Holder continuity of mild solutions of space-time fractional stochastic heat equation driven by colored noise

An initial value problem for space-time fractional stochastic heat equations driven by colored noise partial derivative(s)(t) u(t) +nu/2 (-Delta)(alpha/2)u(t) = sigma(u(t)) (W) over dot has been discussed in this work. Here, partial derivative(s)(t) and (-Delta)(alpha/2) stand for the Caputo's fractional derivative of order s is an element of (0, 1) and the fractional Laplace operator of order alpha is an element of (0, 2), where the second one is also the generator of a strict stable process X-t of order alpha with Ee(i xi).(Xt) = e(-t vertical bar xi vertical bar alpha). The nonlinearity sigma is assumed to be Lip-schitz continuous. A formulation of mild random field solutions is obtained due to the called space-time fractional Green functions G, H, where H contains a singular kernel. We focus on studying the spatially-temporally Holder continuity of mild random field solutions, which can be obtained by constructing relevant moment bounds for increments of the convolutions H circle times b(u) and H circle times sigma(u). Our techniques are based on connecting the space-time fractional Green functions G, H to the fundamental solution of partial derivative(t) P-t + (-Delta)(alpha/2) P-t = 0, P-0 = delta(0) via the Wright-type function M-s.