Classical and generalized solutions of fractional stochastic differential equations
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Classical and generalized solutions of fractional stochastic differential equations S. V. Lototsky1
· B. L. Rozovsky2
Received: 1 November 2018 / Revised: 3 November 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise, there can be a generalized solution in suitable weighted chaos spaces. Presence of fractional derivatives in both time and space leads to various modifications of the stochastic parabolicity condition. Interesting new effects appear when the order of the time derivative in the noise term is less than or equal to one-half. Keywords Anomalous diffusion · Caputo derivative · Chaos expansion · Gaussian Volterra process · Stochastic parabolicity condition Mathematics Subject Classification Primary 60H15 · Secondary 60H10 · 60H40 · 34A08 · 35R15
1 Introduction Given a β ∈ (0, 1), and a smooth function f = f (t), t > 0, the two most popular definitions of the derivative of order β are Riemann–Liouville
Research supported by ARO Grant W911N-16-1-0103.
B
S. V. Lototsky [email protected] http://www-bcf.usc.edu/∼lototsky B. L. Rozovsky [email protected] http://www.dam.brown.edu/people/rozovsky/rozovsky.htm
1
Present Address: Department of Mathematics, USC, Los Angeles, CA 90089, USA
2
Present Address: Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
123
Stoch PDE: Anal Comp
d 1 (1 − β) dt
β
Dt f (t) =
t
(t − s)−β f (s) ds
0
and Caputo t 1 (t − s)−β f (s) ds; (1 − β) 0 +∞ (z) = t z−1 e−t dt.
β ∂˜t f (t) =
(1.1)
0
The Riemann–Liouville derivative can be considered as a true extension of the usual derivative to fractional orders. For example, a function does not have to be continuously differentiable to have Riemann–Liouville derivatives of order β < 1 [20]. On the other hand, the Caputo derivative is more convenient in initial-value problems, with no need for fractional-order initial conditions [19, Section 2.4.1]. The Kochubei extension of the Caputo derivative, β
∂t f (t) =
d 1 (1 − β) dt
t
(t − s)−β f (s) − f (0+) ds,
0
f (0+) = limt→0,t>0 f (t), seems to achieve the right balance between mathematical utility and physical relevance [8] and has been recently used in the study of large classes of stochastic partial differential equations [4,11]. Let w = w(t), t ≥ 0, be a standard Brownian motion on a stochastic basis (, F, {Ft }t≥0 , P) satisfying the usual conditions. The objective of this paper is to address fundamental questions about existence and regularity of solution for equations of the type β
γ
∂t X (t) = a X (t) + ∂t
t
σ X (s) + g(s) dw(s), t > 0, a, b ∈ R.
(1.2)
0
With a suitable choice of a and σ , (1.2) covers the time-fractional versions of the Ornstein-Uhlenbeck process and the geometric Brownian motion, as well as certain evolu
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