DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2021118 DYNAMICAL SYSTEMS SERIES S EXISTENCE AND REGULARITY RESULTS FOR STOCHASTIC FRACTIONAL PSEUDO-PARABOLIC EQUATIONS DRIVEN BY WHITE NOISE Tran Ngoc Thach Applied Analysis Research Group, Faculty of Mathematics and Statistics Ton Duc Thang University Ho Chi Minh City, Vietnam Devendra Kumar ∗ Department of Mathematics University of Rajasthan Jaipur 302004, Rajasthan, India Nguyen Hoang Luc Division of Applied Mathematics Thu Dau Mot University Binh Duong Province, Vietnam Nguyen Huy Tuan 1,2,∗ 1 Division of Applied Mathematics, Science and Technology Advanced Institute Van Lang University Ho Chi Minh City, Vietnam 2 Faculty of Technology, Van Lang University Ho Chi Minh City, Vietnam Abstract. Solutions of a direct problem for a stochastic pseudo-parabolic equation with fractional Caputo derivative are investigated, in which the non- linear space-time-noise is assumed to satisfy distinct Lipshitz conditions includ- ing globally and locally assumptions. The main aim of this work is to establish some existence, uniqueness, regularity, and continuity results for mild solutions. 1. Introduction. Let X ⊂ R d , d ∈ N, be a bounded domain with the boundary is smooth enough. We investigate a direct problem for a stochastic fractional pseudo- parabolic equation      C D β t (u − α∆u)+(−∆) s u = I 1−β t [λϕ(t, u(t)) ˙ W (t)], (t, x) ∈ (0,T ] × X, u(0,x) = u ini (x), x ∈ X, u(t, x) =0, (t, x) ∈ (0,T ] × ∂X. (1) 2020 Mathematics Subject Classification. 60G15, 60G22, 60G52, 60G57. Key words and phrases. Pseudo-parabolic equation, wiener process, existence, regularity, stochastic. ∗ Corresponding authors: Nguyen Huy Tuan (nguyenhuytuan@vlu.edu.vn) and Devendra Ku- mar (devendra.maths@gmail.com). 1