April  2019, 39(4): 2133-2155. doi: 10.3934/dcds.2019089

Memory equations as reduced Markov processes

1. 

Insitut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, D-12489 Berlin, Germany

2. 

Weierstraß-Insitut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117 Berlin, Germany

* Corresponding author: Artur Stephan

Received  May 2018 Revised  October 2018 Published  January 2019

Fund Project: The first author is supported by the Berlin Mathematical School.

A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we propose an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as a change of the type of some quasiparticles along one-way loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realistic modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations like the calculation of the equilibrium state. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations.

Citation: Artur Stephan, Holger Stephan. Memory equations as reduced Markov processes. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2133-2155. doi: 10.3934/dcds.2019089
References:
[1]

W. Arendt, et al., One-parameter Semigroups of Positive Operators, Springer, 1986. doi: 10.1007/BFb0074922.

[2] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511614583.
[3] R. Durrett, Probability: Theory and Examples, Cambridge University Press, 2010.  doi: 10.1017/CBO9780511779398.
[4]

E. B. Dynkin, Markov Processes, Moscow, 1963.

[5]

K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate texts in mathematics, 194. Springer-Verlag, New York, 2000.

[6]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373.

[7]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, 1993. doi: 10.1007/978-1-4612-4342-7.

[8]

U. Hornung and R. E. Showalter, Diffusion models to fractured media, Journal of Mathematical Analysis and Applications, 147 (1990), 69-80.  doi: 10.1016/0022-247X(90)90385-S.

[9]

A. Khrabustovskyi and H. Stephan, Positivity and time behavior of a general linear evolution system, non-local in space and time, Math. Methods Appl. Sci., (2008), 1809–1834. doi: 10.1002/mma.998.

[10]

V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0.

[11]

M. Peszynska, Analysis of an integro-differential equation arising form modelling of flows with fading memory through fissured media, Journal of Partial Differential Equations, 8 (1995), 159-173. 

[12]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

[13]

H. Stephan, A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations, J. Mathematical Physics Analysis Geometry (MAG), 12 (2005), 187-202. 

[14]

H. Stephan, Lyapunov functions for positive linear evolution problems, ZAMM Z. Angew. Math. Mech., 85 (2005), 766-777.  doi: 10.1002/zamm.200410229.

[15] B. Yu. Sternin and V. E. Shatalov, Borel-Laplace Transform and Asymptotic Theory: Introduction to Resurgent Analysis, CRC Press, Boca Raton, FL, 1996. 
[16]

L. Tartar, Memory Effects and Homogenization, Arch. Rational Mech. Anal., 111 (1990), 121-133.  doi: 10.1007/BF00375404.

show all references

References:
[1]

W. Arendt, et al., One-parameter Semigroups of Positive Operators, Springer, 1986. doi: 10.1007/BFb0074922.

[2] A. Bobrowski, Functional Analysis for Probability and Stochastic Processes, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511614583.
[3] R. Durrett, Probability: Theory and Examples, Cambridge University Press, 2010.  doi: 10.1017/CBO9780511779398.
[4]

E. B. Dynkin, Markov Processes, Moscow, 1963.

[5]

K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate texts in mathematics, 194. Springer-Verlag, New York, 2000.

[6]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.  doi: 10.1007/BF00281373.

[7]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, 1993. doi: 10.1007/978-1-4612-4342-7.

[8]

U. Hornung and R. E. Showalter, Diffusion models to fractured media, Journal of Mathematical Analysis and Applications, 147 (1990), 69-80.  doi: 10.1016/0022-247X(90)90385-S.

[9]

A. Khrabustovskyi and H. Stephan, Positivity and time behavior of a general linear evolution system, non-local in space and time, Math. Methods Appl. Sci., (2008), 1809–1834. doi: 10.1002/mma.998.

[10]

V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0.

[11]

M. Peszynska, Analysis of an integro-differential equation arising form modelling of flows with fading memory through fissured media, Journal of Partial Differential Equations, 8 (1995), 159-173. 

[12]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

[13]

H. Stephan, A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations, J. Mathematical Physics Analysis Geometry (MAG), 12 (2005), 187-202. 

[14]

H. Stephan, Lyapunov functions for positive linear evolution problems, ZAMM Z. Angew. Math. Mech., 85 (2005), 766-777.  doi: 10.1002/zamm.200410229.

[15] B. Yu. Sternin and V. E. Shatalov, Borel-Laplace Transform and Asymptotic Theory: Introduction to Resurgent Analysis, CRC Press, Boca Raton, FL, 1996. 
[16]

L. Tartar, Memory Effects and Homogenization, Arch. Rational Mech. Anal., 111 (1990), 121-133.  doi: 10.1007/BF00375404.

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