I am considering the transition semigroup $P_t$ associated with the Ito diffusion process $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$ where the coefficients are assumed to be Lipschitz continuous.

I hope to know under what condition of $b,\sigma$, $P_t$ is a strongly continuous semigroup on $L^2(\mathbb{R},dx)$. Based on my knowledge, one sufficient condition is to show Lebesgue measure is a sub-invariant measure w.r.t $P_t$ in the sense that $$\int_\mathbb{R} P_tf\,dx\le \int_\mathbb{R}f\, dx,\quad f\ge 0,t\ge 0.$$ If above is true, the $P_t$ is not only strongly continuous on $L^2(\mathbb{R},dx)$, but also contractive. I checked Bronwian motion satisfies this criterion.

May I know whether it's true for the general Ito process? Or are there any other ways to check whether $P_t$ is strongly continuous?



1 Answer 1


It turns out solutions to Ito diffusions with Lipschitz coefficients are Feller processes; you can find this in, say, Oksendal's book (on SDE). Moreover, Feller semigroups (semigroups associated to Feller processes) are strongly continuous and even contracting on $C^0$ under the $C^0$ topology, and this is a subtle point. We'll explain this argument and your concern with $L^2(\mathbb{R})$ below.

Let's look at $f \in C^{\infty}_0(\mathbb{R})$ first, and we'll also assume moment bounds. By Taylor's approximation, we'll assume we can write $$f(X_t) \ = \ f(X_0) \ + \ f'(X_0)dX_t \ + \ \frac{f''(X_0)}{2} dX_t^2 \ + \ \ldots$$

Taking expectations (and writing $dX_t$ in terms of its Ito diffusion representation), we see $$P_tf(X_0) \ = \ f(X_0) \ + \ f'(X_0) \mathbb{E} b(X_t) dt \ + \ \frac{f''(X_0)}{2} \mathbb{E} |\sigma(X_t)|^2 dt \ + \ \ldots$$

This is the standard calculation for computing generators of diffusion processes; in particular, we might as well forget the lower order terms above. If we were only concerned about pointwise limits, we'd be done immediately (since the coefficients of the Ito diffusion $dX_t$ are more than nice enough). But we need $L^2$-convergence.

Because of the Lipschitz assumption on the coefficients, we can bound the expectations on the RHS above in absolute value by expectations of $|X_t - X_0|, |X_t - X_0|^2$ respectively. In particular, finding moment estimates for $dX_t$ (which are bounded and continuous in $t$, thinking about quadratic variation of the process $X_t$), the expectation terms on the RHS above are bounded in polynomials in $X_0$. In particular, for each fixed test function $f$, we have the bound $$P_tf(X_0) \ - \ f(X_0) \ \leqslant \ C(|f'(X_0)X_0| \ + \ |f''(X_0)| (|X_0|^2 + |X_0|),$$

I might actually be missing some terms if I computed incorrectly, but this isn't too important. The important idea is that the RHS of the above inequality is in $L^2(\mathbb{R})$ because $f$ and all its derivatives are. Thus, dominated convergence tells us $$\lim_{t \searrow 0} \ \| P_tf - f \|_{L^2(\mathbb{R})} \ = \ \| \lim_{t \searrow 0} \ P_tf - f \|_{L^2(\mathbb{R})} \ = \ 0.$$

To address your question with all functions in $L^2(\mathbb{R})$, note that the semigroup actually acts on $L^2(\mathbb{R})$ and let $f \in L^2(\mathbb{R})$ be any $L^2$-function (since it's Feller and extends by continuity and density). Moreover, suppose $\varphi \in C_0^{\infty}(\mathbb{R})$ approximates $f$ within $\varepsilon$ in the $L^2$-sense. Now consider (I'll suppress all norms explicitly assuming all unspecified norms are in the sense of $L^2$) $$\| P_tf - f \| \ \leqslant \ \|P_tf - P_t\varphi\| \ + \ \|P_t \varphi - \varphi \| + \|\varphi - f\|.$$

We can choose $t$ small enough so that the last two terms on the RHS are bounded by $\varepsilon$. The problem is the first term, which is where the condition you gave would be useful. Note $P_t$ evaluated at an arbitrary point $x$ defines a (positive) continuous distribution on $L^2(\mathbb{R})$ (since it does so on the dense subspace). Thus, choosing $\varphi$ close enough to $f$ in the $L^2$ sense, note first we can assume $f$ is supported on the support of $\varphi$ losing an $\varepsilon$ term (by dominating the tail of $f$ by a tiny Schwartz function), and we know $|P_t\varphi - P_t f|_{L^{\infty}}$ is bounded by $\varepsilon$. Since $\varphi$ has compact support, everything ends up bounded by arbitrarily small $\varepsilon$, which tells you that such diffusions have strongly continuous semigroups on $L^2(\mathbb{R})$.

Just a quick remark: please let me know if anything is not clear or seems false. Thanks for the problem!

  • $\begingroup$ Sorry for the late reply. I just went through your argument carefully and tried to fill in the technical details. I am still not quite clear about the second part of your proof when you argue $\|P_tf-P_t\phi\|$. May you kindly expand your argument and give a more detailed proof? If I understand correctly, in your proof, we have to use that $\|P_tf\|_{L^2}\le M\|f\|_{L^2}$ for $f\in C_c^\infty$, which can be proved by assuming Lebesgue measure is an sub-invariant measure for $P_t$ as stated in the OP. Do you have any other ways to prove the estimate for Ito process? $\endgroup$
    – John
    Commented Feb 9, 2017 at 9:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.