doi: 10.3934/dcdsb.2018303

Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals

Friedrich Schiller University, Institute of Mathematics, Ernst-Abbe-Platz 2, 07743, Jena, Germany

* Corresponding author: Björn Schmalfuss

Dedicated to Peter E. Kloeden on Occasion of his Seventieth Birthday

Received  April 2018 Published  October 2018

We consider a stochastic nonlinear evolution equation where the domain is given by a fractal set. The linear part of the equation is given by a Laplacian defined on the fractal. This equation generates a random dynamical system. The long time behavior is given by an attractor which has a finite Hausdorff dimension. We would like to reveal the connections between upper and lower estimates of this Hausdorff dimension and the geometry of the fractal. In particular, the parameter which determines these bounds is the spectral exponent of the fractal. Especially for the lower estimate we construct a local unstable random Lipschitz manifold.

Citation: Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018303
References:
[1]

C. D. Aliprantis and K. C. Border, Infinite-dimensional Analysis, Springer-Verlag, Berlin, second edition, 1999. A hitchhiker's guide. doi: 10.1007/978-3-662-03961-8.

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

M. T. Barlow, Diffusions on fractals, In Lectures on Probability Theory and Statistics (SaintFlour, 1995), volume 1690 of Lecture Notes in Math., pages 1-121. Springer, Berlin, 1998. doi: 10.1007/BFb0092537.

[4]

M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543-623. doi: 10.1007/BF00318785.

[5]

T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581. doi: 10.1080/03605309808821394.

[6]

T. CaraballoJ. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2041-2061. doi: 10.1098/rspa.2001.0819.

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H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc. (2), 63 (2001), 413-427. doi: 10.1017/S0024610700001915.

[8]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

[9]

A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl. (9), 77 (1998), 967-988. doi: 10.1016/S0021-7824(99)80001-4.

[10]

J. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.

[11]

J. DuanK. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972. doi: 10.1007/s10884-004-7830-z.

[12]

K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications.

[13]

K. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1997.

[14]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.

[15]

U. R. Freiberg, Analysis on fractal objects, Meccanica, 40 (2005), 419-436. doi: 10.1007/s11012-005-2107-0.

[16]

B. M. Hambly, Brownian motion on a random recursive Sierpinski gasket, Ann. Probab., 25 (1997), 1059-1102. doi: 10.1214/aop/1024404506.

[17]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Volume 419 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.

[18]

J. Kigami, Analysis on Fractals, volume 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943.

[19]

M. Rosaria LanciaM. Cefalo and G. Dell'Acqua, Numerical approximation of transmission problems across Koch-type highly conductive layers, Appl. Math. Comput., 218 (2012), 5453-5473. doi: 10.1016/j.amc.2011.11.033.

[20]

M. L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc., 325 (1991), 465-529. doi: 10.1090/S0002-9947-1991-0994168-5.

[21]

K. Lu and B. Schmalfuẞ, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492. doi: 10.1016/j.jde.2006.09.024.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, In V. Reitmann, T. Riedrich, and N. Koksch, editors, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour, (1992), 185-192.

[24]

B. Schmalfuß, The random attractor of the stochastic Lorenz system, Z. Angew. Math. Phys., 48 (1997), 951-975. doi: 10.1007/s000330050074.

[25]

B. Schmalfuss, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998), 91-113. doi: 10.1006/jmaa.1998.6008.

[26]

B. Schmalfuss, Inertial manifolds for random differential equations, In Probability and Partial Differential Equations in Modern Applied Mathematics, volume 140 of IMA Vol. Math. Appl., pages 213-236. Springer, New York, 2005. doi: 10.1007/978-0-387-29371-4_14.

[27]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, volume 143 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[28]

R. S. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006. A tutorial.

[29]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences, Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[30]

T. Wanner, Linearization of random dynamical systems, In Dynamics Reported, volume 4 of Dynam. Report. Expositions Dynam. Systems (N. S. ), pages 203-269. Springer, Berlin, 1995.

[31]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differential-gleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71 (1912), 441-479. doi: 10.1007/BF01456804.

[32]

A. Wouk, A Course of Applied Functional Analysis, Wiley-Interscience [John Wiley & Sons], New York, 1979. Pure and Applied Mathematics.

show all references

References:
[1]

C. D. Aliprantis and K. C. Border, Infinite-dimensional Analysis, Springer-Verlag, Berlin, second edition, 1999. A hitchhiker's guide. doi: 10.1007/978-3-662-03961-8.

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

M. T. Barlow, Diffusions on fractals, In Lectures on Probability Theory and Statistics (SaintFlour, 1995), volume 1690 of Lecture Notes in Math., pages 1-121. Springer, Berlin, 1998. doi: 10.1007/BFb0092537.

[4]

M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543-623. doi: 10.1007/BF00318785.

[5]

T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581. doi: 10.1080/03605309808821394.

[6]

T. CaraballoJ. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2041-2061. doi: 10.1098/rspa.2001.0819.

[7]

H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc. (2), 63 (2001), 413-427. doi: 10.1017/S0024610700001915.

[8]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

[9]

A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl. (9), 77 (1998), 967-988. doi: 10.1016/S0021-7824(99)80001-4.

[10]

J. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.

[11]

J. DuanK. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972. doi: 10.1007/s10884-004-7830-z.

[12]

K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications.

[13]

K. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1997.

[14]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.

[15]

U. R. Freiberg, Analysis on fractal objects, Meccanica, 40 (2005), 419-436. doi: 10.1007/s11012-005-2107-0.

[16]

B. M. Hambly, Brownian motion on a random recursive Sierpinski gasket, Ann. Probab., 25 (1997), 1059-1102. doi: 10.1214/aop/1024404506.

[17]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Volume 419 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.

[18]

J. Kigami, Analysis on Fractals, volume 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943.

[19]

M. Rosaria LanciaM. Cefalo and G. Dell'Acqua, Numerical approximation of transmission problems across Koch-type highly conductive layers, Appl. Math. Comput., 218 (2012), 5453-5473. doi: 10.1016/j.amc.2011.11.033.

[20]

M. L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc., 325 (1991), 465-529. doi: 10.1090/S0002-9947-1991-0994168-5.

[21]

K. Lu and B. Schmalfuẞ, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492. doi: 10.1016/j.jde.2006.09.024.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, In V. Reitmann, T. Riedrich, and N. Koksch, editors, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour, (1992), 185-192.

[24]

B. Schmalfuß, The random attractor of the stochastic Lorenz system, Z. Angew. Math. Phys., 48 (1997), 951-975. doi: 10.1007/s000330050074.

[25]

B. Schmalfuss, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl., 225 (1998), 91-113. doi: 10.1006/jmaa.1998.6008.

[26]

B. Schmalfuss, Inertial manifolds for random differential equations, In Probability and Partial Differential Equations in Modern Applied Mathematics, volume 140 of IMA Vol. Math. Appl., pages 213-236. Springer, New York, 2005. doi: 10.1007/978-0-387-29371-4_14.

[27]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, volume 143 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[28]

R. S. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006. A tutorial.

[29]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences, Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[30]

T. Wanner, Linearization of random dynamical systems, In Dynamics Reported, volume 4 of Dynam. Report. Expositions Dynam. Systems (N. S. ), pages 203-269. Springer, Berlin, 1995.

[31]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differential-gleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71 (1912), 441-479. doi: 10.1007/BF01456804.

[32]

A. Wouk, A Course of Applied Functional Analysis, Wiley-Interscience [John Wiley & Sons], New York, 1979. Pure and Applied Mathematics.

Figure 1.  An approximation of the Sierpinski gasket using a sequence of graphs
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