On the Well-Posedness of Stochastic Partial Differential Equations with Locally Lipschitz Coefficients

We consider the stochastic partial differential equation (SPDE) aligned ∂ ₜ u = 1/2 ∂ ²ₓ u + b(u) + σ (u) Ẇ, aligned ∂ t u = 1 2 ∂ x 2 u + b ( u ) + σ ( u ) W ˙ , where $$u=u(t,x)$$ u = u ( t , x ) is defined for (t,x)∈ (0,∞ )× R ( t , x ) ∈ ( 0 , ∞ ) × R and Ẇ W ˙ denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition u (0) is bounded and measurable, and b and σ σ are locally Lipschitz continuous functions having at most linear growth with regularly behaved local Lipschitz constants. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The novelty of our method is in the pointwise nature of the truncation argument.