On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients

Question

Consider the SPDE $$\frac{\partial}{\partial t}u_t(x) = \frac{\kappa}{2}\frac{\partial^2}{\partial x^2}u_t(x) + u_t(x)(K-u_t(x)) + \sigma u_t(x) \xi(t,x),$$ where $(t,x)\in {\mathbb R}_+\times {\mathbb R}$, $\xi$ is a space-time white noise, and $\kappa, K,$ and $\sigma$ are positive constants. Assume the initial data $u_0(x)$ is measurable, nonnegative, and bounded. Note that the term $u(K-u)$ is not Lip continuous and grows faster than linear.

I wonder whether the above SPDE has a solution or not? And if a solution exists, is it unique, pathwise or in law? All the references that I know do not treat this special case. Many thanks for your help!

Answer 1

Call the drift term $b(u) = u (K- u)$ and for simplicity set $K=1$. Consider the following space of functions: $$C_{\text{tem}}(\mathbb{R},\mathbb{R}) = \{ f:\mathbb{R} \to \mathbb{R}:\, \sup_{x \in \mathbb{R}} |f(x)|e^{-\lambda|x|} < \infty \ \forall \lambda >0, \ f \text{ continuous} \} \, . $$ There are results on existence and uniqueness in $C_{\text{tem}}$ only assuming local Lipschitz conditions and $$ \big( b(x) - b(y)\big) \cdot (x-y) \leq c(|x-y|^2 + |x-y|) , \ x,y \in \mathbb{R} $$ for some constant $c>0$. See Theorem 4.2 of Iwata, An infinite dimensional stochastic differential equation with state space $C(\mathbb{R})$. Note that your $b$ does not satisfy this condition. However $b_1 (u) = b(u) \mathbf{1}(u \geq -1) -2 \mathbf{1}(u< -1) $ does. This allows you to apply Iwata's Theorem. Now you have a solution to the system with $b_1$ instead of $b$.

Next one can should be able to use Theorem 2.3 of Shiga, Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. However, it is only stated assuming global Lipschitz conditions. It should also work in the non-global setting. However, one needs to check his proof carefully. This allows you to deduce that your solution is non-negative: $u \geq 0$ and therefore it does not play a role whether you use $b$ or $b_1$.

Answer 2

flawed, see Martin Hairer's comment below.

Step I) Perform substitution $u_t(x) = \exp(\psi_t(x))$. The SPDE for $\psi$ becomes, after dividing by $u_t(x)$, \begin{equation} \frac{\partial}{\partial t} \psi_t(x) = \frac {\kappa}{2} \left( \left(\frac{\partial \psi_t(X)}{\partial x}\right)^2 + \frac{\partial^2 \psi_t(x)}{\partial x^2} \right) + (K - \exp(\psi_t(x))) + \sigma \xi(t,x).\end{equation} This has the added advantage that your noise has become additive, which is usually easier to analyse.

Step II) Note that $- \psi_t(x) \exp(\psi_t(x)) \leq 0$ for $\psi_t(x) \geq 0$, and $-\psi_t(x) \exp(\psi_t(x)) \leq |\psi_t(x)|$ for $\psi_t(x) \leq 0$. In principle therefore I think you should be able to use an approach similar to the one taken in e.g. X. Mao, Stochastic differential equations & Applications, Theorem 2.3.5, where local Lipschitz property and $x^T f(x) \leq K (1+|x|)^2$ are shown to be sufficient for wellposedness of $d X_t = f(X_t) \ d t + \sigma \ d W_t$. I am afraid I don't know a reference for this result in a Hilbert space context.

Alternatively, you could use Girsanov theorem (Da Prato & Zabczyk, Stochastic equations in infinite dimensions, Chapter 10) to establish existence of solutions without the nonlinear drift part, and then construct an appropriate change of measure.

Hope this helps!