Asymptotic pointwise estimates of fundamental solution for time fractional equations with convolution kernels

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• Elena Zhizhina (Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences)
• Friday 25 March 2022, 09:30-10:00
• Seminar Room 1, Newton Institute.

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FD2W02 - Fractional kinetics, hydrodynamic limits and fractals

In my talk I present results of two papers:1) A. Grigor’yan , Yu. Kondratiev, A. Piatnitski and E. Zhizhina, Pointwise estimates for heatkernels of convolution type operators, Proc. London Math. Soc. (4), 117 (2018), 849-880, and2) Yu. Kondratiev, A. Piatnitski and E. Zhizhina, Asymptotics of fundamental solutions for timefractional equations with convolution kernels, Fract. Calc. Appl. Anal. Vol. 23, No 4 (2020), pp.1161-1187.We studied asymptotic behaviour (as t → ∞) of the heat kernel (=fundamental solution) of theevolution equations with non-local elliptic part:                                                                 ∂tu = Au, where A is a non-local operator given by                                                           Au = a ∗ u − u.                                        (1) The convolution kernel a is such that                                            a(x) ≥ 0; a(x) = a(−x); a(x) ∈ L∞(Rd) ∩ L1(Rd),        (2)                                                     ZRda(x)dx = 1, ZRd|x|2a(x)dx < ∞.                (3) We also assume that convolution kernels a(x) decay at innity at least exponentially.   The large time behaviour of the studied heat kernel depends crucially on the relation between|x| and t.  We consider separately four dierent regions in (x, t) space, namely, (i) |x| ∼ t1/2, (ii)t12 ≪ |x| ≪ t, (iii) |x| ∼ t, (iv) |x| ≫ t, and compare the obtained asymptotics with the heat kernelof the classical heat equation, which is given by the Gauss-Weierstrass function. Notice that the fundamental solution is the same as the transition density for the correspondingcontinuous time jump process in continuum with independent increments generated by the operator(1), and the region (i) corresponds to the standard deviations where the local central limit theoremapplies, (ii) is the region of the moderate deviations, (iii) is the region of large deviation, and (iv)should probably be called the “extra large” deviation region. Using the asymptotic estimates for the non-local heat kernel we study next a solution wα(x, t)of the following fractional time parabolic type problem:                                              ∂αt wα = a ∗ wα − wα                                                 wα |t=0 = δ0, where ∂αt is the fractional derivative (the Caputo derivative) of the order α, 0 < α < 1.

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