Small stochastic perturbations in a general fractional kinetic equation

Márquez-Carreras, David

ESAIM: Probability and Statistics, Tome 19 (2015), pp. 81-99.

In this paper we study some properties of the density for the law of the solution to a generalized multidimensional fractional kinetic equation driven by a Gaussian noise, white in time and correlated in space. The diffusion operator is the composition between the Bessel and Riesz potentials with any fractional parameters. We also establish Varadhan’s estimates for the solution to the equation obtained by perturbing the noise.

J.M. Angulo, V.V. Anh, R. Mcvinish and M.D. Ruiz-Medina, Fractional kinetic equations driven by Gaussian or infinitely divisible noise. Adv. Appl. Prob. 37 (2005) 366–392. | MR | Zbl

J.M. Angulo, M.D. Ruiz-Medina, V.V. Anh and W. Grecksch, Fractional diffusion and fractional heat equation. Adv. Appl. Prob. 32 (2000) 1077–1099. | MR | Zbl

V.V. Ahn, J.M. Angulo and M.D. Ruiz-Medina. Possible long-range dependence in fractional randoms fields. J. Statist. Plan. Infer. 80 (1999) 95–110 | MR | Zbl

V.V. Anh and N.N. Leonenko, Spectral analysis of fractional kinetic equation with random data. J. Stat. Phys. 104 (2001) 1349–1387. | MR | Zbl

V.V. Anh and N.N. Leonenko, Renormalization and homogenization of fractional diffusion equations with random data. Probab. Theory Related Fields 124 (2002) 381–408. | MR | Zbl

R. Azencott, Grandes déviations et applications. École d’Été de Probabilités de Saint-Flour VIII-1978. In vol. 774 of Lect. Notes Math. Springer-Verlag, Berlin (1980) 1–176. | MR | Zbl

L. Boulanba, M. Eddahbi and M. Mellouk, Fractional SPDEs driven by spatially correlated noise: existence of the solution and smoothness of its density. Osaka J. Math. 47 (2010) 41–65. | MR | Zbl

X. Cabré and M. Sanchón, Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian. Commun. Pure Appl. Anal. 6 (2007) 43–67. | MR | Zbl

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Partial Differ. Eq. 32 (2007) 1245–1260. | MR | Zbl

R.C. Dalang and N.E. Frangos, The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998) 187–212. | MR | Zbl

R.C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous spde’s. Electron. J. Probab. 4 (1999) 29. | MR | Zbl

R. Dautray andJ.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. 2. Springer, Berlin (1995). | MR

L. Debbi and M. Dozzi, On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension. Stochastic Processes Appl. 115 (2005) 1764–1781. | MR | Zbl

A. Dembo and O. Zeitouni, Large deviations techniques and applications. Jones and Barlett Publishers, Boston (1993). | MR | Zbl

J.D. Deuschel and D.W. Stroock, Large deviations. In vol. 137 of Pure Appl. Math. Academic Press, Boston (1989). | MR | Zbl

E. Dibenedetto, U. Gianazza and V. Vespri, Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electron. Res. Announc. Amer. Math. Soc. 12 (2006) 95–99. | MR | Zbl

S. Fang, Une inégalité isopérimétrique sur l’espace de Wiener. Bull. Sc. Math. 2e série. 112 (1988) 345–355. | MR | Zbl

M.I. Freidlin and A.D. Wentzell, Random perturbation of dynamical systems. Springer-Verlag, New York (1984). | MR | Zbl

K.J. Hochberg, A signed measure on path space related to Wiener measure. Ann. Probab. 6 (1978) 433–458. | MR | Zbl

B. Jourdain, S. Méléard and W.A. Woyczynski, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws. Bernoulli 11 (2005) 689–714. | MR | Zbl

B. Jourdain, S. Méléard and W.A. Woyczynski, A probabilistic approach for nonlinear equations involving the fractional Laplacian and a singular operator. Potential Anal. 23 (2005) 55–81. | MR | Zbl

V.Y. Krylov, Some properties of the distribution corresponding to the equation $\frac{\partial u}{\partial t}={(-1)}^{q+1}\frac{{\partial }^{2}q}{\partial x^{2q}}$. Soviet Math. Dokl. 1 (1960) 760–763. | MR | Zbl

B.B. Mandelbrot, and J.W. Van Ness, Fractional Brownian motion, fractional noises and applications. SIAM Rev. 10 (1968) 422–437. | MR | Zbl

D. Márquez-Carreras, Contribució a l’estudi de les equacions en derivades parcials estocàstiques. Ph.D. thesis (1998).

D. Márquez-Carreras, Generalized fractional kinetic equations: another point of view. Adv. Appl. Prob. 41 (2009) 893–910. | MR | Zbl

D. Márquez-Carreras, M. Mellouk and M. Sarrà, On stochastic partial differential equations with spatially correlated noise: smoothness of the law. Stochastic Processes Appl. 93 (2001) 269–284. | MR | Zbl

D. Márquez-Carreras and M. Sanz-Solé, Small perturbations in a hyperbolic stochastic partial differential equation. Stochastic Processes Appl. 68 (1997) 133–154. | MR | Zbl

D. Márquez-Carreras and M. Sarrà, Large deviation principle for a stochastic heat equation with spatially correlated noise. Electronic J. Prob. 8 (2003) 1–39. | MR | Zbl

S. Micu, and E. Zuazua, On the controllability of a fractional order parabolic equation. SIAM J. Control Optim. 44 (2006) 1950–1972. | MR | Zbl

A. Millet and M. Sanz-Solé, A stochastic wave equation in two space dimension: Smoothness of the law. Ann. Probab. 27-2 (1999) 803–844. | MR | Zbl

D. Nualart, Analysis on Wiener Space and Anticipating stochastic Calculus. École d’Été de Probabilités de Saint-Flour XXV-1995, in vol. 1690 of Lect. Notes Math. Springer-Verlag (1998) 123–227. | MR | Zbl

D. Nualart, Malliavin Calculus and Related Topics. Springer-Verlag (1995). | MR | Zbl

D. Nualart and L L. Quer-Sardanyons, Existence and smoothness of the density for spatially homogeneous SPDEs. Potential Anal. 27 (2007) 281–299 | MR | Zbl

M.D. Ruiz-Medina, J.M. Angulo, and V.V. Anh, Scaling limit solution of a fractional Burgers equation. Stochastic Processes Appl. 93 (2001) 285–300. | MR | Zbl

S.G. Samko, A.A. Kilbas andO.I. Marichev, Fractional integrals and derivatives. Gordon and Breach Science Publishers, New York (1987). | MR | Zbl

L. Schwartz, Théorie des Distributions. Hermann, Paris (1966). | MR | Zbl

E.M. Stein, Singular integrals and differentiating properties of Functions. Princeton University Press, Princeton (1970). | MR | Zbl

J.B. Walsh, An introduction stochastic partial differential equations. École d’été de probabilités de Saint Flour XIV 1984. In vol. 1180 of Lect. Notes Math. Springer, Berlin (1986) 265–439. | MR | Zbl