Abstract
In light of recent work on particles fluctuating in linear viscoelastic fluids, we study a linear stochastic partial-integro-differential equation with memory that is driven by a stationary noise on a bounded, smooth domain. Using the framework of generalized stationary solutions introduced in McKinley and Nguyen (SIAM J Math Anal 50(5):5119–5160, 2018), we provide conditions on the differential operator and the noise to obtain the existence as well as Hölder regularity of the stationary solutions for the concerned equation. As an application of the regularity results, we compare to analogous classical results for the stochastic heat equation. When the 1d stochastic heat equation is driven by white noise, solutions are continuous with space and time regularity that is Hölder (1/2-epsilon ) and (1/4-epsilon ) respectively. When driven by colored-in-space noise, solutions can have a range of regularity properties depending on the structure of the noise. Here, we show that the particular form of colored-in-time memory that arises in viscoelastic diffusion applications, satisfying what is called the Fluctuation–Dissipation relationship, yields sample paths that are Hölder (1/2-epsilon ) and (1/2-epsilon ) in space and time.
Highlights
With regard to the memory kernel K, we first state the definition of the class of completely monotone functions, of which we will assume that K is a member
We have rigorously analyzed a stochastic integro-partial-differential equation with memory (1.1) satisfying the Fluctuation–Dissipation relationship that arises from statistical mechanical considerations in the study of thermally fluctuating viscoelastic
Using the framework of generalized stationary processes from [24, 29], we obtain stationary solutions of (1.1) when the memory belongs to a large subclass CMb of the completely monotone functions
Summary
Let O be a bounded domain in Rd , d ≥ 1 and denote by H = L2(O), the Hilbert space of square-integrable functions on O. Given a self-adjoint negative operator A : D(A) ⊂ H → H and a memory function K : R → [0, ∞), we are interested in the following equation for u(t, x) : R × O → Rd Communicated by Ilya Spitkovsky.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
