Well-posedness for the fractional Fokker-Planck equations

Wei, Jiaolong; Tian, Rongrong

doi:10.1063/1.4916286

Cited by 14 publications

(9 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When L α t is replaced by Brownian motion B t , the equation becomes For the stochastic differential equations driven by Lévy process, the corresponding Fokker-Planck equation includes a non-local term, namely the fractional Laplacian term, which quanifies the effects of non-Gaussian. Since the equation (3.1) satisfies the condition of the theorem in [41], the equation (3.1) under consideration has a weak L p solution. It is known that the analytic solutions for this kind of Fokker-Planck equations are difficult to obtain, even if the system is very simple.…”

Section: The Summation Symbolmentioning

confidence: 99%

State transitions in the Morris-Lecar model under stable Lévy noise

et al. 2020

View full text Add to dashboard Cite

This paper considers the state transition of the stochastic Morris-Lecar neuronal model driven by symmetric α-stable Lévy noise. The considered system is bistable: a stable fixed point (resting state) and a stable limit cycle (oscillating state), and there is an unstable limit cycle (borderline state) between them. Small disturbances may cause a transition between the two stable states, thus a deterministic quantity, namely the maximal likely trajectory, is used to analyze the transition phenomena in non-Gaussian stochastic environment. According to the numerical experiment, we find that smaller jumps of the Lévy motion and smaller noise intensity can promote such transition from the sustained oscillating state to the resting state. It also can be seen that larger jumps of the Lévy motion and higher noise intensity are conducive for the transition from the borderline state to the sustained oscillating state. As a comparison, Brownian motion is also taken into account. The results show that whether it is the oscillating state or the borderline state, the system disturbed by Brownian motion will be transferred to the resting state under the selected noise intensity.

show abstract

Section: The Summation Symbolmentioning

confidence: 99%

State transitions in the Morris-Lecar model under stable Lévy noise

et al. 2020

View full text Add to dashboard Cite

show abstract

“…This can be justified formally by multiplying the equation by m itself and deriving an usual L 2 -energy estimate (as in (30) below). Since m itself cannot be a test function because of the "asymmetric" integrability requirements on m and ∂ t m, one has to perform a preliminary regularization procedure via convolution (see, e.g., [40,57] and [26]).…”

Section: Relation Between H µ P and W µPmentioning

confidence: 99%

“…[47] and references therein) cannot be directly converted to the nonlocal framework, heuristically because of the gap between the energy terms (−∆) s/2 and the divergence term. Thus, first order techniques as the ones described in [57] for the euclidean case are better suited to work in the nonlocal setting.…”

Section: Introductionmentioning

confidence: 99%

On the Existence and Uniqueness of Solutions to Time-Dependent Fractional MFG

Cirant¹,

Goffi²

2019

SIAM J. Math. Anal.

View full text Add to dashboard Cite

We establish existence and uniqueness of solutions to evolutive fractional Mean Field Game systems with regularizing coupling, for any order of the fractional Laplacian s ∈ (0, 1). The existence is addressed via the vanishing viscosity method. In particular, we prove that in the subcritical regime s > 1/2 the solution of the system is classical, while if s ≤ 1/2 we find a distributional energy solution. To this aim, we develop an appropriate functional setting based on parabolic Bessel potential spaces. We show uniqueness of solutions both under monotonicity conditions and for short time horizons.

show abstract

“…There are few works dealing with the existence and uniqueness of the Fokker-Planck equations. Wei and Tian [10] obtain the existence and uniqueness of weak L p solution on the whole space. AcevesSanchez and Cesbron [11] studied the nonlocal Fokker-Planck problem on R d in fractional Sobolev space with the drift term is a direct proportion function.…”

Section: Introductionmentioning

confidence: 99%