi:10.11948/20210471

2023 Volume 13 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(1): 184-197

Next Article Previous Article

Michael D. Marcozzi. EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE ULTRAPARABOLIC HAMILTON-JACOBI EQUATION[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 184-197. doi: 10.11948/20210471

Citation:

Michael D. Marcozzi. EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE ULTRAPARABOLIC HAMILTON-JACOBI EQUATION[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 184-197. doi: 10.11948/20210471

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE ULTRAPARABOLIC HAMILTON-JACOBI EQUATION

Michael D. Marcozzi^1, ,

1.
Department of Mathematical Sciences, University of Nevada Las Vegas, 89154-4020, USA

Corresponding author: Michael D. Marcozzi, Email: marcozzi@unlv.nevada.edu

Abstract

Abstract

We demonstrate the existence and uniqueness of solutions to the ultra-parabolic Hamilton-Jacobi equation. These solutions maintain the probabilistic interpretation of being the maximal expectation of a controlled ultradiffusion process relative to a discounted performance criteria.
- Ultraparabolic Hamilton-Jacobi equation /
- ultradiffusion process /
- optimal control
MSC: 49L12, 35K70, 93E20

FullText(HTML)

References

[1]	D. Akhmetov, M. Lavrentiev and R. Spigler, Singular perturbations for para-bolic equations with unbounded coefficients leading to ultraparabolic equations, Differential Integral Equations, 2004, 17, 99–118. Google Scholar
[2]	S. Chandrasekhar, Stochastic Problems in Physics and Astronomy, Rev. Mod. Phys., 1943, 15, 1-89; Selected Papers on Noise and Stochastic Processes (Ed. N. Wax), Dover, New York, 2013(Reprint). Google Scholar
[3]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, New York, 1992. Google Scholar
[4]	A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker, New York, 1999. Google Scholar
[5]	A. Favini and G. Marinoschi, Degenerate Nonlinear Diffusion Equations, Lecture Notes in Mathematics, Springer-Verlag, New York, 2012. Google Scholar
[6]	W. H. Fleming and R. W. Rischel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975. Google Scholar
[7]	W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1992. Google Scholar
[8]	T. G. Genčev, Ultraparabolic equations, Dokl. Akad Nauk SSSR, 1963,151,265-268; Soviet Math. Dokl., 1963, 4,979–982 (English Trans.). Google Scholar
[9]	X. Han and J. Mo, The study for generalized asymptotic solution to nonlinear ultra parabolic equation, Applied Mathematics A Journal of Chinese Universities, 2017, 32,156–164. Google Scholar
[10]	B. A. Huberman and M. Kerszberg, Ultradiffusion: the relaxation of hierarchical systems, J. Phys. A. Math. Gen., 1985, 18, L331–L336. doi: 10.1088/0305-4470/18/6/013 CrossRef Google Scholar
[11]	A. M. Il'in, On a class of ultraparabolic equations, Dokl. Akad Nauk SSSR, 1964,159, 1214-1217; Soviet Math. Dokl., 1964, 5, 1673–1676 (English Trans.). Google Scholar
[12]	S. D. Ivasyshen and H. S. Pasichnyk, Ultraparabolic equations withi nfinitely increasing coefficients in the group of lowest terms and degenerations in the initial hyperplane, J. Math. Sci., 2020,249,333–354. doi: 10.1007/s10958-020-04946-3 CrossRef Google Scholar
[13]	V. A. Khoa, T. T. Hung and D. Lesnic, Uniqueness result for an age-dependent reaction–diffusion problem, Applicable Analysis, 2021,100, 2873–2890. doi: 10.1080/00036811.2019.1698730 CrossRef Google Scholar
[14]	A. N. Kolmogorov, Zur Theorie der stetigen zufalligen Progresse, Math. Ann., 1933,108,149–160. doi: 10.1007/BF01452829 CrossRef Google Scholar
[15]	A. N. Kolmogorov, Zufällige Bewegungen, Ann. of Math., 1934, 35,116–117. doi: 10.2307/1968123 CrossRef Google Scholar
[16]	A. I. Kozhanov and Y. A. Kosheleva, Linear Inverse Problems for Ultraparabolic Equations: The Case of Unknown Coefficient of Spatial Type, J. Math. Sci., 2018,230, 67–78. doi: 10.1007/s10958-018-3728-x CrossRef Google Scholar
[17]	E. Lanconelli, A. Pascucci and S. Polidoro, Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance, 243-265, Nonlinear Problems in Mathematical Physics and Related Topics Ⅱ (Eds. Birman, M. S., S. Hildebrandt, V. A. Solonnikov, and N. N. Uraltseva), Kluwer/Plenum, New York, 2002. Google Scholar
[18]	J. L. Lions, Sur certaines équations aux dérivativées partielles, à coefficients opérateurs non bornés, J. Analyse Math., 1958, 6,333–355. doi: 10.1007/BF02790240 CrossRef Google Scholar
[19]	M. D. Marcozzi, Well-posedness of linear ultraparabolic equations on bounded domains, J. Evol. Equ., 2018, 18(1), 75–104. doi: 10.1007/s00028-017-0391-5 CrossRef Google Scholar
[20]	M. D. Marcozzi, Probabilistic interpretation of solutions of linear ultraparabolic equations, Mathematics, 2018, 6(12), 286(16 pages). Google Scholar
[21]	M. D. Marcozzi, An adaptive extrapolation discontinuous Galerkin method for the valuation of Asian options, J. Comput. Appl. Math., 2011,235, 3632–3645. doi: 10.1016/j.cam.2011.02.024 CrossRef Google Scholar
[22]	R. E. Marshak, Theory of the slowing down of neutrons by elastic collisions with atomic nuclei, Rev. Modern Phys., 1947, 19,185–238. doi: 10.1103/RevModPhys.19.185 CrossRef Google Scholar
[23]	N. S. Piskunov, Problémes limits pour les équations du type elliptic-parabolique, Mat. Sb., 1940, 7,385–424. Google Scholar
[24]	A. S. Tersenov, On the global solvability of the Cauchy problem for a quasilinear ultraparabolic equation, Asy. Anal., 2013, 82,295–314. doi: 10.3233/ASY-2012-1146 CrossRef Google Scholar
[25]	G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion, Phys. Rev., 1930, 36,823–841. Google Scholar
[26]	V. S. Vladimirov and J. N. Drožžinov, Generalized Cauchy problem for an ultraparabolic equation, Izv. Akad. Nauk. SSSR Ser. Mat., 1967, 31, 1341-1360; Math. USSR Izv., 1967, 1, 1285–1303 (English Trans.). Google Scholar