On the Well-Posedness of Stochastic Partial Differential Equations with Locally Lipschitz Coefficients
Abstract
Abstract We consider the stochastic partial differential equation (SPDE) $$\begin{aligned} \partial _t u = \tfrac{1}{2} \partial ^2_x u + b(u) + \sigma (u) \dot{W}, \end{aligned}$$ ∂ t u = 1 2 ∂ x 2 u + b ( u ) + σ ( u ) W ˙ , where $$u=u(t,x)$$ u = u ( t , x ) is defined for $$(t,x)\in (0,\infty )\times \mathbb {R}$$ ( t , x ) ∈ ( 0 , ∞ ) × R and $$\dot{W}$$ 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 $$\sigma $$ 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.
Fields of Study & Subjects
Stochastic processes and financial applications
Nonlinear Differential Equations Analysis
Stability and Controllability of Differential Equations
Publication details
- Journal / source
- Journal of Theoretical Probability
- Publisher
- Springer Science and Business Media LLC
- Volume
- 39
- Issue
- 2
- Publication date
- 23 February 2026
- Language
- EN
- Document type
- Book chapter