Non-explosion by Stratonovich noise for ODEs

Positive effects of multiplicative noise on the explosion of nonlinear fractional stochastic differential equations

For the nonlinear stochastic partial differential equation which is driven by multiplicative noise of the form \[D_t^βu = \left[ { - {{\left( { - Δ} \right)}^s}u + ζ\left( u \right)} \right]dt + A\sum\limits_{m \in Z_0^d} {\sum\limits_{j = 1}^{d - 1} {{θ_m}{σ_{m,j}}\left( x \right)} } \circ dW_t^{m,j},\;\; s \ge 1,\;\;\frac{1}{2} < β< 1,\] where $D_{t}^β$ denotes the Caputo derivative, $A>0$ is a constant depending on the noise intensity, $\circ$ represent the Stratonovich-type stochastic differential, we consider the blow-up time of its solutions. We find that the introduction of noise can ef…